On the methods of mechanical non-theorems
André Rognes
April 12, 2013
Contents
0.1
0.2
0.3
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Summary of the chapters . . . . . . . . . . . . . . . . . . . . . . .
0.2.1 Summary of chapter 1 . . . . . . . . . . . . . . . . . . . .
0.2.2 Summary of chapter 2 . . . . . . . . . . . . . . . . . . . .
0.2.3 Summary of chapter 3 . . . . . . . . . . . . . . . . . . . .
What is known . . . . . . . . . . . . . . . . . . . . . . . . . . . .
0.3.1 Classification of the Entscheidungsproblem based on syntax
0.3.2 Beyond syntactically defined classes . . . . . . . . . . . . .
0.3.3 Algebraic logic . . . . . . . . . . . . . . . . . . . . . . . .
0.3.4 Automated reasoning . . . . . . . . . . . . . . . . . . . .
0.3.5 Personal perspective and acknowledgements . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
1 Turning decision procedures into disprovers
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.1 Sets and mappings . . . . . . . . . . . . . . . . . . . . . . .
1.1.2 First-order formulae . . . . . . . . . . . . . . . . . . . . . .
1.1.3 Algebras, homomorphisms, and preservation . . . . . . . . .
1.1.4 Algebras of boolean signature . . . . . . . . . . . . . . . . .
1.2 Algebras of polyadic signature . . . . . . . . . . . . . . . . . . . . .
1.2.1 Operations related to variable substitution . . . . . . . . . .
1.2.2 Algebras of polyadic signature . . . . . . . . . . . . . . . . .
1.2.3 L3 as an algebra of polyadic signature . . . . . . . . . . . . .
1.2.4 Polyadic set algebras of ternary relations . . . . . . . . . . . .
1.2.5 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.6 Additivity . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.7 The Lindenbaum algebra of a theory Γ . . . . . . . . . . . .
1.2.8 Exhaustive search for satisfying interpretations in a Ps3 . . . .
1.3 Algebras of directed many-sorted polyadic signature . . . . . . . . . .
1.3.1 Algebras of directed many-sorted polyadic signature . . . . . .
1.3.2 A conservative reduction class with sub-formulae as an algebra
1.3.3 Directed many-sorted polyadic set algebras . . . . . . . . . .
1.3.4 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.5 The directed many-sorted polyadic closure . . . . . . . . . .
1.3.6 The closure as an algorithm . . . . . . . . . . . . . . . . . .
1.3.7 Exhaustive search for satisfying interpretations in a dMsPs3 .
iii
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
1
1
2
2
3
4
4
6
6
8
8
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
11
12
13
13
14
15
15
16
16
17
18
19
20
21
22
22
22
23
24
25
26
28
29
1.4
1.5
1.6
A construction of Ps3 ’s and dMsPs3 ’s . . . . . . .
1.4.1 A disprover deviced according to the method
Related constructions and algebras . . . . . . . . . .
Concluding remarks . . . . . . . . . . . . . . . . .
.
.
.
.
2 Automata for mechanising consistency proofs
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Outline of paper . . . . . . . . . . . . . . . .
2.1.2 Notation . . . . . . . . . . . . . . . . . . .
2.1.3 p-automata . . . . . . . . . . . . . . . . . .
2.1.4 The Ehrenfeucht-Fraïssé method . . . . . . .
2.1.5 Mappings from and to a finite set . . . . . . .
2.1.6 Vector-spaces over finite fields . . . . . . . . .
2.2 n-fold vector spaces . . . . . . . . . . . . . . . . . .
2.2.1 3-fold vector spaces over the minimal field . .
2.2.2 n-fold vector spaces over a given finite field . .
2.3 p-automata with abstract alphabets . . . . . . . . . .
2.3.1 n-tape p-automata . . . . . . . . . . . . . . .
2.3.2 Infinite tapes with finite support . . . . . . .
2.4 Transitive automata and reachability in general . . . .
2.4.1 A not quite classical notion of equivalence . . .
2.4.2 Transitive p-automata, the two-sorted case . .
2.4.3 Transitive p-automata, the one-sorted case . .
2.5 Reachability refined . . . . . . . . . . . . . . . . . .
2.5.1 Two-sorted multi-automata . . . . . . . . . .
2.5.2 Definable linear transformations . . . . . . .
2.5.3 Projective transitivity . . . . . . . . . . . . .
2.5.4 Projection automata . . . . . . . . . . . . . .
2.5.5 Substitution automata . . . . . . . . . . . . .
2.5.6 Properties of the two-sorted transition function
2.6 Moving to the abstract and to one sort . . . . . . . . .
2.6.1 One-sorted multi-automata . . . . . . . . . .
2.6.2 Properties of the one-sorted transition function
2.7 The automata for a formula and its sub-formulae . . .
2.8 Concluding remarks . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
3 Automata for the computation of finite representable polyadic algebras
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Outline of paper . . . . . . . . . . . . . . . . . . . . . . .
3.1.2 Sets, relations and mappings . . . . . . . . . . . . . . . . .
3.1.3 Finite-dimensional (quasi) polyadic algebras . . . . . . . . .
3.1.4 Purely infinite spectrum . . . . . . . . . . . . . . . . . . .
3.2 The polyadic atom-structure and the h-complex algebra . . . . . . .
3.2.1 Atom-structures and n-homomorphisms . . . . . . . . . .
iv
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
30
31
32
33
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
35
36
37
38
38
40
41
42
43
44
45
47
47
48
49
49
50
51
55
55
56
57
58
60
62
63
64
66
68
73
.
.
.
.
.
.
.
75
76
77
77
78
81
82
82
3.3
3.4
3.5
3.6
3.2.2 The complex algebra tailored for many sorts . . . . . . .
The h-complex algebra of a multi-automaton . . . . . . . . . . .
3.3.1 Concrete PTPS-automata defined . . . . . . . . . . . . .
3.3.2 Properly partitioned automata . . . . . . . . . . . . . . .
3.3.3 Automata as atom-structures . . . . . . . . . . . . . . .
3.3.4 The n-homomorphism induced by an automaton . . . . .
3.3.5 Representability of the h-complex algebra of an automaton
3.3.6 The main result . . . . . . . . . . . . . . . . . . . . . .
Diversity of the h-complex algebras of finite automata . . . . . . .
3.4.1 dMsPsn ’s . . . . . . . . . . . . . . . . . . . . . . . .
3.4.2 Psn ’s . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.3 MsPsn ’s . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.4 Polyadic equality algebras . . . . . . . . . . . . . . . . .
Axiomatisation by a finite set of first order sentences . . . . . . . .
Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . .
v
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. 84
. 86
. 86
. 90
. 90
. 91
. 92
. 95
. 95
. 95
. 96
. 97
. 97
. 98
. 101
0.1
Introduction
The present thesis is on mechanisation of the part of mathematical reasoning that amounts to
establishing that a formal sentence is not a theorem, i.e. it is about disproof. A mathematician typically uses examples in the form of infinite structures to establish that a sentence is not a theorem.
Infinite structures are however unsuitable as objects of mechanical computation by virtue of not
being finite. The present thesis is about a recently discovered class of finite boolean algebras with
operators that can be used in place of infinite structures for establishing that a formal sentence is
a non-theorem. There is no mechanical procedure for constructing all of the finite algebras, but
the algebras can be used to construct mechanical procedures with capabilities that for a mathematician implies pondering structures that necessarily are infinite. Such capability is witnessed when a
procedure establishes the consistency of a formal sentence that is true only of infinite structures. A
formal sentence of this kind is called an infinity axiom.
We work in the framework for formal mechanical procedure and formal theorem, as set forth
and studied by, amongst others the mathematicians D. Hilbert, K. Gödel, A. Church and A. Turing
in the 1930-ies. D. Hilbert posed the problem of whether it is possible to device a mechanical procedure for deciding whether given first-order formulae are (in)consistent. This problem is known
as Hilberts Entscheidungsproblem. We shall mostly be concerned with the Entscheidungsproblem
in its most classical form namely by considering sentences of pure predicate logic, i.e. the logic of
first-order formulae without equality or function symbols. Note that by combining results of J. von
Neumann, P. Bernays and K. Gödel it is possible to axiomatise set-theory with such a sentence, and
therefore questions of derivability from set-theory can be rephrased as questions of consistency of
sentences in pure predicate logic.
For the purpose of experimentation several mechanical disproving procedures, based on the
newly discovered algebras, have been implemented by the present author. They were made publicly
available and presented at the international workshop on disproving held as part of the Federated
Logic Conference in Seattle in 2006. On grounds of the response there, we believe that that these
disprovers were the first implemented procedures with disproving capability for pure predicate logic,
that fully automatically and naturally recognise a class of consistent sentences including infinity
axioms in reasonable time. Natural here means undoctored or non-ad-hoc, i.e., the capability of
recognising infinity axioms stems from the algebra on which the procedure is based alone. The use
of abstract algebra was crucial for solving the practical problem of making the procedures terminate
for at least some infinity axioms in reasonable time.
0.2
Summary of the chapters
Apart from the introduction, the present thesis consists of three chapters, the first of which is based
on a preprint of a journal article, see [Rog09], and the remaining two are manuscripts submitted to
journals for publication.
Before some elaboration, we summarise the chapters in one sentence each as follows. The
algebras and the the way of constructing new algebras from old ones, introduced in chapter 1
provide a novel way of subdividing the consistent sentences into mechanically listable classes of
sentences. In chapter 2 we introduce a novel kind of automaton for the purpose of tackling a
class of consistent sentences, which properly includes those tackled by the method of finite model
1
search. In chapter 3 we show how to use the automata of chapter 2, to compute algebras of the
kind introduced in chapter 1, as well as the well known quasi-polyadic- and substitution-cylindric
algebras, for the purpose of naturally and mechanically tackling infinity axioms in reasonable time.
0.2.1
Summary of chapter 1
In chapter 1 we work with the Entscheidungsproblem in an extremely reduced form, namely in the
form of the subclass of pure predicate logic consisting of finite conjunctions of prenex sentences
using no more than three variables. The class is known to be a conservative reduction class, which
is to say that in a coarse but precise sense of sameness, working with the Entscheidungsproblem
for this subclass is the same as working on the Entscheidungsproblem for first-order language with
an arbitrary supply of variables. D. Scott, in 1962, proved that pure predicate logic with less than
three variables is not, in the above sense, the same as working with an arbitrary supply.
The afore mentioned axiomatisation of set-theory is not known to belong to the three variable
class other than via a non-trivial conservative reduction. However, working with this class facilitates
getting to the core of the Entscheidungsproblem. Restriction to three variables is also beneficial in
that it allows us to compute some of the algebras of interest in a matter of days. Once computed, an
algebra can be used in place of finite structures in a variant of the well known method of establishing
consistency by searching for a finite model for a sentence, i.e. to attempt to find a finite structure of
which the formal sentence is true. The algebras of interest are those where the modified method
yields consistency of infinity axioms. In this case the method of finite model search is strictly
generalised.
Novelties presented in chapter 1 are the newly discovered class of boolean algebras with operators, called directed many-sorted polyadic set algebras together with a result on the class stating that
the finite members suffice for disproof of any consistent sentence of the conservative reduction class.
We have likened this to the downward Löwenheim-Skolem theorem because it states that countable
models suffice for disproof of any consistent sentence of first-order logic, which is a conservative
reduction class trivially. A novel way of constructing new algebras of interest from a given one is introduced and we show how to use this construction to generalise the method of finite model search.
The construction is related to the tensor product of polyadic algebras introduced by A. Daigneault
in 1963, see [Dai63]. The fact that this construction on the well known quasi-polyadic algebras
gives rise to a way of generalising finite model search is also covered in chapter 1 and is novel.
0.2.2
Summary of chapter 2
It is a consequence of the undecidability of the Entscheidungsproblem and of our analogue to the
downward Löwenheim-Skolem theorem that the finite directed many-sorted polyadic set algebras
can not completely be listed by means of a mechanical procedure. Subclasses of the algebras may,
however, be listed and the remaining two chapters are on such a subclass.
In chapter 2 we prepare the ground for mechanically listing algebras, some of which yield
disprovers that recognise classes with infinity axioms. The ground is prepared by introducing a
variant of the finite automata used by J.R. Büchi in the 1960-ies in a procedure for deciding Presburger arithmetic, the first-order theory of addition of natural numbers. This procedure uses finite
automata to represent possibly infinite relations definable in Presburger arithmetic. Operations corresponding to union, negation and projection which are needed to interpret Presburger formulae,
2
can be done on automata computationally. This implies that it is possible to search for models for
given formal sentences amongst, the not necessarily finite, relations definable in Presburger arithmetic. The automata we introduce are the finite models of a first order theory. Moreover we show
that the operations corresponding to union, negation and projection are not only computable but
definable using first-order sentences about the automata. This is relevant for computation, since
a property that is definable by means of a first-order formula can be checked in polynomial time.
Polynomial time is here with respect to the maximum of the size of the state-set and alphabet of
given automata.
The above allows us to transform given sentences in pure predicate logic into equiconsistent
first-order sentences about finite automata. Moreover consistency can be established by the familiar
method of finite model search, provided the pure sentence has a model definable in Presburger
arithmetic.
Novelties of chapter 2 include a fairly general technique for tackling the problem that reachability is not a first-order property over finite automata. It is known that over finite structures this
problem can be tackled by giving a first-order definition of some higher-order constructs and to
define reachability by means of that. The novel technique of chapter 2 avoids defining higherorder constructs by allowing some flexibility in the alphabet of the automata and by postulating
that states that are reachable in two steps also are reachable in one step. We introduce a novel
class of automata, called PTPS-automata. The class is defined by means of a finite set of first-order
formulae. The finite members of the class are shown to be of the same computational strength
as the multi-track automata introduced by J.R. Büchi in that they may replace one another in his
procedure for deciding Presburger arithmetic. The transformation of sentences of pure predicate
logic into equiconsistent first-order statements about automata, without introducing higher-order
constructs, is novel.
0.2.3
Summary of chapter 3
In chapter 3 we turn to the algebras introduced in chapter 1. When using an algebra in place of
a finite structure in the modified method of finite model search, the algebra has to be representable
in order to terminate for consistent sentences only. It is a consequence of the undecidability of
the Entscheidungsproblem and our analogue to the downward Löwenheim-Skolem theorem that
there is no mechanical procedure for telling whether an algebra is representable. In chapter 3 we
introduce a way of turning PTPS-automata, introduced in chapter 2, into directed many-sorted
polyadic algebras that necessarily are representable. Any algebra finitely generated by relations
definable in a slight expansion of Presburger arithmetic is shown to be embeddable in the algebra
of a PTPS-automaton. This class of algebras results in disproving procedures that recognise infinity
axioms. We show some results on criteria making the construction on PTPS-automata result in the
classical one-sorted polyadic algebras and polyadic equality algebras. The criteria and the PTPSautomata are shown to be definable using a finite set of first-order formulae.
The construction of representable algebras from PTPS-automata goes via the well known atomstructure of a finite algebra of polyadic similarity type. The atom-structure plays the same role as
a basis for a vector-space or a topology. Working with atom-structures is beneficial with respect
to mechanical procedures since a polyadic algebra with 2k elements has an atom-structure with k
elements. The afore mentioned implemented disprovers are based on algebras stored in the form of
3
atom-structures.
Novelties of of chapter 3 include the generalisation of the well known complex algebra of an
atom-structure to fit many-sorted algebras and the sufficient conditions for such a complex algebra
to be representable. The application of the generalised complex algebra to finite automata for the
purpose of computing representable algebras of the well known quasi polyadic and substitutioncylindric similarity types with and without diagonals is novel.
0.3
What is known
The present thesis is a contribution to the subjects of the Entscheidungsproblem and of representability in algebraic logic. We briefly go through the most relevant results as gathered from the
two major books on the subjects, namely the book “The Classical Decision Problem” by E. Börger,
E.Grädel and Y. Gurevich [BGG97] and “Cylindric Algebras part II” by J.D. Monk, L. Henkin and
A. Tarski, [MHT85]. As the latter, i.e. [MHT85], may not be easily accessible, we mention that it
is based on some lecture notes that may be available, namely “Cylindric set algebras”, [HMA+ 81].
For algebraic logic the survey articles of I. Németi [Ném97] and of H. Andréka, I. Sain and I.
Németi [ANS01] give a somewhat more updated picture of what has been done. In algebraic logic
the chapters on polyadic algebras are the relevant ones. We work with finite dimensional algebras
and in the present setting polyadic algebras are in all relevant respects the same as quasi-polyadic
algebras, pinter-algebras and substitution-cylindric algebras as encountered in the surveys.
0.3.1
Classification of the Entscheidungsproblem based on syntax
In 1920 the Norwegian mathematician T. Skolem published a proof of the fact that any sentence of
pure predicate logic can be brought to an equisatisfiable, i.e. equiconsistent, sentence of the form
∀x0 . . . ∀xn−1 ∃y0 . . . ∃ym−1 φ
where φ is pure and quantifier free. This reduction is by means of a mechanical procedure as are
the rest of the reductions mentioned in this section.
In 1933 K. Gödel published an improvement of T. Skolems result by showing that sentences
can be brought to the form
∀x0 ∀x1 ∀x2 ∃y0 . . . ∃ym−1 φ
where φ is pure and quantifier free. This is to say that no more than 3 universal quantifiers need
be considered when working with the Entscheidungsproblem. Following the book [BGG97] these
two classes are denoted [∀∗ ∃∗ ] and [∀3 ∃∗ ] respectively.
Then in 1935 T. Skolem published a simpler proof of K. Gödels result and in doing so he
showed that any sentence can be brought to one of the form
(∀x∃y0 . . . ∃yn−1 φ) ∧ (∀x∀y∀zφ0 ) ∧ (
V
i<k
∀x∀y∃zψi )
where φ, φ0 , ψ0 , . . . , ψk−1 are pure, quantifier free and use relation-symbols of arity 2 or less. Such
sentences are examples of conjunctions of three-variable prenex sentences, which are exactly of the
kind the implemented disprovers work on. If we omit the information about the arities, this class
can be written as [∀∃∗ ∧ ∀3 ∧ ∀2 ∃ ∧ · · · ∧ ∀2 ∃].
4
Parallel to this mathematicians worked on syntactic criteria for a decision procedure to exist.
In doing so, sentences of the following two classes were shown to be consistent if and only if the
method of finite model search succeeds.
[∃∗ ∀∗ ] by P. Bernays and M. Schönfinkel (1928)
[∃∗ ∀2 ∃∗ ] proven independently by K. Gödel (1932), L. Kalmár (1933) and K. Schütte (1934).
In 1936 and 1937 respectively, A. Church and A. Turing proved the undecidability of the Entscheidungsproblem. An analysis of A. Turings proof yields the undecidability of the class [∀∃ ∧ ∃∀5 ].
Gradual sharpening of A. Turings result led to proofs that the following two prefix classes are undecidable.
[∀3 ∃] and [∀3 ∧ ∀2 ∃] by J. Surányi (1943) improving a result of J. Pepis (1938)
[∀∃∀] by A. Kahr, E. Moore and H. Wang (1962)
Together with the two decidable classes this yields a complete classification of pure predicate logic
in terms of syntactic criteria on the quantifiers. In short; if we have a maximum of two universal
quantifiers preceding some existential quantifier and the two are consecutive then the method of
finite model search succeeds. Otherwise we have to look for further criteria, or hope that the
sentence has a finite model anyway. Beyond that we may use some of the methods presented in the
present thesis and hope those succeed, which they do when the sentence happens to have a finite
model as well as in several cases where all the models are infinite. We note that the latter two classes
remain undecidable when restricted to relation symbols of arity at most 3.
The undecidable classes can further be divided into decidable and undecidable classes in terms
of syntactic criteria. We mention that the class [∀∃∀] restricted to formulae in which all disjunctions
are binary at most, was proven to be decidable by S.O. Aanderaa in 1971/73, see [Aan71],[AL73].
This class contains the possibly shortest infinity axiom known to mankind, namely
∀x∃y∀z¬Rxx ∧ Rxy ∧ (Rzx → Rzy),
which states that the relation R is irreflexive, moreover it has no maximal elements and is transitive
on connected components. Based on experiments we conjecture that the published implementations of the diprovers described in the present thesis recongnises the consistent members of this
class.
Other infinity axioms where the mentioned implementations, work well in experiments, belong
to a class shown to be decidable by W. Ackermann (1936), namely sentences of the form
∀x∃ySxy ∧ ∀x∀y∀zφ
where S is a binary relation symbol and φ is quantifier-free and uses S only. An infinity axiom of
this class is
∀x∃ySxy ∧ ∀x∀y∀z¬Sxx ∧ (Sxy ∧ Syz → Sxz).
In 1983 S.O. Aanderaa presented a proof sketch of the decidability of a generalisation of W. Ackermanns class [Aan83].
5
0.3.2
Beyond syntactically defined classes
By the completeness theorem of K. Gödel (1930) we may conclude that there is a procedure for
listing the inconsistent sentences. By the undecidability of the Entscheidungsproblem we further
conclude that there is no procedure listing the consistent sentences. A sub-class of pure predicate
logic, where either the sub-class or its complement can not be listed by means of a mechanical
procedure is said to be non-recursive.
In 1950 B. Trakhtenbrot proved that the sentences of pure predicate logic for which the method
of finite model search succeeds is non-recursive. In the same year W. Craig independently announces a proof of the same. In 1953 B. Trakhtenbrot improved his result by showing that any
class of consistent sentences that contains the sentences for which the method of finite model search
succeeds, necessarily is non-recursive.
In 1962 J.R. Büchi showed that the situation is the same for the class [∃ ∧ ∀∃∀] by introducing
the notion now known as a conservative reduction. This is to say that any sentence of pure predicate
logic can, by means of a mechanical procedure, be brought to an equiconsistent sentence of the
class in such a way that the method of finite model search is ’equisuccessful’, i.e., it either succeeds
for a given sentence in both the reduced form and the unreduced form or for neither.
Using a technique introduced by Y. Gurevich (1976), it was possible to prove that there exist
conservative reductions for both the undecidable class of J. Surányi and the class of A. Kahr, E.
Moore and H. Wang. Proofs are available in the book [BGG97].
As a consequence of J.R. Büchis result and of Y. Gurevich’s results on the earlier reduction
class of J. Surányi, the disprovers of the present thesis recognise a non-recursive class of sentences.
Under the fair assumption that syntactic criteria are precise enough to be recursive, we may draw
the following conclusion. The disprovers of the present thesis recognise a set of sentences that is
not contained in any syntactically defined decidable class.
0.3.3
Algebraic logic
The class of finite algebras, proposed as a substitute for infinite structures in the present thesis, is
due to the present author and are a variation of polyadic algebras introduced by P. Halmos’ around
1954. Polyadic algebras are in turn are a variation of cylindric algebras, introduced by A. Tarski
around 1952. The study of such algebras belongs to the branch of mathematics called algebraic
logic. Algebraic logic and indeed mathematical logic as a whole, was pioneered by G. Boole and A.
DeMorgan in the mid eighteen hundreds. What A. Tarski and P. Halmos did was to equip boolean
algebras with extra operators so as to describe how families of n-ary relations behave under the
boolean operations as well as the other operations needed to interpret first-order language. On A.
Tarskis part, this was generalisation of work he had done on relation algebras which are models of a
formalism on binary relations under boolean operations, composition and reciprocation.
A. Tarskis cylindric algebras have extra-boolean operations consisting of cylindrifications and
diagonals which are sufficient to interpret first-order language with equality. P. Halmos’ polyadic
algebras have cylindrifications and operations for substituting variables for variables, which is sufficient for interpreting formulae of pure predicate logic. Since the present thesis is about pure
predicate logic, variants of polyadic algebras are our main concern. In part 3 of the present thesis
we however touch upon polyadic equality algebras which are a common expansion of both cylindric
and polyadic algebras.
6
For an algebra to soundly replace a finite structure in the modified method of finite model
search, it has to be finite and representable. We are therefore interested in computing, or mechanically listing, finite and representable algebras. One method of listing finite representable algebras
is to use n-ary relations over finite sets. Algebras produced thusly do however not give rise to disprovers that recognise infinity axioms. The algebras of interest are finite algebras which give rise to
disprovers that recognise only consistent sentences and some infinity axioms.
If it were possible to finitely axiomatise the representable algebras, we would have a necessary
and sufficient condition by which we could mechanically list finite representable algebras. In chapter 1 we show that finite axiomatisation is not possible in any reasonable language for the variant
of polyadic algebras introduced in [Rog09]. For the classical variants of cylindric and polyadic algebras the situation is as follows. J.S. Johnson in 1969 proved that representable polyadic algebras
of dimension n > 2 can not be axiomatised with a finite set of first-order axioms, see [Joh69]. J.S.
Johnson extended the non-axiomatisability results of J.D. Monk on cylindric algebras from 1965
and 1969, see [Mon69]. In 1969 J.D. Monk also considered infinite dimensional cylindric algebras. But for the modified method of finite model search to work, the representable algebras have
to be finite and therefore finite dimensional.
What remains is to find sufficient and finite conditions for an algebra to be representable. The
book by L. Henkin, J.D. Monk and A. Tarski, [MHT85], has several sufficient conditions for
representability of finite dimensional polyadic algebras and polyadic equality algebras. These are
theorem 5.4.34 on rich algebras, theorem 5.4.36 and 5.4.28 on algebras whose atoms are rectangular and theorem 5.4.39 by L. Henkin on algebras whose characteristic is finite. It is in general
hard to tell from a representation theorem whether the finite algebras, that satisfy a condition, are
representable over finite sets and therefore only of interest in regards to infinite algebras. In the case
of the condition that the atoms be rectangular, theorem 5.4.35 sets up a correspondence between
rectangular atoms and singleton sets which implies that finite algebras with rectangular atoms are
representable over finite sets. Thus using finite algebras that satisfy the condition that atoms be
rectangular gives nothing more than the good old method of finite model search.
As for more recent developments we mention the following. H. Andréka et. al. [AGM+ 98]
introduce a generalised notion of rectangular atoms and a modified version of richness which imply
representability. Also M. Ferenczi and G. Sági [FS06] survey recent developments on two approaches to representability of cylindric algebras. One approach is on relativised set algebras, which
is of interest in regards to proof, but not to disproof in any obvious way. The other approach is
on proving representability using a non-standard set theory whose consistency follows from the set
theory known as ZFC. The latter approach may be of interest in regards to disproof, but there are
serious complexity issues in regards to the use of axiomatisations of set theory as part of sufficient
conditions for representability .
Chapter 3 of the present thesis provides sufficient conditions for an algebra to be representable
provided it is finite. The conditions are in the form of a finite set of first-order sentences about
the atom-structure. Moreover there are finite algebras that satisfy the conditions, which are not
representable over finite sets. Disprovers based on finite algebras that satisfy the conditions, but
that are not representable over finite sets, are capable of recognising infinity axioms.
7
0.3.4
Automated reasoning
In regards to disproof and implementations the branch of computer science called automated reasoning is relevant. In particular the part of automated reasoning called model generation or model
building. In the “Handbook of Automated Reasoning” there is a chapter written by C.G. Fermüller,
A. Leitsch, U. Hustadt and T. Tammet, [FLHT01] which contains a survey of model generation
methods up until 2001. Put very briefly, there have been two basic approaches. One approach is
that of pruning the search tree in regards to the method of finite model search. The other approach
is that of producing a description of a model from the deduced formulae of a complete refutation
procedure, that has terminated without finding an inconsistency. Although exhibiting a model implies consistency, we note that model building is more than disproof in that a model provides more
information than the confirmation we get from a procedure that merely terminates on consistency.
The models resulting from refutation procedures are typically finite descriptions of infinite Herbrand models. However no examples of infinity axioms for which a model can be described is given
in the model generation survey in [FLHT01].
Further developments are reported in the 2004 book “Automated model building” by R. Caferra, A. Leitsch and N. Peltier, which the present author has not had the opportunity to read.
However in 1997 N. Peltier, [Pel97a], [Pel97b], reports on methods capable of recognising infinity
axioms. These appear to be unimplemented methods. In 2000 R. Caferra and N. Peltier, [CP00]
report on a method which has been implemented. The implementation is reported to have succeed on an example of an infinity axiom discovered by W. Goldfarb [Gol84]. The implementation
needed human interaction to succeed, but this is perhaps because of the complexity of the particular
infinity axiom chosen.
In 2009 N. Peltier published on transforming first-order sentences into equiconsistent firstorder descriptions of tree-automata, together with an example of an infinity axiom of pure predicate
logic for which the description of said automaton has a finite model, see [Pel09]. Chapter 2 of the
present thesis is about a similar but distinct transformation. Neither of the two transformations
are known to result in infinity axioms recognisable in reasonable time. We do however in chapter
3 propose to use first-order descriptions of automata to compute atom-structures of polyadic set
algebras. The present author has used polyadic atom-structures, when implementing some of the
methods of the present thesis. The implementations are available on the present authors web-page
and terminate in reasonable time for several infinity axioms of pure predicate logic without human
interaction.
0.3.5
Personal perspective and acknowledgements
For my Cand. Scient. degree I worked on strategies for the method of finite model search under
the supervision of Professor S.O. Aanderaa. After obtaining the Cand. Scient. degree in 1993, I
attempted to enter a Dr. Scient program with S.O. Aanderaa as my supervisor.
I was not accepted for such a program, but had begun work on generalising a decidability result
due to S.O Aanderaa, see [Aan83]. I ended up with the generalisations of the method of finite
model search described in the present thesis. S.O. Aanderaa also found a serious flaw in an early
version of chapter 2 of the present thesis. Early means some time before 1999 in this setting.
As I was not accepted for the Dr. Scient. program I started working full time outside of
academia. In the year 2000 however, I took a part time job for the reason that I wanted go for a
8
Dr. Philos. degree. As life is not always predictable I have had several breaks in my project and my
non-academic career. After one such break I had the opportunity to join an automated reasoning
project with Professor A. Waaler at the Department of Informatics in Oslo for half a year in 2007.
I have also had the opportunity to follow the weekly mathematical logic seminar at the University of Oslo, where besides Professor S.O. Aanderaa, the Professors D. Normann and H.R. Jervell
have been able to follow my progress and to give advice. I hereby thank all of the above for the said
opportunities. I also thank my personal friend Dr. F.A. Dahl for helpful comments.
9
Chapter 1
Turning decision procedures into disprovers
11
1.1
Introduction
In this setting a disprover is a procedure that terminates on input of satisfiable first-order sentences
only. The purpose being to use the disprover in conjunction with a procedure that terminates on
input of inconsistent first-order sentences only, in attempts at deciding whether given sentences are
satisfiable by means of a computer.
The present paper describes a method for devising disprovers, some of which have been implemented. Focus is on why disprovers deviced according to the method work in principle and to what
extent. We rely on established results on the Entscheidungsproblem and use techniques common
in algebraic logic for proofs. A particular class of many-sorted polyadic set algebras suitable for the
task is introduced.
One needs a decision procedure for some first-order theory to devise a disprover according to the
method. By means of the decision procedure a finite many-sorted polyadic set algebra is computed.
Such an algebra forms the basis of one disprover, which on input of a first-order sentence works
by exhaustive search for satisfying interpretations for the sentence in the algebra and in successively
refined versions of the algebra.
Depending on the decision procedure, resulting disprovers can be made to recognise satisfiable
sentences that are not finitely satisfiable. Such sentences are called infinity axioms, and here is an
example.
∀x∃yRxy∧ ∀x¬Rxx∧ ∀x∀y∀zRxy ∧ Ryz → Rxz
The set of sentences recognised by each disprover, turns out to be non-recursive. This is a
consequence of a result of Büchi, sharpening Trakhtenbrots theorem, together with the ability to
recognise any finitely satisfiable sentence of a substantial fragment of first-order language [Büc62].
This implies that no decidable class of sentences covers the set of sentences recognised by any one
of the disprovers described here. Since finite unions of decidable sentence classes are decidable, no
finite union of decidable sentence classes will cover any of the recognised sets either.
As any disprover can be repaired in a most ad hoc way so as to recognise any given infinity axiom,
the following naturalness property is shown. The set of sentences recognised by each disprover is
closed under logical equivalence, within the aforementioned fragment. This property is not shared
with procedures that work by first checking whether the input is syntactically equal to one of a
finite set of satisfiable sentences and then go on with, say, search for satisfying interpretations over
finite sets.
Procedures that do search for satisfying interpretations over finite sets are said to do finite model
search. Each of the disprovers described here externally share the ability to recognise any finitely
satisfiable sentence of a substantial fragment of first-order language and the naturalness property
with finite model search procedures. Also internally there is some resemblance. We therefore use
the term generalised finite model search, for describing how the presented disprovers work. Those
of the disprovers that recognise infinity axioms represent a strict generalisation of finite model
search.
The model search generalisation and the results about it, work for conjunctions of purely relational prenex sentences whose length of quantifier prefix is limited by a constant. For the following
reasons that constant is fixed at 3 throughout. It has made implementation feasible. To some extent
it makes geometric inspection feasible. It economises notation. There is no loss of generality, in
12
terms of computability, satisfiability and finite satisfiability as conjunctions of 3-variable prenex sentences are known to form a conservative reduction class. This is to say; the existence of algorithms
are known that transform any first-order sentence to an equi-satisfiable and equi-finitely-satisfiable
sentence of the required form. The earliest to show the existence of such a transformation was
Büchi [Büc62], and several are known. These transformations are called conservative reductions,
where conservativeness relates to finite satisfiability. A well known example of a conservative reduction is skolemisation, and it is conservative because a model for the transformed sentence can
be defined by expanding a model for the original one. A class of 3-variable prenex sentences that
form a reduction class on their own is the ∀∃∀ class of Kahr Moore and Wang [KMW62]. This
class turned out to be conservative by a result of Gurevich and Koriakov[GK72] utilising a result of
Berger [Ber66]. Proofs of this and related results are accessible in the book [BGG97].
That which is needed about the well known theory of polyadic set algebras, for statement and
proof of the results of the present paper is given in section 1.2. In Section 1.3 a particular class of
many-sorted polyadic set algebras, that is believed to be new, is introduced. They are called directed
many-sorted polyadic set algebras of dimension 3 (dMsPs3 ). A downward Löwenheim-Skolem
type of result is shown, in that any satisfiable sentence of the aforementioned form is satisfiable in
a finite such algebra, even if the sentence is an infinity axiom. In Section 1.4 a construction of
polyadic set algebras, many-sorted or not, is defined. It is for building more refined algebras from
a given one. In particular one can build algebras refined enough to allow a satisfying interpretation
for any finitely satisfiable (in the usual sense) sentence.
1.1.1
Sets and mappings
The numeral 3 denotes {0, 1, 2} and ω the natural numbers. The set of mappings from 3 to 2 is
written both as 23 and as 3 → 2. Formally a mapping f in 3 → 2 is the set of pairs {(u, f (u)) :
u ∈ 3}. The f -image of 2 is the set {f (u) : u ∈ 2}. The restriction of f to the set 2 is the set of
pairs {(u, f (u)) : u ∈ 2}. If f and g are mappings then f ◦ g denotes their composition, applying
g first then f . If f ⊆ g then f is said to be a restriction of g and g an extension of f . Mappings
in 3 → 3 are written as triples of elements of 3, where the value of 0 is leftmost. So the identity
mapping for instance is written 012 and the mapping that subtracts one modulo 3 is written 201.
1.1.2
First-order formulae
First-order formulae in the present paper are taken from pure relation calculus with three variables.
Pure here means being without distinguished equality-, function- and constant-symbols. Relationsymbols are further assumed to be of arity 3. See chapter 6.5 of [DG79] for how to conservatively
reduce ∀∃∀-sentences with relation-symbols of varying arities to those of arity 2. For arities lower
than 3 one can replace each occurrence of Rx for instance, with Rxxx and Rxy with Rxyy. The
obtained language fragment is denoted by L3 .
L3 is assumed to have a countable unlimited supply of distinct ternary relation-symbols. Ternary
relation-symbols are also the only kind of symbol taken from an infinite set. Rxy and xRy are used
as abbreviations for Rxyy. The notation L03 will be used for L3 fragments that have a limited finite
number of relation-symbols. A formula is a sentence if every variable is bound by a quantifier.
Theories are sets of L03 sentences, and the symbol Γ is used for theories. A theory Γ is complete if
for every sentence φ ∈ L03 it is the case that φ ∈ Γ or ¬φ ∈ Γ. If it is the case that for all φ ∈ L03
13
either φ ∈ Γ or ¬φ ∈ Γ, and not both, then Γ is consistent and complete. The set of L03 sentences
true under a fixed interpretation is a complete and consistent theory. A complete theory is said to
be decidable if it is recursive. Two L03 formulae φ and ψ are said to be Γ-equivalent if the sentence
∀x∀y∀z(φ ↔ ψ) is an element of Γ. Note that the variables x, y and z each may occur free, bound
of both in φ and ψ.
For each relation-symbol R and each mapping σ in 3 → 3, Rσ is an atomic formula. An
example of an atomic formula is R012 which is, or denotes, a formula more commonly written as
Rxyz. It is to be understood that the variable x has index 0, y has index 1 and z index 2. The
word atom is used for minimal nonzero elements of boolean algebras and not for atomic formulae.
The literals are the set of formulae generated by the atomic formulae under negation (¬). The open
formulae are the set of formulae generated by the atomic formulae under negation and disjunction
(∨). L3 is the set of formulae generated by the atomic formulae under negation, disjunction and
the existential quantifiers (∃0 , ∃1 , ∃2 ). Again ∃0 φ is, or denotes ∃xφ, etc. The following are used
as abbreviations; ⊥ for some contradiction depending on the language fragment in question, > for
¬(⊥), φ ∧ ψ for ¬(¬φ ∨ ¬ψ), φ ↔ ψ for (¬φ ∨ ψ) ∧ (¬ψ ∨ φ) and ∀i φ for ¬∃i (¬φ). A formula
is said to be a prenex sentence if it is of the form Q0 Q1 Q2 φ where φ is open and each Qi is one
of ∃i or ∀i . Since the languages considered here are equality-free the upward Löwenheim-Skolem
theorem also holds in the finite. This is to say that if φ has a model of cardinality n ∈ ω then φ
also has a model of cardinality n + 1 [Men87](p.73).
1.1.3
Algebras, homomorphisms, and preservation
In the present paper algebras and homomorphisms of several kinds are used. Here we summarise
some properties the kinds have in common. An algebra is a set together with a family of operations
on that set. The given set is called the carrier-set. Operations are mappings under which the
algebra is closed, which is to say that the image of the carrier-set under an operation is a subset of
the carrier-set. Algebras have signatures, these provide information on the arities of the operations,
moreover signatures determine a notion of correspondence between operations in pairs of algebras
of similar kind.
Definition Let A and B be algebras, let f be an n-ary operation of A and g the corresponding
n-ary operation of B. A mapping h from a subset X of the carrier-set of A to the carrier-set of B,
is said to preserve f if for every x0 , . . . , xn ∈ X it is the case that
f (x0 , . . . , xn−1 ) = xn implies g(h(x0 ), . . . , h(xn−1 )) = h(xn ).
A homomorphism from A to B is a mapping from the carrier-set of A to the carrier-set of B that
preserves the operations of A. Embeddings are homomorphisms that are one to one. Isomorphisms
are embeddings that are onto.
The following properties follow by the definition above. Homomorphisms are closed under
composition which is to say that if h0 and h1 are homomorphisms then h1 ◦ h0 is also a homomorphism. If h is a homomorphism from A and if X is a subset of the carrier-set of A, then the
restriction of h to X also preserves the operations of A. If X is closed under the operations of A
then the restriction of h to X is a homomorphism from X.
14
1.1.4
Algebras of boolean signature
Algebras of boolean signature are algebras with signature B = (B, ∨, ¬, ⊥). ∨, ¬, ⊥ are called join,
negation, and bottom respectively. An algebra of boolean signature A is a B-sub-algebra if it is of
the form (X, ∨|, ¬|, ⊥|) where X is a subset of B and ∨|, ¬|, ⊥| are the restrictions of ∨, ¬, ⊥ to
X, moreover X must be such that ∨| ∈ X × X → X and ¬|, ⊥| ∈ X → X. The last requirements here are to exclude X’s that are not closed under ∨|, ¬|, ⊥|. A boolean homomorphism is
a mapping from an algebra of boolean signature to an algebra of boolean signature that preserves
∨, ¬, ⊥.
The algebras that are of interest in the present paper are each expansions of algebras of boolean
signature. This is to say that they are algebras of boolean signature with extra operations on them.
Those of the algebras that are expansions of boolean algebras turn out to be very hard to axiomatise.
We therefore dispose of with the usual axioms all together. By the representation theorem of Stone
the following is way of defining boolean algebras.
Definition An algebra of boolean signature is a boolean algebra if it is embeddable into
(P(U ), ∪, −, ∅), where U is some set,P(U ) is the power-set of U , ∪ is union, − is (absolute)
complement and ∅ is the empty set.
A boolean algebra is finitely generated if it is generated by a finite subset of the carrier-set under the
boolean operations. The following property of boolean algebras is central in the present paper.
Lemma 1.1.1 Finitely generated boolean algebras are finite.
Now if U is infinite then a finitely generated sub algebra of (P(U ), ∪, −, ∅) is finite but its
elements may be infinite sets. As we seek to make computers work with boolean algebras the
following is useful.
Lemma 1.1.2 Let A = ({⊥, >}, ∨, ¬, ⊥) be the usual two element boolean algebra. Every finite
boolean algebra is isomorphic to an algebra with carrier-set n → {⊥, >} where n ∈ ω and where the
operations are defined component-wise.
To say that the operations on a set of mappings are defined component-wise is to say that
the join of f and g for instance, which for now we denote by (f ∨g), is the uniquely determined
operation with the following property. For each i ∈ n it is the case that (f ∨g)(i) = f (i) ∨ g(i).
For the particular case, where the mappings are to a two element set, the term bit-wise is sometimes
used.
1.2
Algebras of polyadic signature
Inspired by Tarskis cylindric algebras, Halmos introduced and studied polyadic algebras. Some
papers on these studies are collected in [Hal62]. A particular class of polyadic algebras are polyadic
set algebras of dimension 3, denoted Ps3 (see definition below). In the present paper these serve
as abstract or generalised models for L3 sentences. The well known Lindenbaum algebra, L03 /Γ,
of a consistent and complete L03 theory Γ can be equipped with extra operations related to variable
substitution and existential quantifiers to make it a Ps3 . It is possible to define interpretation and
15
satisfaction in Ps3 ’s in such a way that finding a satisfying interpretation for an L3 -sentence implies
satisfiability in a usual sense.
Observe that the number of equivalence classes of L03 /Γ may be finite also when Γ does not have
finite models. The theory of a dense order without endpoints is an example of such a Γ. To see this,
note that this Γ can be stated in a language, L03 , with one relation-symbol (<), which together with
the three variables of L03 gives rise to a finite number of atomic formulae. It also turns out to allow
quantifier-elimination within L03 . Any formula in a theory with quantifier elimination is equivalent
to an open formula. An open formula can be brought to an equivalent one in conjunctive normal
form, whose length is limited by a constant depending on the number of atomic formulae. So in
the example, L03 /Γ has a finite number of equivalence classes, each having an infinite number of
formulae defining the same relation over a densely ordered set.
The defined relations are over an infinite set and may be infinite themselves. Importantly
satisfying interpretations for infinity axioms can be found, such as:
∀x∃y∀zRxyy∧ ∀x∀y∀z¬Rxxx∧ ∀x∀y∀zRxyy ∧ Ryzz → Rxzz.
Interpretation and satisfaction in Ps3 are such that; the infinity axiom is satisfied in L03 /Γ by
a uniquely determined interpretation that associates Rxyz with the equivalence class having the
formula x < y as an element. Any association of Rxyz with an element of a Ps3 uniquely
determines an interpretation which may be satisfying or not. Associating Rxyz with an equivalence
class without atomic formulae also gives rise to a unique interpretation for the infinity axiom. L3
formulae with any number of relation-symbols may be interpreted Ps3 ’s. It is in particular the case
that L3 sentences with more relation-symbols than those of L03 have interpretations in L03 /Γ.
Given a decision procedure for Γ where L03 /Γ has a finite number of equivalence classes, one
can compute an abstract finite Ps3 isomorphic to L03 /Γ, by using indices for the equivalence
classes of L03 /Γ, as elements of a carrier-set. In such an abstract Ps3 exhaustive search for satisfying
interpretations can be performed, by means of a computer.
Ps3 ’s are called 0-valued functional algebras in [Hal62]. Some known results about Ps3 ’s
follow. Propositions are given with only hints of proofs. The notation of the present paper is
modernised so as to be close to that found in [Ném91], [AGM+ 98] and [ASN01]. We shall use
slightly fewer operations related to variable substitutions than what is common.
1.2.1
Operations related to variable substitution
Three distinguished mappings in 3 → 3 are are p = 102 called permutation, s = 112 called
substitution and r = 201 called rotation.
Lemma 1.2.1 p,s and r generate all mappings in 3 → 3, under composition.
1.2.2
Algebras of polyadic signature
Definition An algebra of polyadic signature is an A = (B, r, p, s, c0 , c1 , c2 ) such that B has the
signature of a boolean algebra and r, p, s, c0 , c1 , c2 are unary operations on B.
In the intended algebras the boolean operations correspond to boolean connectives. r, s, p relate to
variable substitution. c0 , c1 , c2 correspond to existential quantifiers, and are called cylindrifications.
16
Definition Let A = (B, r, p, s, c0 , c1 , c2 ) and C be algebras of polyadic signature. h is a polyadic
homomorphism from A to C if h is a boolean homomorphism from B to the boolean reduct of C
that preserves each of r, p, s, c0 , c1 , c2 .
1.2.3 L3 as an algebra of polyadic signature
Here the language L3 , seen as an algebra with operations for boolean connectives, is expanded to
make an algebra of polyadic signature. Interpretations of L3 turn out to be the polyadic homomorphisms from this algebra to other algebras of polyadic signature. An interpretation is satisfying for
a sentence if the sentence is interpreted as >. The following defines some syntactic operations on
L3 . Semantics for these operations are to be found further on.
Definition for i ∈ 3 and σ ∈ 3 → 3 define r∗ , p∗ , s∗ ∈ L3 → L3 by
r∗ (Rσ) = R(r ◦ σ)
p∗ (Rσ) = R(p ◦ σ)
s∗ (Rσ) = R(s ◦ σ)
r∗ (¬φ) = ¬r∗ (φ)
p∗ (¬φ) = ¬p∗ (φ)
s∗ (¬φ) = ¬s∗ (φ)
r∗ (φ ∨ ψ) = r∗ (φ) ∨ r∗ (ψ)
p∗ (φ ∨ ψ) = p∗ (φ) ∨ p∗ (ψ)
s∗ (φ ∨ ψ) = s∗ (φ) ∨ s∗ (ψ)
r∗ (∃i φ) = ∃r(i) r∗ (φ)
p∗ (∃i φ) = ∃p(i) p∗ (φ)
s∗ (∃0 φ) = ∃0 φ
s∗ (∃1 φ) = ∃0 p∗ (φ)
s∗ (∃2 φ) = ∃2 s∗ (φ)
Note that r∗ , p∗ and s∗ may be “moved inwards” relative to each of the connectives of L3 . So L3 , the
open and the atomic formulae are each closed under these operations. They don’t quite commute
as s∗ involves renaming only free occurrences of a certain variable, while r∗ and p∗ rename all
occurrences of involved variables. Compare this definition to the axiomatisation of quasi polyadic
algebras in [AGM+ 98] Section 4. We now expand L3 to make it an algebra of polyadic signature.
Definition L3 = (L3 , ∨, ¬, ⊥, r∗ , p∗ , s∗ , ∃0 , ∃1 , ∃2 )
17
1.2.4
Polyadic set algebras of ternary relations
We have seen how to turn a fragment of first-order language into an algebra of polyadic signature.
Here the ternary relations over given sets are are turned into algebras of polyadic signature. A
mapping from the variables of L3 to some set U is called a sequence and an interpretation is an
assignment of each formula of L3 to a set of sequences. This form of interpretation is due to
Tarski. In contrast to other, quite viable, forms of interpretation tarskian interpretation makes
Γ-equivalence coincide with equality of interpretations in given models for Γ.
Since we use 3 variables, a set of sequences is a ternary relation. We refer to the relation assigned
to φ by an interpretation as the relation defined by φ. By the classical Löwenheim-Skolem-Tarski
theorems and the absence of a symbol for equality, satisfiable sentences each have a model over
the reals, R. In such models L3 -formulae define subsets of R3 . This gives rise to a geometric
interpretation of elements of Ps3 ’s that will be appealed to later, instead of very detailed symbolic
proof, see fig.1.1 and fig. 1.2.
Definition Let U be some set. The full Ps3 over U is an algebra
(P(U 3 ), ∪, −, ∅, rU , pU , sU , cU0 , cU1 , cU2 ) where (P(U 3 ), ∪, −, ∅) is the boolean algebra of sets of
triples of elements of U . Operations related to variable substitutions are for V ⊆ U
rU (V ) = {u ∈ U 3 : u ◦ r ∈ V } = {abc ∈ U 3 : cab ∈ V }
pU (V ) = {u ∈ U 3 : u ◦ p ∈ V } = {abc ∈ U 3 : bac ∈ V }
sU (V ) = {u ∈ U 3 : u ◦ s ∈ V } = {abc ∈ U 3 : bbc ∈ V }
Operations for the existential quantifiers are
cU0 (V ) = {abc ∈ U 3 : there is a u ∈ U such that ubc ∈ V }
cU1 (V ) = {abc ∈ U 3 : there is a u ∈ U such that auc ∈ V }
cU2 (V ) = {abc ∈ U 3 : there is a u ∈ U such that abu ∈ V }
An algebra is said to be a full Ps3 if it is the full Ps3 over some set U .
By abuse of notation let P(U 3 ) denote the full Ps3 over U .
Figure 1.1: Cylindrification along x-axis, z-axis is orthogonal to the present paper and the dotted
line marks the xy-diagonal plane which is also orthogonal to the present paper.
18
Figure 1.2: Substitution, cylindrifies that which meets the xy-diagonal plane. Rotation and permutation amount to suitably renaming the axis, then rotating or mirroring the figures respectively.
Definition A Ps3 is an algebra of polyadic signature that is embeddable into a full Ps3 .
The next property relates the defining equations for r∗ , s∗ , p∗ to those of rU , sU , pU . For this
the author finds geometric interpretation helpful, see fig. 1.1 and fig. 1.2.
Lemma 1.2.2 For each i ∈ 3 and V, W ⊆ U
rU (¬V ) = ¬(rU (V )).
pU (¬V ) = ¬(pU (V )).
sU (¬V ) = ¬(sU (V )).
rU (V ∪ W ) = rU (V ) ∪ rU (W ).
pU (V ∪ W ) = pU (V ) ∪ pU (W ).
sU (V ∪ W ) = sU (V ) ∪ sU (W ).
rU (cUi (V )) = cUr(i) V .
pU (cUi (V )) = pUp(i) V .
sU (cU0 (V )) = cU0 (V ).
sU (cU1 (V )) = cU0 (pU (V ).
sU (cU2 (V )) = cU2 (sU (V ).
1.2.5
Interpretation
Here we go some length to see that polyadic homomorphisms from L3 to Ps3 ’s are essentially
the same as interpretations of L3 formulae as ternary relations over some set. The statement is
broken down into a few extension properties, which are of use later in the present paper. Extension
properties are also known as universal properties.
The first extension property says that polyadic homomorphisms from L3 to Ps3 ’s are like
interpretations in that if we interpret the relation-symbols of L3 then an interpretation of each
atomic formula of L3 is determined. We use the notation {R, ...} × {012} for the set of atomic
formulae of L3 in which all variables of L3 occur and in which they occur in the order specified by
012. The following is a consequence of lemma 1.2.1.
19
Lemma 1.2.3 If {R, ...} are the relation-symbols of L3 and if A is a Ps3 , then each mapping f ∈
{R, ...} × {012} → A extends to a unique r∗ , s∗ , p∗ -preserving mapping from the atomic formulae of
L3 to A.
The second extension property has as a consequence that polyadic homomorphisms are like
interpretations in that if an interpretation for each atomic formula of L3 is determined then an
interpretation for each formula of L3 is uniquely determined.
Definition A set of formulae X ⊆ L3 is said to be sub-formula-closed if each atomic formula of
L3 is an element of X and if for each φ ∈ X it is the case that every sub-formula of φ is also in X.
Lemma 1.2.4 Let X ⊆ Y ⊆ L3 be such that X is sub-formula-closed. Let A be a Ps3 . Then each
∨, ¬, ⊥, ∃0 , ∃1 , ∃2 -preserving mapping f from X to A extends to a unique
∨, ¬, ⊥, ∃0 , ∃1 , ∃2 -preserving mapping f ∗ from Y to A.
The above two extension properties can now be combined with lemma 1.2.2 which provides semantics for r∗ , s∗ , p∗ .
Lemma 1.2.5 If {R, ...} are the relation-symbols of L3 and if A is a Ps3 , then each mapping f ∈
{R0 , ...} × {012} → A extends to a unique polyadic homomorphism f ∗ from L3 to A
The following can now be seen by comparing the uniquely determined homomorphisms with
interpretation and satisfaction as found in for example [Men87].
Proposition 1.2.6 A tarskian interpretation of L3 as relations over some set U , is a polyadic homomorphism from L3 to the full Ps3 over U . Also each homomorphism from L3 to some full Ps3 is a
tarskian interpretation of L3 -formulae.
1.2.6
Additivity
The following lemma is not central for understanding why the presented disprovers do what they
are supposed to do. It does however contribute to actually fitting the disprovers into a computers
memory as finite Ps3 ’s mostly are of considerable size. Additivity later turns out to provide a way
of representing finite Ps3 ’s in quite a compact form.
The carrier-set of a Ps3 together with the boolean operations ∨, ¬, ⊥ are by definition a
boolean algebra, called the boolean reduct of the Ps3 .
Definition An operation f on a boolean algebra is called additive if f (⊥) = ⊥ and f (x ∨ y) =
f (x) ∨ f (y).
The following property can to some extent be inspected geometrically, see fig. 1.1 and fig. 1.2.
Lemma 1.2.7 The boolean reduct of a Ps3 is a boolean algebra and each of r, p, s, c0 , c1 , c2 is additive.
20
1.2.7
The Lindenbaum algebra of a theory Γ
A full Ps3 over U is finite if U is finite, and exhaustive search for satisfying interpretations can
readily be done. If U is infinite however, the full Ps3 over U is uncountable and unsuitable for
exhaustive search as such. It turns out that the well known Lindenbaum algebra of a theory Γ
sometimes provides a way of computing and representing finite Ps3 ’s that can not be embedded
into a full Ps3 over any finite U . In these algebras, satisfying interpretations for infinity axioms
can be found. The question of whether a satisfying interpretation in a finite Ps3 exists, can even
be decided by exhaustive search.
We identify the Lindenbaum algebra of a theory Γ with a particular way of representing it. This
representation is chosen so as to be in the form of a finite set of finite objects (formulae), together
with a finite set of tables (finite sets of pairs or triples of formulae), when the Lindenbaum algebra
has a finite number of equivalence classes.
Throughout this paper; fixate a well-ordering on L3 based on a well-ordering of the symbols
of L3 . We assume that the ordering, , is such that if φ is a sub-formula of ψ then φ ψ. We also
assume that (L3 , ) is order-isomorphic to (ω, ≤). Well-orderedness ensures that the following
defines a mapping.
Definition µΓ ∈ L03 → L03 is defined by µΓ (φ) = ψ where ψ is the -minimal L03 formula such
that φ and ψ are Γ-equivalent.
As long as Γ is given we write [φ] in stead of µΓ (φ). We even refer to [φ] as the equivalence class
of φ, and say that ψ is in [φ] when φ and ψ are Γ-equivalent.
Definition L03 /Γ = ([L03 ], ∨0 , ¬0 , ⊥0 , r0 , p0 , s0 , ∃00 , ∃01 , ∃02 ), where for i ∈ 3
[L03 ] is the µΓ -image of L03
[φ] ∨0 [ψ] = [φ ∨ ψ]
¬0 [φ] = [¬φ]
r0 [φ] = [r∗ φ]
p0 [φ] = [p∗ φ]
s0 [φ] = [s∗ φ]
∃0i [φ] = [∃i φ]
Note that by this definition, µΓ is a polyadic homomorphism from L03 to L03 /Γ. Moreover it is
a satisfying interpretation for every logical consequence of Γ.
The following property of consistent and complete theories follows by the completeness theorem of Gödel. Consistency provides a model with universe U , interpretation of L03 -formulae as
relations over U , provides a homomorphism h from L03 to the full Ps3 over U . The restriction of
h to the carrier-set of L03 /Γ is a homomorphism from L03 /Γ to the full Ps3 over U . Completeness
ensures that this homomorphism is an embedding.
Proposition 1.2.8 If Γ is consistent and complete then L03 /Γ is a Ps3
21
1.2.8
Exhaustive search for satisfying interpretations in a Ps3
We finish this section with a summary of the results so far and how they may be applied in disproving. As noted; if L03 is a language with a finite number of relation-symbols and if Γ is a complete
and consistent theory that has quantifier-elimination within L03 , then L03 /Γ has a finite number of
equivalence classes. By the particular representation chosen, this L03 /Γ is a finite object. Given a
decision procedure for Γ, L03 /Γ can be computed and put in the form of tables for the operations
on the carrier-set of L03 /Γ. With such tables one can by exhaustive search decide whether there
exists a satisfying interpretation from L3 to L03 /Γ for any given L3 sentence φ. This is because one
only needs to enumerate mappings f in {R0 , ...Rm } × 012 → L03 /Γ, where {R0 , ...Rm } are the
relation-symbols of φ, to decide whether there exists an interpretation f ∗ from L3 to L03 /Γ that is
satisfying for φ.
If a satisfying interpretation f ∗ for φ in L03 /Γ is found then a satisfying interpretation for φ in
the tarskian sense is obtained by h ◦ f ∗ . Here h is an embedding from L03 /Γ to P(U 3 ) provided
by proposition 1.2.8. Since polyadic homomorphisms and interpretations are the same, h ◦ f ∗ is
an interpretation as polyadic homomorphisms are closed under composition. It is satisfying for φ
as h, like any other polyadic homomorphism, preserves >.
1.3
Algebras of directed many-sorted polyadic signature
We have seen how to do exhaustive search for satisfying interpretations in an L03 /Γ when Γ has
quantifier-elimination within L03 . Theories do not in general have quantifier-elimination like this
however. By an argument found at the end of section 1.5, the Lindenbaum algebra of the theory
of the usual strict order (<) on the natural numbers has an infinite number of equivalence classes.
As the equivalence classes with open formulae in them are a finitely generated boolean algebra, this
theory does not have quantifier elimination within L03 . In this section a particular kind of manysorted polyadic set algebra is introduced that can be computed by means of a decision procedure
for some L03 -theory Γ, regardless of whether Γ has quantifier-elimination within L03 . In algebras of
this kind, exhaustive search for satisfying interpretations can be done for any conjunction of prenex
L3 sentences.
1.3.1
Algebras of directed many-sorted polyadic signature
Here the signatures of the algebras introduced in the present paper are defined, together with suitable notions of homomorphism and sub-algebra.
Definition An algebra of directed many-sorted polyadic signature is an algebra
A = (B3 , B2 , B1 , B0 , r, p, s, c0 , c1 , c2 ), such that B3 , B2 , B1 , B0 have the signatures of boolean
algebras, each with their own join, negation, and bottom. The carrier-sets of B3 , B2 , B1 , B0 are
denoted by B3 , B2 , B1 , B0 respectively and are called the sorts of A. Moreover
r, p, s ∈ B3 → B3 .
c2 ∈ B3 → B2 ,
c1 ∈ B2 → B1 and
c0 ∈ B1 → B0 .
22
In the following definition boolean homomorphisms are used. Note that our definition of a
boolean homomorphisms does not require that the domain is a boolean algebra, only an algebra of
boolean signature.
Definition Let A = (B3 , B2 , B1 , B0 , r, p, s, c0 , c1 , c2 ) be as above and let C be either an algebra of
polyadic signature or an algebra of directed many-sorted polyadic signature. A directed many-sorted
polyadic homomorphism from A to C is a quadruple of boolean homomorphisms
h3 from B3 to C
h2 from B2 to C
h1 from B1 to C
h0 from B0 to C
such that each of r, s, p, c0 , c1 , c2 are preserved.
We define what it means for a many-sorted algebra to be a sub-algebra of a one-sorted algebra.
Definition Let C = (B, r, s, p, c0 , c1 , c2 ) be an algebra of polyadic signature. A C-sub-algebra of
directed many-sorted polyadic signature is an A = (B3 , B2 , B1 , B0 , r|, s|, p|, c0 |, c1 |, c2 |) with sorts
B3 , B2 , B1 , B0 where
B3 , B2 , B1 , B0 are B-sub-algebras of boolean signature,
r|, s|, p| are the restrictions of r, s, p to B3 ,
c0 |, c1 |, c2 | are the restrictions of c0 , c1 , c2 to B1 , B2 , B3 respectively,
r|, p|, s| ∈ B3 → B3 ,
c2 | ∈ B3 → B2 ,
c1 | ∈ B2 → B1 ,
c0 | ∈ B1 → B0 .
1.3.2
A conservative reduction class with sub-formulae as an algebra
Here the set of sub-formulae of every sentence of a conservative reduction class is expanded to make
an algebra of directed many-sorted polyadic signature. What sort a formula is, depends on which
quantifiers occur in it.
Definition The four algebras L33 , L32 , L31 , L30 of boolean signature, with respective carrier-sets
L33 , L32 , L31 , L30 are defined as follows ...
L33 = the L3 -sub-algebra of boolean signature generated by the atomic formulae of L3 , making
L33 the set of open formulae.
L32 = the L3 -sub-algebra of boolean signature generated by {∃2 (φ) : φ ∈ L33 }
L31 = the L3 -sub-algebra of boolean signature generated by {∃1 (φ) : φ ∈ L32 }
L30 = the L3 -sub-algebra of boolean signature generated by {∃0 (φ) : φ ∈ L31 }
23
Lemma 1.3.1 L33 ∪ L32 ∪ L31 ∪ L30 is a sub-formula-closed subset of L3 .
Proof: L33 ∪ L32 ∪ L31 ∪ L30 is built by means of logical connectives beginning with the atomic
formulae.
qed
The above four algebras of boolean signature are now interconnected with operations related to
variable substitution and existential quantifiers.
∗
∗
∗
Definition Lcrc
3 = (L33 , L32 , L31 , L30 , r |, p |, s |, ∃0 |, ∃1 |, ∃2 |) where
r∗ |, p∗ |, s∗ | are the restrictions of r∗ , p∗ , s∗ to L33 .
∃2 | is the restriction of ∃2 to L33 , making it an element of L33 → L32
∃1 | is the restriction of ∃1 to L32 , making it an element of L32 → L31
∃0 | is the restriction of ∃0 to L31 , making it an element of L31 → L30
Proposition 1.3.2 The sort L30 of Lcrc
3 is a conservative reduction class.
Proof: We prove that L30 contains the sentences of the form ∀0 ∃1 ∀2 φ where φ is open. This class
of sentences contains the reduction class of Kahr, More and Wang, which turned out to be conservative by results of Berger, Gurevich and Koriakov. For ∀0 ∃1 ∀2 φ is by definition ¬∃0 ¬∃1 ¬∃2 ¬φ.
As long as φ is open ¬φ ∈ L33 ,
then ¬∃2 ¬φ ∈ L32 ,
then ¬∃1 ¬∃2 ¬φ ∈ L31 ,
then ¬∃0 ¬∃1 ¬∃2 ¬φ ∈ L30 .
qed
The following is virtually the same as above but works for the conservative reduction class of Büchi.
Corollary 1.3.3 The sort L30 of Lcrc
3 has every conjunction of prenex sentences in L3 as an element.
Proof: It can be proven as above that every sentence Q0 Q1 Q2 φ where φ is open is in L30 . The
corollary then follows since L30 has boolean signature and thus has conjunctions.
qed
1.3.3
Directed many-sorted polyadic set algebras
We have seen how to turn a substantial fragment of first-order language into an algebra of directed
many-sorted polyadic signature. Here we define the algebras that are the core of the present paper
and the basis of each disprover deviced. These algebras turn out to be finite if they are finitely
generated.
Definition An algebra A is a dMsPs3 (directed many-sorted polyadic set algebra of dimension 3)
if it is a C-sub algebra of directed many-sorted polyadic signature where C is a Ps3 .
24
1.3.4
Interpretation
We show that one can interpret conjunctions of prenex L3 sentences in dMsPs3 ’s much as in
Ps3 ’s, and that each such interpretation can be used to represent a tarskian interpretation.
Lemma 1.3.4 Let h = (h3 , h2 , h1 , h0 ) be a directed many-sorted polyadic homomorphism from Lcrc
3
to an A in Ps3 . Then h3 ∪ h2 ∪ h1 ∪ h0 is a mapping from a subset of L3 to A that extends to a
unique polyadic homomorphism h∗ from L3 to A.
Proof: Recall that mappings formally are represented by sets of pairs. The boolean homomorphisms
h3 , h2 , h1 , h0 are defined on disjoint sets, distinguished by what quantifiers occur in the elements.
Thus h3 ∪ h2 ∪ h1 ∪ h0 is a mapping from L33 ∪ L32 ∪ L31 ∪ L30 . Which is easily seen to be a
sub-formula-closed subset of L3 . By lemma 1.2.4, letting X denote L33 ∪ L32 ∪ L31 ∪ L30 and
Y denote L3 , the mapping h3 ∪ h2 ∪ h1 ∪ h0 uniquely extends to a ∨, ¬, ⊥, ∃0 , ∃1 , ∃2 -preserving
mapping h∗ from L3 . It remains to show that r∗ , s∗ , p∗ are preserved by h∗ to show that h∗ is a
polyadic homomorphism. Let f denote the restriction of h∗ to the atomic formulae of the form
{R, ...} × {012}. By definition, h∗ extends f and must be the uniquely determined polyadic
homomorphism f ∗ of lemma 1.2.5.
qed
The following says that homomorphisms are like interpretations in the usual sense in that if an interpretation of the relation-symbols is given then an interpretation of each formula in our fragment
is determined.
Lemma 1.3.5 If {R, ...} are the relation-symbols of L3 and if A is a dMsPs3 and B3 of A has
carrier-set B3 , then each mapping f ∈ {R, ...} × {012} → B3 extends to a unique directed manysorted polyadic homomorphism (f3∗ , f2∗ , f1∗ , f0∗ ) from Lcrc
3 to A.
Proof: We display boolean homomorphisms f3∗ , f2∗ , f1∗ , f0∗ from L33 , L32 , L31 , L30 to the boolean
reducts B3 , B2 , B1 , B0 of A respectively, we show that the extra-boolean operations of Lcrc
3 are preserved and that this is a unique directed many-sorted polyadic homomorphism. By the definition
of dMsPs3 there is a C in Ps3 such that A is a C-sub-algebra of directed many-sorted signature.
Lemma 1.2.5 provides a unique polyadic homomorphism f ∗ from L3 to C that extends f . Define
f3∗ , f2∗ , f1∗ , f0∗ as the restrictions of f ∗ to L33 , L32 , L31 , L30 respectively. These preserve the required
operations as they are restrictions of f ∗ , which preserves them.
Let (h3 , h2 , h1 , h0 ) be an arbitrary directed many-sorted polyadic homomorphism of the required form. To show uniqueness we show that h3 ∪ h2 ∪ h1 ∪ h0 and f3∗ ∪ f2∗ ∪ f1∗ ∪ f0∗ are
the same. By lemma 1.2.3, the two are the same on atomic formulae. Letting X be the atomic
formulae and Y = L33 ∪ L32 ∪ L31 ∪ L30 , in lemma 1.2.4 we get the desired result. Note that f3∗
for instance trivially preserves ∃0 , ∃1 , ∃2 , as there is no pair of open formulae φ and ψ such that the
syntactic equality ∃i φ = ψ holds.
qed
As before a homomorphism is satisfying for a sentence if it is mapped to >.
Definition Let L30 be the sort of Lcrc
consisting of sentences. A directed many-sorted polyadic
3
crc
homomorphism (h3 , h2 , h1 , h0 ) from L3 to an A in dMsPs3 is satisfying for a sentence φ ∈ L30
if h0 (φ) = >.
25
The following allows us to use homomorphisms as representatives of tarskian interpretations.
to an A in dMsPs3 corresponds to a tarskian
Proposition 1.3.6 Each homomorphism from Lcrc
3
interpretation of L3 as relations over some set U . Moreover a homomorphism is satisfying for a sentence
φ if and only if the corresponding interpretation is satisfying for φ.
Proof: The correspondence is set up in two steps. First we compose the given homomorphism
h = (h3 , h2 , h1 , h0 ) from Lcrc
to A with an embedding f from A to a full Ps3 obtaining
3
f ◦ h = (f ◦ h3 , f ◦ h2 , f ◦ h1 , f ◦ h0 ). The definitions of dMsPs3 and Ps3 provide such an f .
Secondly we use lemma 1.3.4 and extend f ◦ h3 ∪ f ◦ h2 ∪ f ◦ h1 ∪ f ◦ h0 to a homomorphism
(f ◦ h)∗ from L3 to the full Ps3 . By proposition 1.2.6 this is an interpretation in the usual sense.
If h0 (φ) = > then (f ◦h)∗ is a satisfying interpretation since f ◦h0 is a boolean homomorphism
which preserves > and since (f ◦ h)∗ extends f ◦ h3 ∪ f ◦ h2 ∪ f ◦ h1 ∪ f ◦ h0 . For the other
direction: if φ ∈ L30 and if (f ◦ h)∗ is satisfying for φ then f ◦ h is satisfying for φ. Finally h is
satisfying for φ since f is an embedding.
qed
By the above we may refer to homomorphisms from Lcrc
3 to dMsPs3 ’s as interpretations. A
computer can decide whether there exist satisfying interpretation for given prenex sentences in a
finite dMsPs3 by means of exhaustive search. The following proposition, provides a naturalness
property for disprovers based on such exhaustive search. The property does in particular say that if
one has two logically equivalent conjunctions of prenex sentences whose status regarding satisfiability one wishes to decide, running a disprover once, for any one of the two sentences suffices. This
property is not shared with procedures that compare the input with sentences from a given finite
set of sentences, as to each sentence there is an infinite number of logically equivalent sentences.
Proposition 1.3.7 Let A be a (finite) dMsPs3 , let φ be a sentence of Lcrc
3 and h an interpretation
crc
in A such that h is satisfying for φ. If ψ is a sentence of L3 such that φ and ψ are logically equivalent
then h is a satisfying interpretation for ψ
Proof: To say that φ and ψ are logically equivalent is to say that h(φ) = h(ψ). To say that h is
satisfying for φ is to say that h(φ) = h(>). These two equations yield h(ψ) = h(>), which is to
say that h is satisfying for ψ.
qed
1.3.5
The directed many-sorted polyadic closure
Here we show that any satisfiable finite conjunction of prenex sentences is satisfied in a finite
dMsPs3 , even if such a conjunction is an infinity axiom.
Let O30 denote the open formulae of L03 , and O30 /Γ the appropriate sub-boolean algebra of
L03 /Γ. Recall that O30 is closed under the operations r∗ , p∗ , s∗ .
Definition This defines the directed many-sorted polyadic closure of the atomic formulae of L03
in L03 /Γ. The closure defines boolean algebras B3 , B2 , B1 , B0 , with carrier-sets B3 , B2 , B1 , B0 and
and operations r, p, s, c0 , c1 , c2 , which constitute a dMsPs3 .
B3 = the boolean algebra O30 /Γ
B2 = the sub-boolean-algebra of L03 /Γ generated by the ∃02 -image of B3
26
B1 = the sub-boolean-algebra of L03 /Γ generated by the ∃01 -image of B2
B0 = the sub-boolean-algebra of L03 /Γ generated by the ∃00 -image of B1
r = {(φ, ψ) ∈ B3 × B3 : ∀0 ∀1 ∀2 (r∗ (φ) ↔ ψ) ∈ Γ}
p = {(φ, ψ) ∈ B3 × B3 : ∀0 ∀1 ∀2 (p∗ (φ) ↔ ψ) ∈ Γ}
s = {(φ, ψ) ∈ B3 × B3 : ∀0 ∀1 ∀2 (s∗ (φ) ↔ ψ) ∈ Γ}
c2 = {(φ, ψ) ∈ B3 × B2 : ∀0 ∀1 (∃2 (φ) ↔ ψ) ∈ Γ}
c1 = {(φ, ψ) ∈ B2 × B1 : ∀0 (∃1 (φ) ↔ ψ) ∈ Γ}
c0 = {(φ, ψ) ∈ B1 × B0 : ∃0 (φ) ↔ ψ ∈ Γ}
With the above definition of closure finitely generated dMsPs3 ’s are finite.
Proposition 1.3.8 Let Γ be a consistent and complete L03 theory. As long as the number of atomic
formulae of L03 is finite, their directed many-sorted polyadic closure in L03 /Γ is finite.
Proof: The initial boolean algebra B3 is O30 /Γ, the boolean algebra generated by the equivalence
classes that have atomic formulae in them. It is finite since it is a finitely generated boolean algebra.
The subsequent boolean algebras are generated by images of finite ones.
qed
Note that in the above proposition, Γ needs only be complete enough to contain formulae needed
for the closure. Since the closure is finite there is to each satisfiable L03 sentence a finite Γ sufficient
for the closure to be defined. There is no general way of computing consistent Γ sufficient for the
closure to be defined but, as it turns out, sufficiency can be established computationally. In the
following we appeal to Lindenbaums lemma which says that any consistent first-order theory has a
completion.
Corollary 1.3.9 A finite conjunction of prenex L3 sentences, is satisfiable iff it is satisfiable in a finite
dMsPs3 .
Proof: Let L03 denote the language whose relation-symbols are those of the given conjunction.
View such a conjunction as an L03 theory by itself, and let Γ denote one of its consistent completions. The sentence is then satisfied in L03 /Γ by the homomorphism that maps each formula to
it’s equivalence class. The sentence is also satisfied in the dMsPs3 that is the closure of its atoms
in L03 /Γ. For the other direction; a satisfying interpretation in a finite dMsPs3 does by lemma
1.3.4 extend to a satisfying interpretation in the Ps3 of which the finite dMsPs3 is a directed
many-sorted sub-algebra.
qed
The following sets things straight with fundamental results of Church and Turing. It also excludes
the possibility of defining the class of dMsPs3 by means of a finite number of axioms in a language
where one can check whether an axiom holds in a finite algebra recursively.
In the following the letter R stands for representable, a notion we have not defined in the
present paper but see [Ném91] or [AGM+ 98]. Letting R denote the finite dMsPs3 ’s and |= the
notion of satisfaction between finite algebras of suitable signature and Lcrc
3 -sentences used in the
present paper, the corrolary states that the finite dMsPs3 ’s are not recursively enumerable.
27
Corollary 1.3.10 Let |= be a recursively enumerable binary relation between conjunctions of prenex L3
sentences and algebras of directed many-sorted polyadic signature. Let R be a class of algebras of directed
many-sorted polyadic signature, containing the finite dMsPs3 ’s. Let R and |= have the following two
properties for each finite conjunction of prenex L3 sentences φ:
if A is a finite dMsPs3 and there is a satisfying homomorphism for φ in A then A |= φ.
if A is in R and A |= φ then φ is satisfiable in the tarskian sense.
Then R is not recursively enumerable.
Proof: For the purpose of arriving at a contradiction assume that there is a way of recursively
enumerating the class R One could then write a procedure that recognised only satisfiable and all
satisfiable first-order sentences working as follows. This procedure would first transform an input
to an equi-satisfiable conjunction of prenex L3 sentences φ by the reduction of Büchi. Then the
procedure would enumerate algebras A in R, and whilst doing so seek to find if A |= φ. If A |= φ
were found hold for some A the procedure would terminate.
Since every satisfiable finite conjunction of prenex sentences has a satisfying interpretation in
a finite A in dMsPs3 , (corollary 1.3.9), and since this implies A |= φ this procedure would
terminate for every satisfiable sentence. It would only terminate for satisfiable sentences as A |= φ
implies that φ is satisfiable in the tarskian sense.
Combining this hypothetical procedure with a known to exist complete procedure for recognising inconsistent sentences, results in a procedure for the Entscheidungsproblem. Assuming the
existence of such, was shown to lead to a contradiction by Church and Turing.
qed
1.3.6
The closure as an algorithm
Assume that Γ is a complete and consistent theory in a language L03 , with a finite number of
relation-symbols. Also assume that there is a decision procedure for Γ. The directed many-sorted
polyadic closure of the atomic formulae of L03 in L03 /Γ can then be seen as an algorithm. First
of all, the definition of the closure has the overall structure of a procedure. Moreover B3 for
instance, can be computed by starting with Γ-distinct (not Γ-equivalent) atomic formulae and
generating new formulae by the boolean operations. As new formulae φ are generated these are kept
or discarded, depending on whether they are Γ-equivalent to some already generated formula ψ.
That is; depending on what the decision procedure has to say about the sentence ∀0 ∀1 ∀2 (φ ↔ ψ).
The kept formulae serve as indices for distinct elements of O30 /Γ, and the process terminates because
the number of equivalence classes of O30 /Γ is finite. Thus the procedure so far is an algorithm. One
can also show that all of O30 /Γ is generated like this by induction on the well-ordering used in the
definition of L03 /Γ. The rest of the closure is seen to be algorithmic in a similar fashion.
The outlined, straight forward approach, which is presenting the sub-boolean algebras of L03 /Γ
with one formula for each equivalence class and corresponding tables for the operations (finite
sets of triples or pairs of formulae), is less than optimal with respect to size of the tables. By the
additivity of Ps3 ’s (lemma 1.2.7), all one needs to store, is information about how r, p, s, c0 , c1
and c2 behave on the atoms of the sub-boolean algebras. Let, for example, [φ] and [ψ] be atoms
of O30 /Γ. Note that [φ] and [ψ] need not contain atomic formulae. Then r0 ([φ ∨ ψ]) for instance
is determined by r0 ([φ]) ∨0 r0 ([ψ]). By lemma 1.1.2 one may represent O30 /Γ by an algebra with
28
carrier-set n → {⊥, >}. This can be done in such a way that the atoms are represented by the
mappings that equal ⊥ in all but one component. The representative of r0 ([φ]) ∨0 r0 ([ψ]) is then
determined by the bit-wise join of the representatives of r0 ([φ]) and r0 ([ψ]). This allows for a
logarithmic reduction of the size of the tables for the corresponding algebra. For instance, O30 /Γ of
the theory of a dense order without endpoints turns out to have 213 elements and 13 atoms.
It can be quite enlightening actually defining the mentioned 13 atoms by means of a knife,
in some physical medium such as a reasonably cubic and homogenous piece of fruit. The cube
represents R3 . One should make 3 slices, one for each of the planes defined by x = y, x = z
and y = z, see fig. 1.3. One ought then end up with 6 pieces that are 3 dimensional, and thus
visible. Wedged between these pieces there are another 6 triangular pieces, and finally there is a
13th, 1-dimensional piece where the 3 defining planes meet. These 13 pieces correspond to subsets
of R3 which in turn correspond to the atoms of O30 /Γ by the embedding obtained by restricting
the usual interpretation of O30 , as ternary relations over the dense order (R, <), to the image of
µΓ (definition 1.2.7). The relations over (R, <) definable by open formulae are now unions of the
subsets that correspond to the atoms of O30 /Γ. By appeal to fig. 1.3 we now claim that the theory of
the dense order without endpoints has quantifier-elimination within L03 . See also [Men87](p.90).
Figure 1.3: Three planes of a cube defined by ¬(x < y ∨ y < x), ¬(x < z ∨ z < x) and
¬(y < z ∨ z < y).
1.3.7
Exhaustive search for satisfying interpretations in a dMsPs3
Let L03 be a language with a finite number of relation-symbols and Γ a consistent L03 theory that is
complete or sufficiently complete for the directed many-sorted polyadic closure to be defined. If Γ
is given then the closure can be computed and put in the form of tables for operations over a finite
set of formulae.
With such tables one can by exhaustive search decide whether there exists a satisfying interpretation for any given conjunction of prenex sentences of L3 in the closure of the atomic formulae
of L03 in L03 /Γ. If a satisfying interpretation is found, that interpretation represents a satisfying
interpretation in the usual sense by proposition 1.3.6.
To compare search for interpretations in directed many-sorted polyadic closures with finite
model search, we use the closure of x < y in the Lindenbaum algebra of the theory of a dense order
without endpoints. B3 of that closure turned out to have 13 atoms. By proposition 1.2.6 finite
model search is search for satisfying interpretation in full Ps3 ’s over finite sets U . In these algebras
the atoms are the one-element subsets of U 3 . Therefore, the full Ps3 over 2 has 8 atoms and
29
the full Ps3 over 3 has 27 atoms. So in terms of size of search-space, doing exhaustive search for
satisfying interpretations in an algebra whose B3 has 13 atoms, is a lesser task than that of searching
for satisfying interpretations over a 3 element set. Exhaustive search in such a space is well within
reach for present day computers, for sentences of some length.
1.4
A construction of Ps3’s and dMsPs3’s
In the previous section effort was put into keeping only finite parts of polyadic set algebras, as this
makes them suitable as components of disprovers. The resulting algebras are however somewhat
coarse. For example; to each such algebra there exists a finitely satisfiable sentence not satisfiable in
that algebra. Such a sentence may be constructed in a language built out of more relation-symbols
than there are elements in the algebra, by stating that the named relations are pairwise different.
This section describes a way of constructing a new and more refined finite Ps3 or dMsPs3
from a given one, by finite means. The construction is such that any finitely satisfiable sentence
is satisfiable in an iterate of the construction. This can be used as part of a disprover to ensure
termination in case of finite satisfiability. Here it is defined for Ps3 ’s only. The construction is
analogous for directed many-sorted ones.
Definition Let A = (B, r, p, s, c0 , c1 , c2 ) be a Ps3 . Equip the set of mappings in 23 → A with
polyadic structure as follows. Boolean operations are defined component-wise. For t ∈ 23 → A
define r, p, s, c0 , c1 , c2 ∈ (23 → A) → (23 → A) by:
r(t)(abc) = r(t(abc ◦ r)),
p(t)(abc) = p(t(abc ◦ p)),
s(t)(abc) = s(t(abc ◦ s)),
c0 (t)(abc) =
W
u∈2 c0 (t(ubc)),
c1 (t)(abc) =
W
u∈2 c1 (t(auc)),
c2 (t)(abc) =
W
u∈2 c2 (t(abu)).
By abuse of notation let 23 → A denote the algebra just defined. Two propositions about the
construction follow.
Proposition 1.4.1 If A is a Ps3 then 23 → A is a Ps3 .
Proof: Since A is a Ps3 there is a polyadic embedding h from A to a full Ps3 over some set U .
Using the fact that there is a correspondence between sets of elements of U 3 and their characteristic
functions (U 3 → 2), (23 → A) is embeddable into 23 → (U 3 → 2) by the mapping t 7→ h ◦ t.
Now there is a natural isomorphism from 23 → (U 3 → 2) to (2 × U )3 → 2 which is the set
of characteristic functions of the elements of a full Ps3 . Regard 3-dimensional figures with one
quadrant, or “octant” rather, for each element of 23 for details of this, fig. 1.4.
qed
Proposition 1.4.2 Any finitely satisfiable sentence is satisfied in an iterate of the above construction
beginning with an arbitrary Ps3 .
30
Figure 1.4: Cylindrification along x-axis of an element of 23 → P([0, 1i3 ). Here [0, 1i is a halfopen interval of the reals and the element represents a relation in [0, 2i3 visualised by suitably
stacking 8 copies of [0, 1i3 .
Proof: Given an A in Ps3 , iterates are of the form 23 → (. . . → (23 → A)) which are naturally
isomorphic to those of the form (2 × . . . × 2)3 → A. Since the top and bottom of A can serve
as image of a characteristic function, the latter algebra contains (2 × . . . × 2)3 → {⊥, >}, which
is isomorphic to the full polyadic set algebra over some set with 2n elements for a suitable n ∈ ω.
The proposition follows since any finitely satisfiable equality-free sentence is satisfiable over a set
with 2n elements, for some n ∈ ω.
qed
1.4.1
A disprover deviced according to the method
To device a disprover according to the presented method, one needs a decision procedure for a
theory Γ, or alternatively a, known to be satisfiable, finite set of sentences sufficiently complete
for a directed many-sorted polyadic closure to be defined. For implementation the present author
has used a publicly available decision procedure for Presburger arithmetic by Karlund, Møller and
Schwartzbach [KMS02]. For instance a computer can, over night, produce the closure of the atomic
formulae of L03 in L03 /Γ, where L03 has exactly one relation-symbol, <, and Γ is the theory of the
usual strict order on the natural numbers.
The closure A is put in the form of finite tables which form an integral part of the resulting
disprover. When the input is a prenex sentence φ ∈ L3 , whose relation-symbols may be disjoint
from, and many more than, those of L03 , the disprover proceeds as follows:
if there exists a satisfying interpretation for φ in A terminate!
if there exists a satisfying interpretation for φ in 23 → A terminate!
if there exists a satisfying interpretation for φ in 23 → (23 → A) terminate!
..
.
A brief analysis, in terms of size of search-space follows. By Büchis sharpening of Trakhtenbrots
theorem and by proposition 2.5.3 the sentences recognised by this procedure form a non-recursive
31
set. In the example, the initial sort B3 of A, turned out to have 13 atoms, which can be verified
by a geometric argument similar to the one depicted in fig. 1.3, when viewing ω as a subset of
R. B3 of 23 → A has 8 times that number. Multiplication with 8 proceeds, so after i iterations
search is done in an algebra with 13 · 8i atoms. By lemma 1.1.2 and the possibility of storing the
extra-boolean operations of A in the form of tables (finite sets of pairs), the question of whether
there exists a satisfying interpretation for a sentence with m relation-symbols, in an algebra with
13 · 8i atoms, can be phrased as a constraint satisfaction problem with m · 13 · 8i variables ranging
over {⊥, >}.
If the disprover is based on the full Ps3 over a finite set then it is a finite model search procedure, which at each step doubles the size of the set over which satisfying interpretations are sought.
Various disprovers have been implemented, based on the presented method. They are publicly
available with source code included. 1 .
1.5
Related constructions and algebras
The construction of definition 1.4 is rather similar to what was called the cardinal multiple of theories by Feferman and Vaught [FV59] (Section 4.7), when bearing in mind that polyadic set algebras
correspond to complete and consistent theories. The constructions of [FV59] were carried over
to the setting of polyadic algebras and generalised by Daigneault [Dai63]. The main construction
is called the tensor product of polyadic algebras. Daigneaults paper is mainly about infinite, not
necessarily atomic nor complete, polyadic algebras so the definition goes via Stone-spaces. Finite
polyadic algebras are all atomic and complete, so it is worth noting that this tensor product can also
be constructed by finite means. It turns out to be the polyadic version of the Kronecker product,
⊗, of finite-dimensional vector spaces. This can be seen by regarding elements of a finite (directed
many-sorted) polyadic set algebra as vectors of zeros and ones and using boolean operations instead
of ring-operations. In this view 23 → A is isomorphic to P(23 ) ⊗ A. Under one such isomorphism the function values of an element of 23 → A appear as rows of the corresponding matrix in
P(23 ) ⊗ A. This product works for arbitrary pairs of set algebras and provides a way of combining
any two disprovers devised as presented.
An essential property of the construction, is that the polyadic set algebras are closed under
it, and that one gets more than what one had to begin with. The direct product of polyadic
algebras is not such a construction, as a sentence that is satisfiable in a product, by projection,
already is satisfiable in one of the factors. There are no projections of this kind for Ps3 ’s. They are
simple. By an argument involving lemma 1.3.4, dMsPs3 ’s are also simple. The direct product does
however give rise to the extensively studied representable polyadic algebras, which together with
representable relation algebras and cylindric algebras, provide a potential source for dMsPs3 ’s,
besides decision procedures, and sufficiently complete finite sets of sentences.
Regarding dMsPs3 ’s, the idea of using partial and many-sorted variants of algebras for interpretation of languages with quantifiers is far from new. An early one is by Bernays [Ber59], more are
mentioned by Németi [Ném91]. Connections to category theory approaches are also mentioned
there. The approaches the present author is aware of, are each different from dMsPs3 ’s in at least
one of the following two respects. Firstly, classes of partial or many-sorted algebras are not gener1
http://flipper.berlios.de
32
ally such that each arity respecting mapping of non-logical symbols into an A extends naturally to
an interpretation in A of each sentence of an entire reduction class as in lemma 1.3.5. Secondly,
known many-sorted variants, which obviously can be chopped off at some finite dimension, typically allow the definition of cylindrification within each sort, which in the present vocabulary is to
say that they are not directed. Undirectedness can ruin the property that finitely generated algebras
are finite. As an example of how little it takes for undirected closures to be infinite, consider P(ω 2 ),
the full Ps2 over the natural numbers. A signature for this algebra is (B, s, p, cx , cy ). Both cx and
s may even be left out for the following. Let y < x denote the set of pairs of numbers whose
second component is less than the first. Moreover let for each natural a, a < x denote the set
of pairs whose first component is greater than a, and a < y denote the set of pairs whose second
component is greater than a. The claim is that for each natural a, a < x lies in the closure of
y < x. Now 0 < x lies in the closure as cy (y < x) = 0 < x. Proceed by assuming that a < x lies
in the closure. Consider cy (p(a < x) ∩ y < x), here p(a < x) = a < y so the term is true if there
is a y such that x is greater than y and y is greater than a, which is when a + 1 < x.
1.6
Concluding remarks
The author believes that the adaption of the polyadic algebras of Tarski and Halmos to the conservative reduction class of Büchi so as to form the class dMsPs3 is a contribution to algebraic logic.
Some potential in automated reasoning has also been made likely.
Distinguishing features of the dMsPs3 ’s are the naturalness property of lemma 1.3.5, their
having a finite dimension higher than two and their being directed. Directedness enables the further
downward Löwenheim-Skolem property (corollary 1.3.9), for each sentence of the conservative
reduction class. This property makes it possible to do exhaustive search for, and to sometimes find,
satisfying interpretations for infinity axioms in finite dMsPs3 ’s. Various finite dMsPs3 ’s can be
computed by means of given decision procedures for first-order theories. The above makes finite
dMsPs3 ’s quite suitable as components of disprovers.
Each finite dMsPs3 can together with the construction of definition 1.4 be used to devise
a disprover that behaves naturally and that works by a generalisation of finite model search for a
substantial fragment of first-order language. The fragment is substantial since it is a conservative
reduction class. The disprover behaves naturally by proposition 1.3.7. The disprover works by
a generalisation of finite model search since, by proposition 2.5.3, it does search through a set of
interpretations containing a representative for every interpretation over any finite set. As long as the
given dMsPs3 allows a satisfying interpretation for an infinity axiom, the generalisation is strict.
33
Chapter 2
Automata for mechanising consistency
proofs
35
2.1
Introduction
This is the first of two papers on a kind of automaton that is suitable for consistency proofs and for
the computation of atom-structures of finite representable polyadic algebras, including some which
have purely infinite spectrum. Finite representable algebras with infinite spectrum are of interest in
regards to Hilberts Entscheidungsproblem as the algebras can be used to construct semi-decision
procedures that recognise not only finitely satisfiable sentences as consistent but also some infinity
axioms, see A. Rognes [Rog09]. Infinity axioms are consistent first-order sentences that have infinite
models only. Devising reasonably natural semi-decision procedures that terminate on input of even
a single infinity axiom is a challenge. Note, for instance, that an eventually periodic infinite branch
of semantic tree implies finite satisfiability. Similar effects occur with finitely presented Herbrand
models.
Polyadic algebras were introduced by P. Halmos who was inspired by A.Tarskis cylindric algebras. Finite and representable polyadic algebras are mathematical objects that can be used much like
structures and models when recognising given first-order sentences as consistent. As with structures
we may interpret relational sentences in these algebras and only consistent sentences have satisfying
interpretations in representable algebras. Curiously some of the finite and representable algebras
have satisfying interpretations for infinity axioms. In computations involving infinity axioms explicitly presented models are generally unsuitable as arguments to computable functions by virtue
of being infinite. Finite representable algebras with satisfying interpretations for infinity axioms
however, work well by virtue of being finite. The atom-structures of finite algebras are even more
suitable as they are considerably smaller that the algebra it self. An atom-structure plays the same
role for a finite polyadic algebra as does a basis for a vector space or a topology.
The present paper, i.e. part one of two papers, can be read independently of the subsequent
part and makes no use of algebraic logic. It introduces the automata mentioned in the title and
shows how these can be used to construct semi-decision procedures that tackle not only finitely
satisfiable sentences but also some infinity axioms. The paper considers several classes of automata
but culminates in a class that is basic elementary, i.e. the class is defined by a finite set of first-order
sentences. The significance of a class being basic elementary is that this provides a sense in which
the class is natural, moreover it is useful when computing, i.e., recursively enumerating, those of
the automata that are finite.
A recurring theme in the two papers is the need to express reachability in the automata at
hand, whilst keeping things basic elementary. It is known from model-theory that notions which
are naturally defined by means of transitive closure, are not definable in first-order language over
structures in general. By the Ehrenfeucht-Fraïssé method one can prove that this remains true over
finite structures. Reachability in finite automata is an example of such a notion.
As we shall see in the present paper, a reason for the impossibility of defining reachability is the
fact that one considers classes of automata all of which have the same fixed alphabet. By allowing
some flexibility in the alphabet however, we can show the existence of classes of automata with the
following properties.
1. The class of automata is definable with a finite set of first-order axioms.
2. There is in the language that defines the class of automata a first-order formula that defines
the reachability relation in each automaton of the class.
36
3. The finite automata of the class are as versatile as finite automata with fixed alphabets, at least
in regards to deciding the theory of automatic structures such as Presburger arithmetic.
The first two properties are needed to obtain a basic elementary class of automata in which reachability is first-order. The third property is sufficient for a semi-decision procedure, based on an
automaton, to recognise some infinity axiom. The latter is because Presburger arithmetic contains
infinity axioms. To fulfil the second property we will simply postulate that the relations of which we
need to take the transitive closure are transitive. This makes expressing reachability trivial. What
remains then is to make sure that we have not lost computational power, i.e., we need to show
that those of the automata that fulfil the transitivity postulates suffice for deciding the theories of
automatic structures.
We introduce and show some results on two classes with the three properties. Firstly the class
of transitive automata, made to be a simple and transparent example of such a class. Automata of
the second class, called PTPS automata, consist of transitive automata where one is able to express
various refinements of the reachability relation in first-order language. We are in particular able
to give first-order definitions of constructions on automata, that correspond to logical connectives,
quantifiers and substitutions, internally in a finite PTPS automaton.
2.1.1
Outline of paper
In section 1 we repeat well known definitions and results to make the present paper fairly self
contained. In section 2 we introduce n-fold vector-spaces. These are vector-spaces with two extra
operators. Whether this particular subclass of groups with operators has been described before is
unknown to the present author. The n-fold vector-spaces serve as alphabets for all the automata
introduced in the present paper.
In section 3, n-tape p-automata which have n-fold vector-spaces as alphabets are introduced.
These are believed to be new. The rest of the automata described in the present paper are variations
over these. We define what it means for an automaton, with a possibly abstract n-fold vector-space
as alphabet, to recognise an n-ary relation on natural numbers.
In section 4 a subclass of n-tape p-automata is introduced. They are called transitive automata.
We provide a finite set of first-order axioms for transitive automata, and show that in finite transitive
automata the reachability relation is first-order definable, quite trivially. Less trivial is the fact that
finite transitive n-tape p-automata are as versatile as finite automata in general when it comes to
their use as parts of decision procedures for theories of automatic structures.
In section 5 the definitions and proof techniques of section 4 are elaborated upon and we
introduce PTPS automata. They are also definable by a finite set of first-order formula. In PTPS
automata we are able to express, in first-order language, various forms of reachability between states.
For example we are able to express that one state is reachable from another using a tape whose second
track represents the number 0.
In section 6 we look at how PTPS automata, used to decide a given sentence about an automatic
structure, are in relationship to one another. We show that the relationship between an automaton
for a formula and the automata for its sub-formulae is first-order definable over finite structures.
Finally we see how this can be used for proving consistency computationally.
37
2.1.2
Notation
We assume familiarity with automata theory. N denotes the natural numbers. A one-sorted structure, or just a structure, is a tuple (X, R0 , . . . , Rk−1 , f0 , . . . , fl−1 ) where X is a set and where
R0 , . . . , Rk−1 are relations and f0 , . . . , fl−1 are functions on X of fixed finite arities. A two-sorted
structure is a tuple
(X, Y, R0 , . . . , Rk−1 , f0 , . . . , fl−1 ) where X and Y are sets and where the relations and functions
are on X or Y or both. Two structures are said to be of the same similarity-type if they are both
one-sorted or both two-sorted and if they have the same number of relations and functions and
these correspond in arities.
Moreover a matrix is a rectangular array of elements from a field. If X = (X, . . .) and X 0 =
(X 0 , . . .) are structures then their product, denoted X × X 0 , is the structure with carrier-set X × X 0
and where the functions are defined component-wise, and where a tuple is in a relation if both
projections are in the corresponding relations. The k-th power of X , denoted X k , is the product of
X with it self k times.
The definition of a first-order language suitable for a class of structures is assumed to be known.
If φ is a sentence (formula without free variables), and A a suitable structure we write A |= φ for φ
is true in A, (alternatively A satisfies φ).
For mappings f we use f k to denote f composed with it self k times. We use p to denote prime
powers. The definition of groups, rings and fields are assumed to be known. For each prime power
p there is a unique field, Fp , of that order, see a text on algebra such as Herstein [Her75].
2.1.3 p-automata
We use the word effective to mean computable and not necessarily fast. J.R. Büchi [B6̈0] observed
that effective constructions on finite synchronous automata could be used to decide Presburger
arithmetic, the set of sentences true of the structure (N, +). Central in J.R. Büchis observation
is to view strings of tuples of the form {0, . . . , p − 1}n as elements of Nn written in base p, and
considering automata that have alphabets of the form {0, . . . , p − 1}n .
Example This is an example of a 3-tape with entries from {0, 1} which can be viewed as the
binary representation of the tuple (7, 2, 9).







1
1
1
0
0
0







 0  1  0  0  0  0 







1
0
0
1
0
0
Here the binary expansion of 7 is at the top row and the most significant end is to the right.
Observe that one may add any number of zero-vectors at the most significant end of this tape,
without changing the tuple of numbers it represents. See one of the survey articles of V.Bruyère
et al. [BHMV94] or W.Thomas [Tho96] for more on this encoding of tuples of numbers for use
with automata.
In accordance with the article of V.Bruyère et al. [BHMV94], automata that work in base p are
referred to as p-automata. The relations on natural numbers recognised by p-automata are called
p-recognisable. We shall in the next subsection recall the Ehrenfeucht-Fraïssé method and discuss
first-order definable properties of finite p-automata, for a fixed alphabet {0, . . . , p − 1}n . When
being specific about n ∈ N here, we will call these n-tape p-automata. By the Büchi-Bruyère
38
theorem [BHMV94], the p-recognisable relations are exactly those relations we can define in firstorder language over the structure (N, +, |p ). Here x|p y is true when x is a power of p dividing y.
We define automatic structure as a structure which can be interpreted in the structure (N, +, |p ),
see A. Blumensath E. Grädel [BG04] theorem 4.5. An automatic model for a set of sentences is an
automatic structure that is also a model for the sentences.
Büchis procedure to decide a sentence in the theory of (N, +, |p ), and thus Presburger arithmetic and the theory of the other automatic structures, can be outlined as follows. It is an easy
exercise to construct two automata; one 3-tape automaton for the summation relation + and one
2-tape automaton for the |p relation. Using these two automata one can, by manipulating rows in
the alphabet, effectively construct automata that recognise the interpretation of each atomic formula of the theory of (N, +, |p ). Similarity there are constructions on automata for each of the
logical connectives so that we get an association between formulae and automata. The association is
such that an automaton that corresponds to a formula recognises the interpretation of the formula
in (N, +, |p ). To check whether a sentence is true in (N, +, |p ) one checks whether every reachable
state in an associated automaton is a terminal state.
It is well known that regular languages are closed under reversal of direction, so when working
with automata for automatic structures one has to decide which end one reads tapes from. We chose
to read tapes from the least significant end. Likewise one can chose to work with deterministic or
non-deterministic automata. We chose to use deterministic automata.
There is a number of ways for defining the class of automata, over a fixed alphabet, so as to
make it a class of structures suitable for first-order language. We shall consider two ways.
Firstly we provide a definition that is perhaps the most standard for deterministic automata.
These automata are two-sorted. The equivalence that forms the sole proper axiom of the definition
relates to the convention that words are fed to our automata with the least significant end first, and
that adding zeros at the most significant end does not change the represented numbers.
Definition Let n, p ∈ N. A two-sorted n-tape p-automaton with classical alphabet is a structure
({0, . . . , p − 1}n , K, δ, ι, T ) with carrier-sets {0, . . . , p − 1}n and K where
K is a set whose elements are called states,
{0, . . . , p − 1}n is called the alphabet,
δ : K × {0, . . . , p − 1}n → K is called the transition function,
ι ∈ K is called the initial state,
T ⊆ K is called the set of terminal states.
Moreover T is invariant under adding zero-vectors at the most significant end, i.e.

q ∈ T iff



δ(q, 


0
0
..
.
0




)


∈T .
Secondly we provide a definition that is intended to be the same as above in every relevant respect
except for the number of sorts. These automata are one-sorted and which means we can apply
methods of finite model-theory to them directly.
39
Definition Let n, p ∈ N and let I = {0, . . . , p − 1}n . A one-sorted n-tape p-automaton with
classical alphabet is a structure
(K, {δi }i∈I , ι, T ) with carrier-set K where
K is a set whose elements are called states,
For each symbol i of the alphabet {0, . . . , p − 1}n the mapping
δi : K → K is called the i-transition.
ι ∈ K is called the initial state,
T ⊆ K is called the set of terminal states.
Moreover the following axiom holds, with the zero-vector transposed for typographical reasons,
q ∈ T iff δ(0,0,...,0) (q) ∈ T .
We now define reachability, the central relation of the present paper.
Definition Let (K, {δi }i∈I , ι, T ) be an automaton. The reachability relation is the ⊆-minimal
relation, R ⊆ K × K, such that:
If there is an i ∈ I such that δi (q) = q 0 then (q, q 0 ) ∈ R.
If (q, q 0 ) ∈ R and i ∈ I then (q, δi (q 0 )) ∈ R.
2.1.4
The Ehrenfeucht-Fraïssé method
The Ehrenfeucht-Fraïssé method of finite model-theory is based on an application of one of two
closely related theorems, one due to Fraïssé and another to Ehrenfeucht. Which of these theorems
one uses is a mater of taste, in the setting of first-order logic.
We use Fraïssés theorem, and we briefly recall the Ehrenfeucht-Fraïssé method to show that
the reachability relation is not first-order definable over finite automata in general. This is for
clarification as the present paper is about a class of automata where the reachability relation is
indeed first-order definable.
We shall, for this section alone, use the terms quantifier rank and m-isomorphism. For precise
definitions of these see a text on finite model-theory such as Ebbinghaus and Flum [EF06]. The
quantifier rank of a formula is a natural number that serves as a measurement of the complexity of
the formula with respect to occurrences of quantifiers. The relation of m-isomorphism is a weak
form of isomorphism between pairs of structures. We write ∼
=m for m-isomorphism. The existence
of an m-isomorphism is a sufficient condition for a pair of structures to satisfy the same sentences
of quantifier rank m. If the language and structures in question are purely relational it is also a
necessary condition. See e.g. Ebbinghaus and Flum [EF06] definition 2.3.1 and corollary 2.3.4.
We now state Fraïssé’s theorem in the sufficiency direction. It is usually proven for purely
relational sentences, but in the sufficiency direction it is easily generalised to full first-order language.
Theorem 2.1.1 (Fraïssé) If φ is a first-order sentence, with or without constant- and function-symbols,
then there exists a k ∈ N, namely the quantifier rank of φ, such that for all structures A, B we have; if
both A |= φ and A∼
=k B then B |= φ.
40
Using this theorem contrapositively we can, by for each k ∈ N showing the existence of a pair
of structures A∼
=k B where A has a given property that B does not, conclude that this is not a
first-order property. This remains a valid way of reasoning when we restrict attention to a sub-class
of all structures, such as the class of finite structures.
Proposition 2.1.2 The reachability relation in one-sorted n-tape p-automata with classical alphabet
is not first-order definable.
Proof: We prove this in three steps. In the first two steps we show that connectedness is not
definable. This is done pretty much as with graphs, see a finite model-theory text such as [EF06]
for details.
Firstly we show that the property “having an even number of states” is not first-order definable.
Call an automaton trivial if each δi is the identity mapping and T = ∅. For each k ∈ N let Ak be
the trivial automaton with 2k states and let Bk be the trivial automaton with 2k + 1 states. For each
k we now have that Ak has and even number of states and that Ak ∼
=k Bk but Bk does not have an
even number of states. By Fraïssés (or Ehrenfeuchts) theorem we conclude that the automata with
an even number of states is not first-order definable (over finite structures).
Secondly one assumes that connectedness is first-order definable and shows that having an even
number of states then becomes first-order definable. This is in contradiction with the first step so
connectedness is not first-order definable.
Thirdly assume for the purpose of arriving at a contradiction that the reachability relation, R,
is first-order definable. We can then define connectedness as follows ∀q∀q 0 R(q, q 0 ). This is in
contradiction to the second step, therefore reachability is not first-order definable over the finite
automata.
qed
2.1.5
Mappings from and to a finite set
Lemma 2.1.4, soon to follow, is central in various profs in the present paper. The lemma is on
mappings from and to a finite set. The collection of such mappings is well known to form a
semigroup under composition. As one may expect then, the lemma follows fairly immediately
from a result known to semigroup-theorists. See J-E.Pin [Pin97] the section on idempotents. The
role of the lemma in the present paper justifies a proof.
Lemma 2.1.3 If X is a finite set with a point α ∈ X and a function f : X → X. Then there exist
numbers a > 0 and b > 0 s.t. for all l, m ∈ N we have f m+b (α) = f a·l+m+b (α).
Proof: Intuitively the successive application of f will bend in on it self at a point we refer to as
f b (α). By bending in we mean that there is an a > 0 such that f a+b (α) = f b (α). The number
a is here the length of a cycle we may repeat. We may therefore without loss of generality assume
that 0 < b. By using the cycle we get the property that for all l ∈ N we have f a·l+b (α) = f b (α).
We may also jump into the cycle any number m of successions and do a-cycles from there so, for
all l and m we have f a·l+b+m (α) = f b+m (α).
qed
Lemma 2.1.4 If X is a finite set with an α ∈ X and a function f : X → X, then there exists a
k ∈ N s.t. 0 < k and f k (α) = f 2·k (α).
41
Proof: Use a and b from the lemma above. Let l = b and m = a · b − b and k = a · b. Note that
from the conditions that 0 < a and 0 < b we get 0 < k. Moreover:
f k (α) = f a·b (α) by definition of k,
= f a·b−b+b (α) = f m+b (α) by basic arithmetic,
= f m+b (α) = f a·l+m+b (α) by the lemma,
= f a·b+(a·b−b)+b (α) by definition of m and n,
= f a·b+a·b (α) = f 2·k by basic arithmetic and the definition of k.
qed
2.1.6
Vector-spaces over finite fields
The definition of a vector-space over a finite field is a ready made, finite and first-order definition
of the notion of a set of tuples with entries from a finite set. When the vector-space has dimension
m ∈ N, it is necessarily of the same form as the set of m-tuples with entries from the underlying
finite field. These vector-spaces can accordingly be assembled to n-tapes with entries from the
underlying field, for a chosen n.
A one-sorted definition of vector space over the field of integers modulo two follows.
Definition A vector space V over the minimal field is a tuple (V, +, −, 0, 0, 1) where V is a set, +
is a binary operation, −, 0, 1 are unary operations and where 0 is a constant. Moreover V is
1. an abelian group, i.e.
V |= ∀xyz[(x + y) + z = x + (y + z)],
V |= ∀x[x + 0 = x],
V |= ∀x[x + −(x) = 0],
V |= ∀xy[x + y = y + x],
2. left distributive, i.e.
V |= ∀xy[0(x + y) = 0(x) + 0(y)],
V |= ∀xy[1(x + y) = 1(x) + 1(y)],
3. right distributive, i.e.
V |= ∀x[0(x) = 0(x) + 0(x)],
V |= ∀x[0(x) = 1(x) + 1(x)],
V |= ∀x[1(x) = 0(x) + 1(x)],
V |= ∀x[1(x) = 1(x) + 0(x)],
42
4. associative, i.e.
V |= ∀x[0(0(x)) = 0(x)],
V |= ∀x[0(1(x)) = 0(x)],
V |= ∀x[1(0(x)) = 0(x)],
V |= ∀x[1(1(x)) = 1(x)],
5. in possession of a unit, i.e.
V |= ∀x[1(x) = x],
V |= ∀x[0(x) = 0]. (*)
The axiom with the (∗) is dependent i.e. it follows from the rest of the definition, in particular the
axiom 0(x) = 0(x) + 0(x). Mappings between vector-spaces over the same field that preserve the
vector-space operations are called linear transformations. If there is a bijective linear transformation
between two vector-spaces they are said to be of the same form, or isomorphic. It is known that
there is a minimal field. It has two elements. Also all minimal fields are isomorphic, therefore we
say the minimal field and write F2 .
Lemma 2.1.5 Every finite vector space over F2 is of the form F2 × · · · × F2
Proof: It is known that finite dimensional vector-spaces in general are of this form. See an algebra
text such as Herstein [Her75].
qed
We write h(V) for the image of a vector-space V under a linear transformation h.
Lemma 2.1.6 If V and h are as as above then h(V) is a vector-space.
Proof: see [Her75].
2.2
qed
n-fold vector spaces
We introduce vector spaces with two additional operations. The operations and their accompanying
axioms allow us to view the elements of the vector-spaces as n × m matrices with entries from
the underlying field. These matrices will serve as symbols of alphabets for synchronous n-tape
automata.
43
2.2.1 3-fold vector spaces over the minimal field
We begin with an axiomatisation of vector-spaces which can be assembled to 3-tapes with entries
from {0, 1}. Later we do this for n-tapes with entries from larger sets.
Definition A 3-fold vector space over F2 is a tuple V = (V 0 , π, r) where
V 0 = (V, +, −, 0, 0, 1) is a vector space over F2
π : V → V called the projection,
r : V → V called the rotation.
Moreover
1. π is a linear transformation, i.e.
V |= [π(0) = 0], (*)
V |= ∀x[π(−(x)) = −(π(x))], (*)
V |= ∀xy[π(x + y) = π(x) + π(y)],
V |= ∀x[π(0(x)) = 0(π(x))],
V |= ∀x[π(1(x)) = 1(π(x)))],
2. r is a linear transformation, i.e.
V |= ∀x[r(0) = 0], (*)
V |= ∀x[r(−(x)) = −(r(x))], (*)
V |= ∀xy[r(x + y) = r(x) + r(y)],
V |= ∀x[r(0(x)) = 0(r(x))],
V |= ∀x[r(1(x)) = 1(r(x))],
3. r3 is the identity mapping, i.e.
V |= ∀x[r(r(r(x))) = x],
4. π is idempotent, i.e.
V |= ∀x[π(π(x)) = π(x)],
5. V is the sum of 3 isomorphic copies of π(V), i.e.
V |= ∀x[π(r(r(r(x)))) + r(π(r(r(x)))) + r(r(π(r(x)))) = x].
44
Again the axioms with a (*) behind them are dependent, which means they follow from those that
don’t have a (*), see [Her75] lemma 2.7.2.
Example Consider the set of 3 × 5 matrices whose entries are from F2 . Following convention
this means an array of 3 rows and 5 columns. Letting 0 and 1 denote the elements of F2 , the entries
of the matrix are 0’s and 1’s. The top row and leftmost column have lowest index. It is clearly a
vector-space when the vector-space operations are defined entry-wise. It is a 3-fold vector-space
when r is the operation that moves the top row to the bottom of a matrix, and when π preserves
the top row and replaces every entry off the top row with a 0.
2.2.2 n-fold vector spaces over a given finite field
Using the fact that for each prime power p there exists a unique field, Fp , of that order, the axiomatisation of 3-fold vector spaces over the minimal field ,F2 , can clearly be generalised to 3-fold
vector spaces over each Fp . Likewise we may generalise from 3-fold to n-fold as follows.
Definition Let n ∈ N. An n-fold vector-space over Fp is a tuple V = (V 0 , π, r) where
V 0 = (V, +, −, 0, 0, 1, . . .) is a vector space over Fp ,
π : V → V is the projection,
r : V → V is the rotation.
Moreover
1. π is a linear transformation,
2. r is a linear transformation,
3. rn is the identity mapping, i.e
V |= ∀x[rn (x) = x]
4. π is idempotent, i.e.
V |= ∀x[π(π(x)) = π(x)]
5. V is the sum of n isomorphic copies of π(V), i.e.
V |= ∀x[π(rn (x)) + r(π(rn−1 (x))) + · · · + rn−1 (π(r(x))) = x]
We recall a result on vector-spaces, n-fold or not.
Lemma 2.2.1 Every finite vector-space over Fp is of the form Fp × · · · × Fp .
Proof: See an algebra text such as Herstein [Her75].
qed
Definition A linear transformation between two n-fold vector-spaces is said to be an n-fold linear
transformation if in addition to the vector-space operations it preserves π and r.
45
As usual we will say that two n-fold vector-spaces are isomorphic (as n-fold vector-spaces) if
there is a bijection between them that is also an n-fold linear transformation.
Definition We write 0n×m for the n × m matrix whose entries are all zero. Also write On×m for
the unique n-fold vector-space whose sole element is 0n×m
Definition Mp (n, m) is the n-fold vector-space of n × m matrices where the vector-space operahave
tions are defined entry-wise and where for rows Ai in Fm
p we








A1
A0
A0
A0







.. 


 A1 
 01×m 
 A1 







. 


)
=


and
r(


)
=

π(
.. 
.. 
.. 

 A



.
.
.
n−1

An−1


01×m


An−1


A0

Lemma 2.2.2 Let (V, π, r) be a finite n-fold vector-space over Fp . When m is greater than or
equal to the dimension of the vector-space π(V) then there exists a n-fold linear transformation h from
Mp (n, m) to (V, π, r) that is onto, i.e. surjective.
Proof: Since the dimension of π(V) is less than or equal to m there exists a surjective linear
to π(V). From this we conclude that there is a mapping h0 from matrices
transformation from Fm

p
A0


 01×m 


of the form  .  onto π(V). Write M for this sub-vector-space of Mp (n, m). Let r0 denote

.. 


01×m
the operation of Mp (n, m) that corresponds to r in the lemma.
Let h0 map M onto π(rn (V)).
Let h1 map r0 (M) onto r(π(rn−1 (V))).
..
.
Let hn−1 map r0n−1 (M)) onto rn−1 (π(r(V))).
Now every n × m matrix is the sum of n matrices that are non-zero in at most one row so there is
an isomorphism f from Mp (n, m) to the vector-space M × r0 (M) ×· · ·× r0n−1 (M)). Using this
isomorphism we easily turn this product into an n-fold vector-space in such a way that f becomes
an n-fold linear transformation. Now h0 + h1 + · · · + hn−1 is an n-fold linear transformation from
M × r0 (M) × · · · × r0n−1 (M)) to π(rn (V)) + r(π(rn−1 (V))) + · · · + rn−1 (π(r(V))). This
mapping is onto by the axiom for n-fold vector-spaces which reads: π(rn (x)) + r(π(rn−1 (x))) +
· · · + rn−1 (π(r(x))) = x. Therefore (h0 + h1 + · · · + hn−1 ) ◦ f is the desired surjective n-fold
linear transformation.
qed
We add the following slightly less general version of the above.
Lemma 2.2.3 Let V = (V 0 , π, r) be a finite n-fold vector-space over Fp such that elements of V are
determined by their projections, i.e.,
V |= π(rn (x)) = π(rn (y))∧r(π(rn−1 (x)) = r(π(rn−1 (y))∧· · ·∧rn−1 (π(r(x)) = rn−1 (π(r(y)) →
x = y.
46
Let m be equal to the dimension of the vector-space π(V). Then V is isomorphic to Mp (n, m).
Proof: Imediate from the lemma 2.2.2 and the statement that elements of V are determied by
their projections.
qed
2.3
p-automata with abstract alphabets
We now define p-automata using possibly abstract n-fold vector-spaces as alphabets. The pautomata of section 2.1.3 now become a special case with concrete vector-spaces Mp (n, 1) as
alphabets. In light of lemmas 2.1.5 and 2.2.1 we have a sense in which this is isomorphic.
By lemma 2.2.2 we can view the elements of an n-fold vector-space over Fp as a set of n × m
matrices where n · m is the dimension. We shall define n-tapes on which automata operate by
concatenating such matrices.
Definition Let A and B be n × m matrices over Fp . Then A _ B denotes the n × (2 · m) matrix
obtained by concatenating A and B. Likewise if A0 _ · · · _ Ak−1 is a n × (k · m) matrix and B is
as before then A0 , _ · · · _ Ak−1 _ B is the n × ((k + 1) · m) matrix obtained by concatenation.
We now define the set of n-tapes as a set of matrices closed under concatenation.
Definition Let n ∈ N. Then for each m ∈ N a n × m matrix with entries from Fp is called an
n-symbol or an n-tape depending on the context. In a context where Mp (n, m) is the set of symbols
S
the set of n-tapes is the set k∈N Mp (n, k · m) for which we write Mp (n, m)∗ .
2.3.1 n-tape p-automata
We give a two-sorted definition of p-automata with abstract alphabets.
Definition An n-tape p-automaton is a structure (V, K, δ, ι, T ) where
V = (V, +, −, 0, . . .) is an n-fold vector-space over Fp , V is called the alphabet,
K is a set whose elements are called states,
δ : K × V → K is called the transition function,
T ⊆ K is called the set of terminal states,
ι ∈ K is called the initial state,
moreover the following axiom holds
q ∈ T iff δ(q, 0) ∈ T .
To see how these relate to the classical p-automata as in e.g. V.Bruyère et al. [BHMV94] or
W.Thomas [Tho96], note that these would be n-tape p-automata with the concrete vector-space
Mp (n, 1) as alphabet.
We extend the transition function to map from tapes to states, rather than symbols to states.
We give a slightly more abstract version than the standard in automata theory. Symbols are in
now n × m matrices for some m depending on the size of the carrier-set. So for each k ∈ N an
n × (k · m) matrix is a tape.
47
Definition Let (V, K, δ, ι, T ) be an n-tape p-automaton, where V is an n-fold vector-space over
Fp . Let h be an n-fold linear transformation from Mp (n, m) onto V. Then δh∗ : K×Mp (n, m)∗ →
K is defined by recursion as follows:
δh∗ (q, 0n×0 ) = q, here 0n×0 is the unique n × 0 matrix, i.e. the tape of length 0,
δh∗ (q, B _ A) = δ(δh∗ (q, B), h(A)), here B ∈ Mp (n, m)∗ and A ∈ Mp (n, m).
2.3.2
Infinite tapes with finite support
Here we define infinite n-tapes with finite support and what it means for an automaton to accept
such a tape. We also prove a lemma which eventually provides us with a sense in which automata
with slightly different alphabets can be said to accept the same finitely supported n-tapes.
Definition 0n×N denotes the n × N matrix whose entries are all zero, and On×N the unique n-fold
vector-space whose carrier set consists of 0n×N .
Definition Let n, m ∈ N be positive. Then Mp (n, m)∗ _ 0n×N denotes the set of infinite ntapes with m-support. When A is an element of Mp (n, m)∗ , then A _ 0n×N denotes an element
Mp (n, m)∗ _ On×N .
It is left to the reader to verify that the following defines an n-fold vector-space.
Definition Let n, m ∈ N be positive. Then Mp (n, m)∗ _ On×N denotes the n-fold vector-space
((Mp (n, m)∗ _ 0n×N , +, −, 0, . . .), π, r), where the vector-space operations are defined
component-wise and where π and r operate on rows.
Lemma 2.3.1 If m, m0 ∈ N are positive then Mp (n, m)∗ _ On×N and Mp (n, m0 )∗ _ On×N are
isomorphic as n-fold vector-spaces.
0
Proof: The dimension of Mp (n, m)m is n · m · m0 . Likewise the dimension of
Mp (n, m0 )m is n · m0 · m. By lemma 2.2.3 we may therefore conclude that the mentioned nfold vector-spaces are isomorphic. In the same manner we conclude that there are isomorphisms
0
between Mp (n, m)k·m and Mp (n, m0 )k·m for each k ∈ N. Being isomorphic they are for each k
the same set of matrices. For this proof only, write Xk for this set. The sought after isomorphism
is obtained by taking the colimit of these isomorphisms under the inclusions A 7→ A _ 0n×(m·m0 ) ,
where A ∈ Xk and A _ 0n×(m·m0 ) ∈ Xk+1
qed
By the last lemma the following is well-defined.
Definition For p, n ∈ N the finitely supported n-tapes (over Fp ) is the vector-space
Mp (n, m)∗ _ On×N for any positive m ∈ N.
The following is well defined since we have defined our automata in such a way that the terminal
states T are invariant under adding zero-vectors at the most significant end of tapes.
Definition Let W = (V, K, δ, ι, T ) be an n-tape p-automaton and h an n-fold linear transformation from Mp (n, m) onto the alphabet V. We say that the pair (W, h) accepts A _ 0n×N if
A ∈ Mp (n, m) and δh∗ (ι, A) ∈ T .
48
Definition Let W = (V, K, δ, ι, T ) be an n-tape p-automaton and h an n-fold linear transformation from Mp (n, m) onto the alphabet V. Let L ⊆ Mp (n, m)∗ _ 0n×N . We say that the pair
(W, h) recognises L if L is the set of A _ 0n×N accepted by (W, h).
It should be clear that for positive m, n ∈ N there is a one-to-one correspondence f : Nn →
Mp (n, m)∗ _ 0n×N . Using this we also say that (W, h) recognises L ⊆ Nn if (W, h) recognises
f (L) ⊆ Mp (n, m)∗ _ 0n×N .
2.4
Transitive automata and reachability in general
In this section transitive automata are introduced. We look at one- and two-sorted variants. Twosorted automata appear easier to work with. For given n, p ∈ N the class of one-sorted transitive
n-tape p-automata is formally axiomatised by a finite set of first-order sentences. In one-sorted
transitive n-tape p-automata the reachability relation is definable using a first-order formula.
First we define a notion of equivalence with the property that every automaton is equivalent to
a transitive automaton. Equivalent automata accept the same finitely supported tapes.
2.4.1
A not quite classical notion of equivalence
Classically two automata are equivalent if they have the same alphabet, and they accept the same
tapes. We introduce a slightly looser notion of equivalence where the alphabets aren’t required
to be exactly the same. The alphabets of classical automata are of the form Mp (n, 1). We allow
for comparison where one automaton has an alphabet of the form Mp (n, m) and the other an
alphabet of the form Mp (n, m0 ) where m, m0 ∈ N are possibly different. By lemma 2.3.1 it is still
meaningful to talk about recognising the same finitely supported tapes.
We shall only be concerned with equivalence of automata with concrete alphabets, i.e. the
alphabets of the form Mp (n, m). These are a sub-class of the class of n-tape p-automata. To make
the concreteness explicit consider pairs (W, h) where h is an n-fold linear transformation from a
concrete n-fold vector-space to the possibly abstract alphabet of W.
Definition Let W = (V, K, δ, ι, T ) and W 0 = (V 0 , K 0 , δ 0 , ι0 , T 0 ) be two n-tape p-automata. Let
m, m0 ∈ N and h, h0 be two n-fold linear transformations:
h from Mp (n, m) onto V and
h0 from Mp (n, m0 ) onto V 0
Then (W, h) and (W 0 , h0 ) are equivalent provided that for tapes A of Mp (n, 1)∗ we have δh∗ (ι, A) ∈
0
0
T if and only if δh∗0 (ι0 , A) ∈ T 0 whenever δh∗ and δh∗0 are both defined.
0
Note that in the definition δh∗ and δh∗0 are both defined on Mp (n, k) whenever k is a common
multiple of m and m0 .
Proposition 2.4.1 If (W, h) and (W 0 , h0 ) are equivalent then they accept the same finitely supported
n-tapes.
Proof: Left to the reader.
qed
49
2.4.2
Transitive p-automata, the two-sorted case
We single out a class of concrete n-tape p-automata in which a state is reachable from another iff
it is reachable in exactly one step. This makes reachability first-order definable quite trivially. Less
trivially the class turns out to suffice for deciding Presburger arithmetic and the theories of the other
automatic structures.
For the rest of this section we fixate n, p ∈ N where p is a prime power. Let W = (V, K, δ, ι, T )
be a finite n-tape p-automaton. Moreover let m ∈ N and let h be an n-fold linear transformation
from Mp (n, m) onto V.
Definition Let W and h be as above. Let k ∈ N. Then the k-outreach of (W, h) is the the
pair (W 0 , h0 ) where W 0 = (Mp (n, m · k), K, δ 0 , ι, T ). Here δ 0 is the restriction of δh∗ to K ×
Mp (n, m · k) and h0 : Mp (n, m · k) → Mp (n, m · k) is the identity mapping and thus an n-fold
linear transformation.
Note that the k-outreach (W 0 , h0 ) of (W, h) is a finite automaton as long as W is.
Proposition 2.4.2 Let W and h be as above. Let k ∈ N. Then the k-outreach (W 0 , h0 ) of (W, h) is
equivalent to (W, h).
0
Proof: Observe that δ 0 and h0 are defined by means of δh∗ in such a way that δh∗ (q, A) and δh∗0 (q, A)
0
qed
are both defined and elements of T whenever δh∗0 (q, A) is defined.
We now define a notion of reachability that depends on an n-fold linear transformation from a
concrete n-fold vector-space to the alphabet of an automaton.
Definition Let W and h be as above. Let q, q 0 ∈ K be two states. Then q 0 is h-reachable from q if
there is a k ∈ N and a matrix A of Mp (n, k · m) such that δh∗ (q, A) = q 0 .
We now define the notion that makes reachability first-order definable.
Definition Let W and h be as above. Then (W, h) is said to be transitive if
for all states q ∈ K and tapes A, B ∈ Mp (n, m)
there exists a tape C ∈ Mp (n, m) such that
δh∗ (q, A _ B) = δh∗ (q, C).
Now the main result on two-sorted transitive automata.
Proposition 2.4.3 Let W and h be as above. Then there exists a k ∈ N such that the k-outreach
(W 0 , h0 ) of (W, h) is transitive. This implies that if a state is h0 -reachable from another, then it is
h0 -reachable in exactly one step.
Proof: To find the sought k we shall use lemma 2.1.4. Consider therefore X = P(K)K , the set of
mappings from the set of states to the power-set of the set of states. Define α ∈ X by α(q) = {q}.
We now define a function f : X → X that, when beginning with α and iterating, we can use to
keep track of the states reachable from a given state q. This is to say that [f i (α)](q) is the set of
50
states reachable in i steps from the state q. Formally for ξ ∈ X let [f (ξ)](q) = {δ(q 0 , h(A)) : q 0 ∈
ξ(q) ∧ A ∈ Mp (n, m)}.
By lemma 2.1.4 we obtain a k such that f k (α) = f 2·k (α). Using this k, consider the koutreach (W 0 , h0 ) of (W, h). Let q ∈ K be any state of W 0 and let A, B ∈ Mp (n, k · m) be any
symbols in the alphabet of W 0 . Now δh∗ (q, A _ B) ∈ [f 2·k (α)](q). Since by lemma 2.1.4 we have
f k (α) = f 2·k (α), it is the case that δh∗ (q, A _ B) ∈ [f k (α)](q). Accordingly by the definition of
f and α there is a symbol C ∈ Mp (n, k · m), the alphabet, such that δh∗ (q, A _ B) = δh∗ (q, C).
Since δ 0 , the transition function of W 0 , is defined as the restriction of δh∗ to K × Mp (n, k · m) the
proposition follows.
qed
Corollary 2.4.4 Every concrete n-tape p-automaton is equivalent to a transitive automaton.
Proof: Using the k of proposition 2.4.3, the k-outreach of a classical automaton is a transitive
automaton, moreover it is equivalent to the classical one by proposition 2.4.2.
qed
Corollary 2.4.5 Finite transitive n-tape p-automata, are versatile enough to replace classical automata
in Büchis decision procedure.
Proof: This follows from the previous corollary and the fact that classical p-automata are sufficient
for deciding Presburger arithmetic and the theories of the other automatic structures. Here a classical automaton is an n-tape p-automaton where the alphabet is the concrete vector-space Mp (n, 1).
qed
2.4.3
Transitive p-automata, the one-sorted case
By corollary 3.3.5 a class of two-sorted automata sufficient for deciding theories of automatic structure was singled out. Namely the transitive and hence concrete n-tape p-automata. In these automata a state is reachable from another iff it is reachable exactly one step.
Recall that we applied the Ehrenfeucht-Fraïssé method to one-sorted automata. To make sure
that first-order definability of reachability is not a property of two-sorted automata or of concreteness, we provide a finite set of axioms for a one-sorted version of transitive automata. In these,
transitivity and reachability is first-order definable, independently of a particular concrete alphabet.
The elements of the carrier-set serve both as states and as symbols of the alphabet. The initial state
and the 0 symbol are identified.
Definition Let n, p ∈ N where p is a prime power. A (one-sorted) n-tape p-automaton is a structure
W = (V, δ, T ) where
V = (V, +, −, 0, . . .) is an n-fold vector-space over the field Fp where the elements of V serve both
as symbols and states,
δ : V × V → V is the transition function,
T ⊆ V , is the set of terminal states.
Moreover
T is closed under adding zero-vectors at the most significant end, i.e.
51
W |= ∀q[T (q) ↔ T (δ(q, 0))].
Definition An n-tape p-automaton W = (V, δ, T ) is said to be transitive if every state that is
reachable from a given state q, is reachable from q in one step, i.e.
W |= ∀qxy∃z[δ(δ(q, x), y) = δ(q, z)].
Note that, as opposed to the two sorted case, one-sorted transitive automata need not be concrete.
Proposition 2.4.6 Over structures of suitable similarity-type transitive n-tape p-automata are axiomatisable by a finite set of first-order axioms.
Proof: Stating the first axiom, “V is an n-fold vector-space over the field Fp ” with a finite number
of first-order sentences can be done as in section 2.2. The rest of the axioms are obviously finite in
number and first-order.
qed
We will now regard various classes of automata we have defined in this paper, and provide notation
for them. For the rest of this section fixate n, p ∈ N such that p is a prime power.
Definition The following denote classes of finite one-sorted n-tape p-automata.
K1,cla classical automata, i.e those with alphabet Mp (n, 1)
K1,con concrete automata, i.e those of the form (W, h)
K1,ctr concrete and transitive automata.
The following denote two-sorted versions of the same.
K2,cla classical automata, i.e those with alphabet Mp (n, 1)
K2,con concrete automata, i.e those of the form (W, h)
K2,ctr concrete and transitive automata.
We will now define equivalence for members of K1,con in such a way that equivalent automata accept the same n-tapes. We could do this by adapting the definitions of equivalence and recognition
we have on K2,con , but we shall instead use a trick and embed K1,con into K2,con and have the
needed notions reflected by the embedding.
Definition The mapping ∆ : K1,con → K2,con is defined as follows. Let W = (V, δ, T ) an
automaton and h an n-fold linear transformation making (W, h) concrete. Let V have carrier-set
V and zero-vector 0. Then ∆((V, δ, T ), h) = ((V, V, δ, 0, T ), h)
Definition (W, h), (W 0 , h0 ) ∈ K1,con are equivalent iff ∆(W, h), ∆(W 0 , h0 ) ∈ K2,con are equivalent.
Definition Let (W, h) ∈ K1,con Let V be the alphabet of W and Mp (n, m) the domain of h. Let
L ⊆ Mp (n, m)∗ _ 0n×N . Then (W, h) recognises L iff ∆(W, h) recognises L.
Proposition 2.4.7 If (W, h), (W 0 , h0 ) ∈ K1,con are equivalent then they accept the same finitely
supported n-tapes.
52
Proof: Left to the reader.
qed
Definition Let KX and KY be two classes of automata and let f : KX → KY . Then f is said
to be recognition-invariant if for all W ∈ KX and all A ∈ Mp (n, 1)∗ we have that W accepts
A _ 0n×N iff f (W) accepts A _ 0n×N .
Again effective means computable and not necessarily very fast, assuming some reasonable way of
encoding automata.
Proposition 2.4.8 There are effective recognition-invariant mappings for each of the types:
1. K1,cla → K2,cla
2. K2,cla → K2,con
3. K2,con → K2,ctr
4. K2,ctr → K1,ctr
5. K1,ctr → K2,ctr
6. K2,ctr → K2,con
7. K2,con → K2,cla
8. K2,cla → K1,cla
Proof:
1. This is a statement about classical automata and left to the reader.
2. K2,cla ⊆ K2,con so the identity mapping will do.
3. For this we map automata to their k-outreach where k is as in proposition 2.4.3. Such a
mapping is recognition-invariant by proposition 2.4.2.
4. Given a two-sorted transitive ((V, K, δ, ι, T ), h) we leave V and h as they are and define a
one-sorted counterpart ((V, δ 0 , T 0 ), h). We are only interested in states that are reachable in
one step and we shall use the members of V as states. First we choose some well-ordering ≤
on V making 0 minimal. Second define δ 0 and T 0 as follows.
For each pair x, y of V let δ 0 (x, y) be the ≤-minimal z s.t. δ(δ(ι, x), y) = δ(ι, z).
Moreover let x ∈ T 0 iff δ(ι, x) ∈ T .
5. This is the opposite direction. Here we use the recently defined mapping ∆ which maps
structures
((V, δ, T ), h) to ((V, V, δ, 0, T ), h). One easily verifies that transitivity is preserved and that
∆ is recognition-invariant.
6. K2,ctr ⊆ K2,con so the identity mapping will do.
53
7. It is known from classical automata theory that nondeterministic and partially defined automata can effectively be transformed into equivalent deterministic and totally defined automata. The members of K2,con can be seen as partially defined members of K2,cla , and the
mentioned determination-operation as the sought mapping.
8. Again this is a statement about classical automata and left to the reader.
qed
Now two corollaries. Note that by the proposition we have effective recognition-invariant transformations to and from classical automata. Since there are effective constructions on classical automata
to do negation, disjunction, variable substitutions and existential quantification we get the same for
concrete n-tape p-automata, be they transitive or not. Therefore:
Corollary 2.4.9 Transitive n-tape p-automata are versatile enough to replace classical automata in
Büchis decision procedure. More specifically transitive automata that are both concrete and finite suffice.
By the definition of transitive automaton we get the following.
Corollary 2.4.10 In concrete finite transitive n-tape p-automata (W, h), the relation of being hreachable is definable by the first-order formula ∃z[δ(q, z) = q 0 ].
Note that there is no reference to the homomorphism h in the first-order formula. This means that
the formula defines reachability regardless of which surjection h one chooses.
54
2.5
Reachability refined
Here we introduce three properties of automata, namely being Projectively Transitive, having Projections
and having Substitutions. We refer to automata with all three of these properties as PTPS automata.
We look at one- and two-sorted variants. For given n, p ∈ N the class of one-sorted n-tape p-entry
PTPS automata is definable by a finite set of first-order sentences. In each one-sorted PTPS automaton we are able to express, in first-order language, that states are reachable from one another
using tapes that represent various projections of Nn . We are for instance be able to express, in firstorder language, that the state q 0 is reachable from q using an n-tape whose second row represents
the number 0.
We introduce PTPS automata as multi-automata, i.e. automata with several sets of terminal
states. We don’t actually use the fact that PTPS automata are multi-automata before the next
section. So on a first reading one may assume that PTPS automata have one set of terminal states,
in which case PTPS automata form a subclass of the transitive automata.
2.5.1
Two-sorted multi-automata
We introduce multi-automata to make it possible to compare automata with the same transition
function quite directly. For example we will see how an automaton recognising the relation defined
by the formula φ ∨ ψ, compares to the automata recognising the relations defined by φ and ψ.
Definition An n-tape p-multi-automaton is a tuple (V, K, δ, ι, T0 , . . . , Tt−1 ) where each
(V, K, δ, ι, T0 ), . . . , (V, K, δ, ι, Tt−1 ) is a two-sorted n-tape p-automaton.
Each Wi = (V, K, δ, ι, Ti ) is said to be a component of W.
We can now carry defined notions to multi-automata via their components.
Definition Let W = (V, . . .) and W 0 = (V 0 , . . .) be n-tape p-multi-automata. Let h be an n-fold
linear transformation from Mp (n, m) onto V. Let h0 be an n-fold linear transformation from
Mp (n, m) onto V 0 . Then
(W, h) is concrete if each component (Wi , h) is concrete.
(W 0 , h0 ) is the k-outreach of (W, h) if each component (Wi0 , h0 ) is the k-outreach of (Wi , h).
(W, h) is equivalent to (W 0 , h0 ) if W and W 0 are tuples of the same length and each corresponding
pair of components (Wi , h) and (Wi0 , h) is equivalent.
The following multi-automaton version of Proposition 2.4.2 is now a fairly immediate consequence.
Proposition 2.5.1 Let (W, h) and (W 0 , h0 ) be concrete n-tape p-multi-automata. Let k ∈ N. Then
the k-outreach (W 0 , h0 ) of (W, h) is equivalent to (W, h).
Proof: Immediate from the last definition and proposition 2.4.2.
qed
The following lemma allows us to merge multi-automata in a way that resembles concatenation.
The present author was unable to adapt the proof of the lemma to the one-sorted counterpart of
multi-automata. This is the reason we bother with two-sorted automata at this point.
55
Lemma 2.5.2 If (W, h) and (W 0 , h0 ) are finite concrete n-tape p-multi-automata, then there exists a
finite concrete n-tape p-multi-automaton (W 00 , h00 ) such that
(W0 , h) is equivalent to (W000 , h00 )
..
.
00
, h00 )
(Wl−1 , h) is equivalent to (Wl−1
(W00 , h0 ) is equivalent to (Wl00 , h00 )
..
.
00
00
(Wl00 −1 , h0 ) is equivalent to (Wl+l
0 −1 , h )
Moreover h00 can be chosen so as to make it an isomorphism.
Proof: To introduce required notation, let (W, h) = (V, K, δ, ι, T0 , . . . , Tl−1 , h) and
(W 0 , h0 ) = (V 0 , K 0 , δ 0 , ι, T00 , . . . , Tl00 −1 , h0 ).
By definition of concrete automaton there are m, m0 ∈ N s.t. h and h0 are surjective linear transformations from Mp (n, m) onto V and Mp (n, m0 ) onto V 0 respectively. Let m00 be
the least common multiple of m and m0 . We now define the concrete automaton (W 00 , h00 ) =
00
00
(V 00 , K 00 , δ 00 , ι00 , T000 , . . . , Tl+l
0 −1 , h ) which proves the lemma.
V 00 = Mp (n, m00 )
. . . the n-fold vector-space of n × m00 matrices, which is finite
K 00 = K × K 0
. . . cartesian product of sets which preserves finiteness
δ 00 ((q, q 0 ), A) = (δh∗ (q, A), δh0∗0 (q 0 , A))
ι00 = (ι, ι0 )
(q, q 0 ) ∈ T000 iff q ∈ T0
..
.
00
(q, q 0 ) ∈ Tl−1
iff q ∈ Tl−1
(q, q 0 ) ∈ Tl00 iff q ∈ T00
..
.
00
(q, q 0 ) ∈ Tl+l
0 −1 iff q ∈ Tl0 −1
h00 (A) = A
. . . the identity mapping on Mp (n, m00 ), clearly an isomorphism.
The crucial thing to note here is that δ 00 is well defined, which it is by choice of m00 . The equivalences then are immediate by the definition of the Ti00 ’s.
qed
2.5.2
Definable linear transformations
Here we introduce two kinds of linear transformations that allow us to define certain semigroups
of symbols for n-tapes in the language of n-fold vector-spaces. We will eventually be able to give
first-order definitions of reachability by tapes built from these semigroups.
The first kind comprises nn special linear transformations. When the alphabet happens to be
concrete, i.e. a set of matrices, a given linear transformation of this class corresponds to a particular
way of swapping and/or overwriting rows with other rows.
56
Definition For each n-fold vector-space V = (V, +.−, 0, . . . , π, r) and mapping σ : n → n we
define σ̂ : V → V as follows. σ̂(x) = r−0 (π(rσ(0) (x))) + . . . + r−(n−1) (π(rσ(n−1) (x))).
The second kind comprises n linear transformations. When the alphabet happens to be concrete, i.e. a set of matrices, the i’th linear transformation replaces each entry on the i’th row with a
0.
Definition For for each n-fold vector-space V = (V, +, −, 0, . . . , π, r) and i < n we define
π̂i : V → V as follows
π̂0 (x) = 0 + r1 (π(r−1 (x))) + · · · + rn−1 (π(r−(n−1) (x)))
π̂1 (x) = r0 (π(r−0 (x))) + 0 + · · · + rn−1 (π(r−(n−1) (x)))
..
.
π̂n−1 (x) = r0 (π(r−0 (x))) + · · · + rn−2 (π(r−(n−2) (x))) + 0
The σ̂’s and π̂i ’s are clearly linear transformations since they are built using sums and the linear
transformations π and r. They need not be n-fold linear transformations.
Given an n-fold vector-space the σ̂’s and π̂i ’s each generate a semigroup under composition.
We introduce notation for the carrier set of these two semigroups.
Definition Let V = (V, . . .) be an n-fold vector space over Fp . Then we define SL(V) ⊆ V V as
follows
SL(V) = {σ̂ : σ ∈ nn }
The letters SL stand for Substitution and Linear.
Definition Let V = (V, . . .) be an n-fold vector space over Fp . Then we define P L(V) ⊆ V V as
follows
P L(V) = the set of mappings generated by {π̂i : i < n} under composition.
The letters P L stand for Projection and Llinear.
Lemma 2.5.3 For finite n-fold vector-spaces V = (V, . . .) over Fp , the sets SL(V) and P L(V) are
finite.
Proof: Since SL(V) ⊆ V V and P L(V) ⊆ V V and V V is finite.
2.5.3
qed
Projective transitivity
Here we define a property that enables us to express notions such as: the state q 0 is reachable from q
using an n-tape whose second row represents the number 0. We will be using linear transformations
in P L(Mp (n, m)) for this. We shortly prove a variant of Proposition 2.4.3. A definition is handy
first.
Definition Let W = (V, K, δ, . . .) be an n-tape p-multi-automaton. Let h be an n-fold linear
transformation from Mp (n, m) onto V. Then (W, h) is said to be projectively transitive if
57
for each t ∈ P L(Mp (n, m)),
for all states q ∈ K and tapes A, B ∈ Mp (n, m),
there exists a C ∈ Mp (n, m) such that
δh∗ (δh∗ (q, t(A)), t(B)) = δh∗ (q, t(C)).
By repeated use of this definition, projectively transitive automata do for example have the property
that if a state is h-reachable by a tape in Mp (n, m)∗ whose second row represents the number 0,
then it is h-reachable by a symbol in Mp (n, m) whose second row represents 0.
Lemma 2.5.4 Let W = (V, K, δ, . . .) be a finite n-tape p-multi-automaton. Let h be an n-fold
linear transformation from Mp (n, m) onto V. Then there exists a k ∈ N such that the k-outreach
(W 0 , h0 ) of (W, h) is projectively transitive.
Proof: To find the sought k we shall use lemma 2.1.4. Consider therefore
X = P(K)K×P L(Mp (n,m)) , the set of mappings from the set of pairs of states and projections to
the power-set of the set of states. The set X is a finite set since K is finite and by lemma 2.5.3,
P L(Mp (n, m)) is finite. Define the mapping α ∈ X by α(q, t) = {q}. We now define a function
f : X → X that, when beginning with α and iterating, we can use to keep track of the states
reachable form a given state q using symbols of the form t(A). This is to say that [f i (α)](q, t) is
the set of states reachable in i steps from the state q using symbols of the form t(A). Formally for
ξ ∈ X let [f (ξ)](q, t) = {δ(q 0 , h(t(A))) : q 0 ∈ ξ(q, t) ∧ A ∈ Mp (n, m)}.
By lemma 2.1.4 we obtain a k such that f k (α) = f 2·k (α). Using this k consider the k-outreach
(W 0 , h0 ) of (W, h). Let q ∈ K be any state of W 0 . Let A, B ∈ Mp (n, k · m) be any symbols in
the alphabet of W 0 . Let t ∈ P L(Mp (n, k · m) be a transformation on the alphabet of W 0 . Now
δh∗ (q, t(A) _ t(B)) ∈ [f 2·k (α)](q, t). Since by lemma 2.1.4 we have f k (α) = f 2·k (α), it is the case
that δh∗ (q, t(A) _ t(B)) ∈ [f k (α)](q, t). Accordingly, by the definition of f and α, there exists a
symbol C ∈ Mp (n, k · m), the alphabet, such that δh∗ (q, t(A) _ t(B)) = δh∗ (q, t(C)). Since δ 0 by
definition of W 0 , is the restriction of δh∗ to K × Mp (n, k · m), the lemma follows.
qed
Keep in mind that the k-outreach of a finite automaton is a finite equivalent automaton. Hence
with each finite automaton, we may associate an equivalent finite automaton that is projectively
transitive. By examining the proof of the previous proposition we obtain the following.
Corollary 2.5.5 For k ∈ N the k-outreach of a projectively transitive automaton is again projectively
transitive.
2.5.4
Projection automata
For each linear transformation in P L(Mp (n, m)) we now define a relation on pairs of states of
a given automaton which for example we can use as follows. From the fact that q and q 0 are in
relation, we may infer that there exist two symbols,that differ only in the the second row, which,
from the initial state ι, we can use to reach q and q 0 respectively. This property turns out to be useful
when comparing an automaton recognising an interpretation of the formula φ to an automaton
recognising an interpretation of the formula ∃vi φ.
58
Definition Let W = (V, K, δ, ι, . . .) be an n-tape p-multi-automaton. Let h be an n-fold linear
t
transformation from Mp (n, m) onto V. Let t ∈ P L(Mp (n, m)). Then ∼⊆ K × K is defined as
follows.
t
The relation q ∼ q 0 holds if
there exist A, B ∈ Mp (n, m) such that
t(x) = t(y) and
δh∗ (ι, A) = q and
δh∗ (ι, B) = q 0 .
t
Note that if q ∼ q 0 then the states q and q 0 can both be reached from the initial state in one step.
Definition Let W = (V, K, δ, ι, . . .) be an n-tape p-multi-automaton. Let h be an n-fold linear
transformation from Mp (n, m) onto V. Then (W, h) is said to be a projection automaton or to
have projections if
for each t ∈ P L(Mp (n, m))
t
t
{(q, q 0 ) : q ∼ q 0 } = {(δh∗ (q, A), δh∗ (q 0 , B)) : q ∼ q 0 ∧ A, B ∈ Mp (n, m) ∧ t(A) = t(B)}
t
So in a projection automaton if q ∼ q 0 then the states q and q 0 can both be reached from the initial
state in one step, moreover both can be reached in exactly two steps, and so forth. We now use
lemma 2.1.4 to show that every automaton is equivalent to a projection automaton.
Lemma 2.5.6 Let W = (V, K, δ, ι, . . .) be a finite n-tape p-multi-automaton. Let h be an n-fold
linear transformation from Mp (n, m) onto V. Then there exists a k ∈ N such that the k-outreach
(W 0 , h0 ) of (W, h) is a projection automaton.
Proof: To use lemma 2.1.4 consider X = P(K × K)P L(Mp (n,m)) , the set of mappings from
P L(Mp (n, m)) to the set of binary relations on states. The set X is finite since by lemma 2.5.3,
P L(Mp (n, m)) is finite. Define the mapping α ∈ X by α(t) = {(ι, ι)}. We now define a
function f : X → X that, when beginning with α and iterating, we can use to keep track of pairs
of states that are reachable from the initial state by a pair of tapes that differ in, for example, the
second row only. This is to say that for t ∈ P L(Mp (n, m)) we have
(q, q 0 ) ∈ [f i (α)](t)
iff
there exist A0 , . . . , Ai−1 , B0 , . . . , Bi−1 ∈ Mp (n, m)
such that
t(A0 ) _ · · ·
_
t(Ai−1 ) = t(B0 ) _ · · ·
δh∗ (ι, A0 _ · · ·
_
Ai−1 ) = q and
δh∗ (ι, B0 _ · · ·
_
Bi−1 ) = q 0 .
_
t(Bi−1 ) and
59
A way of defining f is for ξ ∈ X to let [f (ξ)](t) = {(δ(q, h(A)), δ(q 0 , h(B))) : (q, q 0 ) ∈
ξ(t) ∧ A, B ∈ Mp (n, m) ∧ t(A) = t(B)}.
By lemma 2.1.4 we obtain a k such that f k (α) = f 2·k (α). Using this k consider the koutreach (W 0 , h0 ) of (W, h). With each t ∈ P L(Mp (n, m)) associate t0 ∈ P L(Mp (n, k · m))
by the equation t0 (A0 _ · · · _ Ak−1 ) = t(A0 ) _ · · · _ t(Ak−1 ). Note that this association sets up
a bijection between P L(Mp (n, m)) and P L(Mp (n, k · m)), since both are sets of operations
on n rows. We now have what we need to show that (W 0 , h0 ) fulfils the requirement of being a
projection automaton. So letting δ 0 be as in the definition of k-outreach we have:
t0
{(δ 0 (q, A), δ 0 (q 0 , B)) : q ∼ q 0 ∧ A, B ∈ Mp (n, k · m) ∧ t0 (A) = t0 (B)}
= {(δh∗ (ι, A _ A0 ), δh∗ (ι, B _ B 0 )) : A, A0 , B, B 0 ∈ Mp (n, k · m) ∧ t0 (A) = t0 (B) ∧ t0 (A0 ) =
t0
t0 (B 0 )}
by definition of δ 0 and ∼
= {(δh∗ (ι, A0 _ · · · _ A2k−1 ), δh∗ (ι, B0 _ · · · _ B2k−1 )) : A0 _ · · ·
Mp (n, m)
∧t(A0 ) = t(B0 ) ∧ · · · ∧ t(A2k−1 ) = t(B2k−1 )}
= [f 2·k (α)](t)
_
A2k−1 , B0 _ · · ·
_
B2k−1 ∈
by definition of t0
by design of f
= [f k (α)](t)
by choosing the k of lemma 2.1.4
t0
t0
= {(q, q 0 ) : q ∼ q 0 }
again by definition of ∼
The left and right side of this sequence of equations is what we require of a projection automaton.
qed
As before the k-outreach of a finite automaton is a finite equivalent automaton. Hence with each
finite automaton, we may associate an equivalent finite automaton that is a projection automaton.
By examining the proof of the previous proposition we gather the following.
Corollary 2.5.7 For k ∈ N the k-outreach of a projection automaton is again a projection automaton.
2.5.5
Substitution automata
For each linear transformation in SL(Mp (n, m)) we now define a relation on pairs of states of
a given automaton which for example we can use as follows. From the fact that q and q 0 are
in relation, we may infer that there exist two symbols, one whose rows are a permutation of the
other symbols rows, which from the initial state we can use to reach q and q 0 respectively. This
property turns out to be useful when comparing the automata that recognise the interpretations of
for example the atomic formulas R(v0 , v1 ) and R(v1 , v0 ).
Definition Let (V, K, δ, ι, . . .) be a two-sorted n-tape p-multi-automaton. Let h be an n-fold
linear transformation from Mp (n, m) onto V. Let σ̂ ∈ SL(Mp (n, m)). Then .σ̂ ⊆ K × K is
defined as follows.
q .σ̂ q 0 if
there exists a symbol A ∈ Mp (n, m) such that
δh∗ (ι, A) = q and
60
δh∗ (ι, σ̂(A)) = q 0
Also σ̂ ⊆ K × K is defined by
q 0 σ̂ q 00 if
there exists a q ∈ K such that
q .σ̂ q 0 and
q .σ̂ q 00
Note that if q .σ̂ q 0 then the states q and q 0 are h-reachable from the initial state in one step.
Definition Let W = (V, K, δ, ι, . . .) be a two-sorted n-tape p-multi-automaton. Let h be an nfold linear transformation from Mp (n, m) onto V. Then the concrete automaton (W, h) is said
to be a substitution automaton or to have substitutions if
for each σ̂ ∈ SL(Mp (n, m))
{(q, q 0 ) : q .σ̂ q 0 } = {(δh∗ (q, A), δh∗ (q 0 , σ̂(A)) : q .σ̂ q 0 ∧ A ∈ Mp (n, m)}
So in an automaton which has substitutions, if q .σ̂ q 0 then the states q and q 0 are h-reachable from
the initial state in one step, moreover they are h-reachable from the initial state in exactly two steps,
and so forth.
Lemma 2.5.8 Let W = (V, . . .) be a finite two-sorted n-tape p-multi-automaton. Let h be an n-fold
linear transformation from Mp (n, m) onto V. Then there exists a k ∈ N such that the k-outreach
(W 0 , h0 ) of (W, h) has substitutions.
Proof: We use lemma 2.1.4, so consider X = P(K × K)SL(Mp (n,m)) , the set of mappings
from SL(Mp (n, m)) to the set of binary relations on states. The set X is finite since by lemma
2.5.3, the set SL(Mp (n, m)) is finite. Define the mapping α ∈ X by α(t) = {(ι, ι)}. We
now define a function f : X → X that, when beginning with α and iterating, we can use to
keep track of pairs of states that are reachable from the initial state by a pair of tapes of the form
(A0 , . . . , Ai−1 , σ̂(A0 ), . . . , σ̂(Ai−1 )).
This is to say that for σ̂ ∈ SL(Mp (n, m)) we have
(q, q 0 ) ∈ [f i (α)](σ̂)
iff
there exist A0 , . . . , Ai−1 ∈ Mp (n, m) such that
δh∗ (ι, A0 _ · · ·
_
Ai−1 ) = q
δh∗ (ι, σ̂(A0 ) _ · · ·
_
σ̂(Ai−1 )) = q 0
A way of defining such an f is for ξ ∈ X to let
[f (ξ)](σ̂) = {(δ(q, h(A)), δ(q 0 , h(σ̂(A)))) : (q, q 0 ) ∈ ξ(σ̂) ∧ A ∈ Mp (n, m)}.
61
By lemma 2.1.4 we obtain a k ∈ N such that f k (α) = f 2·k (α). Using this k consider the koutreach (W 0 , h0 ) of (W, h). For σ ∈ nn let σ̂ denote the corresponding operation on Mp (n, m)
ˆ the same on Mp (n, k · m).
and σ̂
We now have what we need to show that the k-outreach (W 0 , h0 ) fulfills the requirement of
being a substitution automaton. So letting δ 0 be as in the definition of k-outreach we have.
ˆ
{(δ 0 (q, A), δ 0 (q 0 , σ̂(A)))
: q .σ̂ˆ q 0 ∧ A ∈ Mp (n, k · m)}
_ˆ
ˆ
= {(δ ∗ (ι, A _ A0 ), δ ∗ (ι, σ̂(A)
σ̂(A0 ))) : A, A0 ∈ Mp (n, k · m)}
h
h
= {(δh∗ (ι, A0 _ · · · _ A2·k−1 ), δh∗ (ι, σ̂(A0 ) _ · · ·
ˆ
by definition of σ̂ and σ̂
_
by definition of δ 0 and .σ̂ˆ
σ̂(A2·k−1 ))) : A0 , . . . , A2·k−1 ∈ Mp (n, m)}
= [f 2·k (α)](σ̂)
by design of f
= [f k (α)](σ̂)
by choosing the k of lemma 2.1.4
{(q, q 0 ) : q .σ̂ˆ q 0 }
by definition of .σ̂ˆ
The left and right side of this sequence of equations is what we require of a substitution automaton.
qed
By examining the proof of the previous proposition we obtain the following.
Corollary 2.5.9 For k ∈ N the k-outreach of a substitution automaton is again a substitution automaton.
2.5.6
Properties of the two-sorted transition function
Here we prove two properties of two-sorted automata with projections, which shortly will be used
as defining properties for the one-sorted version of having projections.
Lemma 2.5.10 Let W = (V, K, δ, ι, . . .) be an n-tape p-multi-automaton. Let h be an n-fold linear
transformation from Mp (n, m) onto V. Let (W, h) be a projection automaton. Then
1. for each t ∈ P L(Mp (n, m))
for each pair of states q, q 0 ∈ K and symbols A, B ∈ Mp (n, m)
t
if q ∼ q 0 and t(A) = t(B) then
t
δh∗ (q, A) ∼ δh∗ (q 0 , B) .
2. for each t ∈ P L(Mp (n, m))
for each pair of states q, q 0 ∈ K and symbol A ∈ Mp (n, m)
t
if δh∗ (q, A) ∼ q 0
then there exist q 00 ∈ K and B ∈ Mp (n, m) such that
t
q ∼ q 00 and t(A) = t(B) and δh∗ (q 00 , B) = q 0
Proof:
62
1. Immediate from the definition of projection automaton.
t
2. Note that, by definition, the relation ∼ is an equivalence relation on δh∗ (ι, Mp (n, m)), the set
of states h-reachable from ι in one step. Therefore δh∗ (ι, Mp (n, m)) can be partitioned into
equivalence classes. We use [q] to denote the equivalence class of q ∈ δh∗ (ι, Mp (n, m)).
By the definition of projection automaton we get the following equality: [δh∗ (q, A)] =
t
t
{δh∗ (q 00 , B) : q ∼ q 00 ∧ t(A) = t(B)}. To prove the lemma, assume now that δh∗ (q, A) ∼ q 0 .
t
Then q 0 ∈ [δh∗ (q, A)] and by the equality there exist q 00 and B such that q ∼ q 00 and
t(A) = t(B) and δh∗ (q 00 , B) = q 0 .
qed
Here we prove two properties of two-sorted automata with substitutions which shortly will be
used as defining properties for the one-sorted version of having substitutions.
Lemma 2.5.11 Let W = (V, K, δ, ι, . . .) be a two-sorted n-tape p-multi-automaton. Let h be an nfold linear transformation from Mp (n, m) onto V. Let the concrete automaton (W, h) have substitions.
Let σ̂ ∈ SL(Mp (n, m)). Then
1. for each pair of states q, q 0 ∈ K and symbol A ∈ Mp (n, m)
if q .σ̂ q 0
then δh∗ (q, A) .σ̂ δh∗ (q 0 , σ̂(A))
2. for each pair of states q, q 0 ∈ K and symbol A ∈ Mp (n, m)
if δh∗ (q, A) .σ̂ q 0
then there exists a q 00 ∈ K such that
q .σ̂ q 00 and δh∗ (q 00 , σ̂(A)) = q 0 .
Proof:
1. By definition of (W, h) having substitutions we conclude that
{(δh∗ (q, A), δh∗ (q 0 , σ̂(A)) : q.σ̂ q 0 ∧A ∈ Mp (n, m)} ⊆ {(q, q 0 ) : q.σ̂ q 0 }. Using this inclusion
together with the assumptions A ∈ Mp (n, m) and q .σ̂ q 0 we conclude that δh∗ (q, A) .σ̂
δh∗ (q 0 , σ̂(A)).
2. By definition of (W, h) having substitutions we also conclude that
{(q, q 0 ) : q .σ̂ q 0 } ⊆ {(δh∗ (q, A), δh∗ (q 0 , σ̂(A)) : q .σ̂ q 0 ∧ A ∈ Mp (n, m)}. By the assumptions A ∈ Mp (n, m) and δh∗ (q, A) .σ̂ q 0 of the lemma, we conclude that (δh∗ (q, A), q 0 ) ∈
{(δh∗ (q, A), δh∗ (q 0 , σ̂(A)) : q .σ̂ q 0 ∧ A ∈ Mp (n, m)}. If we rename variables we conclude
that (δh∗ (q, A), q 0 ) ∈ {(δh∗ (q, A), δh∗ (q 00 , σ̂(A)) : q .σ̂ q 00 ∧ A ∈ Mp (n, m)}. It follows that
there exists a q 00 ∈ K such that q .σ̂ q 00 and δh∗ (q 00 , σ̂(A)) = q 0 .
qed
2.6
Moving to the abstract and to one sort
We now define a class of one-sorted automata that are projectively transitive, have projections and
substitutions, namely PTPS automata. As opposed to two-sorted automata, one-sorted automata
need not be concrete to be in possetion of either of these tree properties.
63
2.6.1
One-sorted multi-automata
Definition A (one-sorted) n-tape p-multi-automaton is a tuple W = (V, δ, {Ti }i∈I ) where
V = (V, +, −, 0, . . .) is an n-fold vector-space over Fp
δ :V ×V →V
I is a finite set and
for each i ∈ I it is the case that Ti ⊆ V .
moreover the terminal states are invariant under adding zero-vectors at the most significant end, i.e.
for each i ∈ I
W |= ∀q[Ti (q) ↔ Ti (δ(q, 0))].
We use the variables q, q 0 , q 00 and x to range over the carrier-set of one-sorted automata. We use
q, q 0 , q 00 when the elements are used as states and x, y, . . . when they are used as symbols.
We adapt some notions from two-sorted automata to one-sorted n-tape p-automata.
Definition If W = (V, δ, {Ti }i∈I ) is an n-tape p-multi-automaton and i ∈ I then the automaton
Wi = (V, δ, Ti ) is said to be a component of W.
We use (W, h) to denote concrete automata also in the one sorted case. The definitions of δh∗ ,
acceptence and recoginition are as in the two-sorted case.
We define the relation q .σ̂ q 0 for pairs of states q, q 0 and mappings σ̂ ∈ SL(Mp (n, m)). This
time without reference to a linear transformation h.
Definition Let W = (V, δ, . . .) be a one-sorted n-tape p-multi-automaton.
Let V = (V, . . . , 0, . . .). Let σ̂ ∈ SL(Mp (n, m)). Then .σ̂ ⊆ V × V is defined as follows.
The relation q .σ̂ q 0 holds if
there exists an x ∈ V such that
δ(0, x) = q and
δ(0, σ̂(x)) = q 0 .
Also σ̂ ⊆ V × V is defined as follows.
q 0 σ̂ q 00 if
there exists a q ∈ V such that
q .σ̂ q 0 and
q .σ̂ q 00 .
t
We again define the relation q ∼ q 0 for pairs of states q, q 0 and projections t. This time without
reference to a linear transformation h.
Definition Let W = (V, δ, . . .) be an n-tape p-multi-automaton. Let V = (V, . . . , 0, . . .). Let
t
t ∈ P L(V). Then ∼⊆ V × V is defined as follows.
t
The relation q ∼ q 0 holds if
64
there exist x, y ∈ V such that
t(x) = t(y) and
δ(0, x) = q and
δ(0, y) = q 0 .
Using this binary relation we are able to define the one-sorted counterpart of projection automaton.
In the following we treat terms like t(x), where t ∈ P L(V), as shorthand for a longer defining
term such as for instance: π̂1 (x) = r0 (π(r−0 (x))) + 0 + · · · + rn−1 (π(r−(n−1) (x))). Moreover
t
we treat expressions like q ∼ q 0 as shorthand for a longer defining formula such as for instance:
∃z, z 0 [t(z) = t(z 0 ) ∧ δ(0, z) = q ∧ δ(0, z 0 ) = q 0 ].
Definition Let W = (V, δ, . . .) be a one-sorted n-tape p-multi-automaton. Then W is said to be
a PTPS automaton if
1. it is projectively transitive, i.e. for each t ∈ P L(V),
W |= ∀qxy∃z[δ(δ(q, t(x)), t(y)) = δ(q, t(z))].
2. it has the property of lemma 2.5.10.1, i.e. for each t ∈ P L(V),
t
t
W |= ∀qq 0 xy[q ∼ q 0 ∧ t(x) = t(y) → δ(q, x) ∼ δ(q 0 , y)].
3. it has the property of lemma 2.5.10.2, i.e. for each t ∈ P L(V),
t
t
W |= ∀qq 0 x[δ(q, x) ∼ q 0 → ∃q 00 , y[q ∼ q 00 ∧ t(x) = t(y) ∧ δ(q 00 , y) = q 0 ]].
4. it has the property of lemma 2.5.11.1, i.e. for each σ ∈ nn ,
W |= ∀xqq 0 [q .σ̂ q 0 → δ(q, x) .σ̂ δ(q 0 , σ̂(x))],
5. it has the property of lemma 2.5.11.2, i.e. for each σ ∈ nn ,
W |= ∀xqq 0 [δ(q, x) .σ̂ q 0 → ∃q 00 [q .σ̂ q 00 ∧ δ(q 00 , σ̂(x)) = q 0 ]].
As with one-sorted transitive automata, one-sorted PTPS automata are finitely axiomatisable.
Proposition 2.6.1 Let n, p ∈ N and I ⊂ N where p is a prime power and I is finite. PTPS
automata are axiomatisable by a finite set of first-order axioms in the language of (V, δ, {Ti }i∈I ).
Proof: The fact that PTPS automata are n-tape p-multi-automata can be finitely axiomatised
much as the case of one-sorted transitive automata. To state the remaining properties of PTPS
automata recall that SL(V) and P L(V) are finite for finite V. Also recall that P L(V) is the
closure, under composition, of operations definable in the language for PTPS automata.
qed
We will soon focus on concrete automata (W, h) where h is an isomorphism. It is therfore worth
noting that the property of being isomorphic to some Mp (n, m) is expressible with a first-order
axiom, provided W is finite.
Corollary 2.6.2 Let n, p ∈ N and I ⊂ N where p is a prime number and I is finite. Over finite
structures, the class of PTPS automata whose alphabet is isomorphic to Mp (n, m) for some m ∈ N, is
axiomatisable by a finite set of first-order axioms in the language of (V, δ, {Ti }i∈I ).
65
Proof: To prove this we we add to the axioms of PTPS automata that the elements of the alphabet
are determined by their components as in lemma 2.2.3.
qed
We now prove a counterpart of proposition 2.4.8 for PTPS automata.
Proposition 2.6.3 There exists an effective mapping between finite two-sorted concrete n-tape p-multiautomata (W, h) and finite one-sorted concrete PTPS automata (W 0 , h0 ) that is recognition-invariant
in the components. Moreover h0 can be chosen so as to be an isomorphism.
Proof: We describe a procedure to transform a given two-sorted multi-automaton (W, h) into one
with the desired properties. First transform (W, h) into its k-outreach (W 0 , h0 ) where k ∈ N is the
result of doing the k-outreach three times. First as in lemma 2.5.4 to make it projectively transitive,
then as in lemma 2.5.6 to make it a projection automaton and finally as in lemma 2.5.8 to make
it have substitutions. The resulting automaton (W 0 , h0 ) has all three of the above properties by the
corollarys to the three lemmas. Moreover it is finite when (W, h) is finite. This transformation is
recognition-invariant in the components by proposition 2.5.1. Moreover h0 is an isomorphism by
the definition of k-outreach. We then transform (W 0 , h0 ) into a one-sorted automaton (W 00 , h00 ), as
in proposition 2.4.8, the step for K2,ctr → K1,ctr . Here h00 = h0 and therefore is an isomorphism.
qed
As with proposition 2.4.8 we have two corollaries. The first regarding the versatility of PTPS
automata.
Corollary 2.6.4 Finite one-sorted PTPS automata are versatile enough to replace classical automata in
Büchis decision procedure. More specifically finite and concrete one-sorted projection automata (W, h)
where h is an isomorphism suffice.
The second regarding reachability.
Corollary 2.6.5 Let W = (V, . . .) be a finite concrete projection automaton. Let h from Mp (n, m)
onto V be an isomorphism. Define for each t ∈ P L(V) a corresponding t0 ∈ P L(Mp (n, m))
by t0 = h−1 (t(h(A))). Then in (W, h) the relation of being h-reachable by a word of the form
t0 (A0 ) _ · · · _ t0 (Al−1 ) is definable by the first-order formula ∃z[δ(q, t(z)) = q 0 ].
Note that there is no reference to the isomorphism h in the first-order formula. This means that
the formula defines the relevant reachability relation regardless of which particular isomorphism, h,
one chooses.
2.6.2
Properties of the one-sorted transition function
By proposition 2.6.3 one-sorted PTPS automata (W, h) where h is an isomorphism are as versatile
as the other classes of concrete automata we have looked at so far. Also by corollary 2.6.2 having
an alphabet that is isomorphic to some Mp (n, m) is first-order definable over finite structures. We
may therefore restrict attention to (W, h) where h is an isomorphism from now on.
Lemma 2.6.6 Let W = (V, δ, . . .) be a one-sorted n-tape p-multi-automaton. Let h be an n-fold
isomorphism from Mp (n, m) to V. Let the concrete automaton (W, h) have substitutions. Let σ̂ ∈
SL(Mp (n, m)). Let A0 , . . . , Ak−1 ∈ Mp (n, m). Then both the following hold for positive k ∈ N.
66
1. δh∗ (0, A0 _ · · ·
_
2. If δh∗ (0, A0 _ · · ·
Ak−1 ) .σ̂ δh∗ (0, σ̂(A0 ) _ · · ·
_
_
σ̂(Ak−1 )).
Ak−1 ) .σ̂ q then δh∗ (0, σ̂(A0 ) _ · · ·
_
σ̂(Ak−1 )) σ̂ q.
Proof:
1. Trivial by repeated use of 1. in the definition of having substitution.
2. This follows from the definition of σ̂ , i.e. definition 3.3.1, which by a suitable instantiation
of the variables says that
(a) if δh∗ (0, A0 _ · · ·
_
(b) δh∗ (0, A0 _ · · ·
Ak−1 ) .σ̂ q then
_
(c) δh∗ (0, σ̂(A0 ) _ · · ·
Ak−1 ) .σ̂ δh∗ (0, σ̂(A0 ) _ · · ·
_
_
σ̂(Ak−1 )) and
σ̂(Ak−1 )) σ̂ q
Here (a) follows from part 1. of this lemma and (b) is the assumption of part 2. of this
lemma.
qed
The following two lemmas are useful when in the next section we look at automata for recognising interpretations of formulae beginning with an existential quantifier.
Lemma 2.6.7 Let (V, δ, . . .) be a PTPS automaton. Let h be an isomorphism from Mp (n, m) to V.
Let i < n and A0 , . . . , Ak−1 ∈ Mp (n, m). Then for each state q the following are equivalent.
1. There exist B0 , . . . , Bl−1 ∈ Mp (n, m) s.t.
π̂i (A0 ) _ · · · _ π̂i (Ak−1 ) _ 0n×m _ · · · _ 0n×m =
π̂i (A0 ) _ · · · _ π̂i (Ak−1 ) _ π̂i (B0 ) _ · · · _ π̂i (Bl−1 ) and s.t.
δh∗ (0, A0 _ · · · _ Ak−1 _ B0 _ · · · _ Bl−1 ) = q.
2. There exists C ∈ Mp (n, m) s.t.
π̂i (A0 ) _ · · · _ π̂i (Ak−1 ) _ 0n×m =
π̂i (A0 ) _ · · · _ π̂i (Ak−1 ) _ π̂i (C) and s.t. δh∗ (0, A0 _ · · ·
_
Ak−1 _ C) = q.
Proof: 1. ⇒ 2. : Let q = δh∗ (0, A0 _ · · · _ Ak−1 _ B0 _ · · · _ Bl−1 ). Recall that π̂i replaces every
entry on the i’th row with a 0, therefore from 1. we may conclude that B0 _ · · · _ Bl−1 is an n-tape
whose entries are 0 except possibly on the i’th row. Let t ∈ P L(V) be the linear transformation that
replaces every entry not on the i’th row with a 0. Then t(B0 ) _ · · · _ t(Bl−1 ) = B0 _ · · · _ Bl−1 .
Now q is reachable from δh∗ (0, A0 _ · · · _ Ak−1 ) by a word of the form
t(B0 ) _ · · · _ t(Bl−1 ). Since (W, h) is projectively transitive there exists a C s.t. t(C) = C and
such that δh∗ (0, A0 _ · · · _ Ak−1 _ C) ∈ Tφ .
2. ⇒ 1. : Immediate when letting l = 1 and B0 = C.
qed
Lemma 2.6.8 Let (V, δ, . . .) be an PTPS automaton. Let h be an isomorphism from Mp (n, m) to
V. Associate t0 ∈ P L(Mp (n, m)) with each t ∈ P L(V) by the equation
t
t0 (A) = h−1 (t(h(A))). Let A0 , . . . , Ak−1 ∈ Mp (n, m). Then δh∗ (0, A0 _ . . . _ Ak−1 ) ∼ q iff
there exist B0 , . . . , Bk−1 ∈ Mp (n, m) such that
t0 (A0 ) _ · · · _ t0 (Ak−1 ) = t0 (B0 ) _ · · · _ t0 (Bk−1 ) and δh∗ (0, B0 _ . . . _ Bk−1 ) = q.
67
t
Proof: (If ): For tapes of length 1 we prove this as follows. By the definition of ∼ there is an x s.t.
δ(0, x) = q. Since h is an isomorphism this case of the lemma follows by letting B0 = h−1 (x).
For longer tapes we combine this with repeated use of the property of lemma 2.5.10.2.
(Only if ): This follows by repeated use of the property of lemma 2.5.10.1.
qed
2.7
The automata for a formula and its sub-formulae
Here we investigate how PTPS automata used in Büchis procedure for deciding a first-order sentence about the structure (N, +, |p ) relate. We are in particular interested in the relationship between the automaton associated with a formula and the automata associated with its sub-formulae.
We provide a sense in which this relationship is first-order.
The fact that PTPS automata are multi-automata is now put to use. We switch to using a set
of first-order formulae as index-set for the terminal states, as this is convenient when we set up a
correspondence between automata and relations defined by formulae. The following proposition
states that concrete one-sorted projection multi-automata are as versatile as finite sets of classical
automata.
Definition Let W = (V, δ, T ) be an n-tape p-automaton. Let h be an isomorphism from
Mp (n, m) to V. Then L(W, h) ⊆ Nn denotes is the relation recongised by (W, h).
Proposition 2.7.1 There is an effective mapping from the set of finite sets X of classical finite ntape p-automata (one- or two-sorted) to finite concrete PTPS automata, (W, h), such that for each
(W 0 , h0 ) ∈ X there exists a component (Wφ , h) of (W, h) such that L(W 0 , h0 ) = L(Wφ , h).
Proof: Let X be a finite set of classical finite n-tape p-automata. We now describe a procedure
to build a one-sorted n-tape p-multi-automaton out of X. By proposition 2.4.8 there exists a
recognition-invariant mapping, f from the classical automata (one- or two-sorted) to the finite
two-sorted concrete automata K2,con . By lemma 2.5.2 we are able to concatenate the elements of
f (X) into a finite two-sorted n-tape p-multi automaton, which by proposition 2.6.3 we can turn
into a one-sorted PTPS automaton with the desired property.
qed
Part of the procedure to decide a given n-variable formula, φ, in the language of (N, +, |p ), is
to associate one concrete n-tape p-automaton (Wψ0 , h0 ) with each sub-formula ψ of φ. By the just
proven proposition 2.7.1 there is for all φ in the language of (N, +, |p ), a concrete PTPS automaton
(W, h) such that for each sub-formula ψ of φ there is a component (Wψ , h) that recognises the
interpretation of ψ in (N, +, |p ). The components of (W, h) that recognise interpretations for
formulae and immediate sub-formulae are related as follows.
Proposition 2.7.2 Let W = (V, δ, {Tφ }φ∈Γ ) be an PTPS automaton. Let h be an isomorphism from
Mp (n, m) to V. Let R be the set of states reachable from the initial state, which since W is projectively
π̂i
transitive is the set of states q defined by the formula ∃x[δ(0, x) = q]. Consider the operations ∼,
.σ̂
and σ̂ as defined in the language of PTPS automata.
1. Negation: Let {φ, ¬φ} ⊆ Γ. Then
W |= ∀q ∈ R[T¬φ (q) ↔ ¬Tφ (q)]
iff
(W¬φ , h) recognises the complement of the relation recognised by (Wφ , h).
68
2. Disjunction: Let {φ, ψ, φ ∨ ψ} ⊆ Γ. Then
W |= ∀q ∈ R[Tφ∨ψ (q) ↔ (Tφ (q) ∨ Tψ (q))]
iff
(Wφ∧ψ , h) recognises the union of the relations recognised by (Wφ , h) and (Wψ , h).
3. Substituting variables for variables:
Let σ̂ ∈ SL(Mp (n, m)). Let {R(v0 , . . . , vn−1 ), R(vσ(0) , . . . , vσ(n−1) )} ⊆ Γ then
W |= ∀q ∈ R[TR(vσ(0) ,...,vσ(n−1) ) (q) ↔ ∃q 0 [q .σ̂ q 0 ∧ TR(v0 ,...,vn−1 ) (q 0 )]]
∧∀q 0 q 00 ∈ R[q 0 σ̂ q 00 → (TR(v0 ,...,vn−1 ) (q 0 ) ↔ TR(v0 ,...,vn−1 ) (q 00 ))]
iff
L(WR(vσ(0) ,...,vσ(n−1) ) , h) = {(x0 , . . . , xn−1 ) : (xσ(0) , . . . , xσ(n−1) ) ∈ L(WR(v0 ,...,vn−1 ) , h)}
4. Existential quantification:
Let i ≤ n. Let {φ, ∃vi φ} ⊆ Γ. Then
π̂
W |= ∀q ∈ R[(T∃vi φ (q) ↔ ∃q 0 , x[q ∼i q 0 ∧ 0 = π̂i (x) ∧ Tφ (δ(q 0 , x))]]
iff
(W∃vi φ , h) recognises the set of (x0 , . . . , xn−1 ) ∈ Nn s.t. there exists a
(y0 , . . . , yn−1 ) ∈ Nn that is equal to (x0 , . . . , xn−1 ) except possibly in the i’th component and
s.t. (Wφ , h) recognises (y0 , . . . , yn−1 ).
Proof: We treat n-ary relations on N as subsets of Mp (n, m)∗ _ On×N . Recall that the elements
of
Mp (n, m)∗ _ On×N are n-tapes where the entries on each row signifies the p-nary expansion of a
natural number. In the following we let A0 , . . . , Ak−1 ∈ Mp (n, m).
1. Negation: Make the assumptions of the lemma. Then (V, δ, T¬φ , h) recognises the tuple represented by A0 _ · · · _ Ak−1 iff δh∗ (0, A0 _ · · · _ Ak−1 ) ∈ T¬φ , which by the assumptions
of the lemma is iff it is not the case that δh∗ (A0 _ · · · _ Ak−1 ) ∈ Tφ , which is iff the tuple
represented by A0 _ · · · _ Ak−1 is not recognised by (V, δ, Tφ , h).
2. Disjunction: This is fairly similar to the negation case and left to the reader.
3. Substituting variables for variables: Let Rv and Rσv be short for the two atomic formulae
occurring in the proposition.
(if ) We prove the equality by showing that the sets are included in one another under the
assumption that W |= ∀q ∈ R[TRσv (q) ↔ ∃q 0 [q .σ̂ q 0 ∧ TRv (q 0 )]]. and the assumption that
W |= ∀q 0 q 00 ∈ R[q 0 σ̂ q 00 → (TR(v0 ,...,vn−1 ) (q 0 ) ↔ TR(v0 ,...,vn−1 ) (q 00 ))] We refer the first of
these this as the assumed equivalence and the second as the assumed coherence.
(⊆) Let A0 _ · · · _ Ak−1 represent a tuple (x0 , . . . , xn−1 ) ∈ L(WRσv , h). By definition
of acceptance and recognition this translates to δh∗ (0, A0 _ · · · _ Ak−1 ) ∈ TRσv . By the
assumed equivalence there exists a q 0 such that δh∗ (0, A0 _ · · · _ Ak−1 ) .σ̂ q 0 and such that
q 0 ∈ TRv . By part two of lemma 2.6.6 it is the case that δh∗ (0, σ̂(A0 ) _ · · · _ σ̂(Ak−1 )) σ̂ q 0
and by the assumed coherence δh∗ (0, σ̂(A0 ) _ · · · _ σ̂(Ak−1 )) ∈ TRv .
Hence (xσ(0) , . . . , xσ(n−1) ) ∈ L(WRv , h).
69
(⊇) Let σ̂(A0 ) _ · · · _ σ̂(Ak−1 ) represent a tuple (xσ(0) , . . . , xσ(n−1) ) ∈ L(WRv , h). By
definition of acceptance and recognition this translates to δh∗ (0, σ̂(A0 ) _ · · · _ σ̂(Ak−1 )) ∈
TRv . By part one of lemma 2.6.6 we get
δh∗ (0, A0 _ · · · _ Ak−1 ) .σ̂ δh∗ (0, σ̂(A0 ) _ · · · _ σ̂(Ak−1 )). By the assumed equivalence we
conclude that δh∗ (0, A0 _ · · · _ Ak−1 ) ∈ TRσv and hence that (x0 , . . . , xn−1 ) ∈ L(WRσv , h).
(only if ) For this direction we assume that
L(WR(σv) , h) = {(x0 , . . . , xn−1 ) : (xσ(0) , . . . , xσ(n−1) ) ∈ L(WRv , h)}. Then we show
coherence, i.e. that W |= ∀q 0 q 00 ∈ R[q 0 σ̂ q 00 → (TR(v0 ,...,vn−1 ) (q 0 ) ↔ TR(v0 ,...,vn−1 ) (q 00 ))]
and finally the rest. Our assumption we translate into a statement about infinite tapes as
follows: A _ 0n×N ∈ L(WRσv , h) iff σ̂(A) _ 0n×N ∈ L(WRv , h).
To show coherence we proceed as follows. Let q be an arbitrary reachable state, and q 0 , q 00 such
that q 0 σ̂ q 00 . By def of there must be symbols A and B such that δ(0, hA) = δ(0, hB) = q
and such that δ(0, σ̂hA) = q 0 and δ(0, σ̂hB) = q 00 . Now
q 0 ∈ TRv
by choosing δ(0, σ̂hA) = q 0
iff δ(0, σ̂hA) ∈ TRv
iff δh∗ (0, σ̂A) ∈ TRv
iff σ̂A _ 0n×N ∈ L(WRv , h)
iff A _ 0n×N ∈ L(WRσv , h)
by the translated assumption
iff δh∗ (0, A) ∈ TRσv
iff δh∗ (0, B) ∈ TRσv
since by choice of A and B we have δ(0, hA) = δ(0, hB) = q
iff B _ 0n×N ∈ L(WRσv , h)
iff σ̂B _ 0n×N ∈ L(WRv , h)
iff δh∗ (0, σ̂B) ∈ TRv
iff δ(0, σ̂hB) ∈ TRv
iff q 00 ∈ TRv
by choosing δ(0, σ̂hB) = q 00
Now we show that each of the directions in the equivalence in W |= ∀q ∈ R[TRσv (q) ↔
∃q 0 [q .σ̂ q 0 ∧ TRv (q 0 )]] holds.
(→) Let q be a reachable state such that q ∈ TRσv . Let A ∈ Mp (n, m) be a symbol by
which q is reached. This means that δ(0, h(A)) = q. From q ∈ TRσv we infer that the
number represented by A is in L(WRσv , h). We now define the q 0 that fulfils the right side
of ↔ by q 0 = δ(0, h(σ̂(A))). By definition of .σ̂ we have q .σ̂ q 0 . By the assumed equality
the tuple of natural numbers represented by σ̂(A) is a member of L(WRv , h). Therefore
δ(0, h(σ̂(A))) ∈ TRv . Since q 0 = δ(0, h(σ̂(A))) we get q 0 ∈ TRv .
70
(←) Again we let q be reachable and A ∈ Mp (n, m) such that δ(0, h(A)) = q. For this
direction we assume that there exists a q 0 such that q .σ̂ q 0 and q 0 ∈ TRv . By the definition of σ̂ we get δ(0, h(σ̂(A))) σ̂ q 0 and by the recently shown coherence it follows
that δ(0, h(σ̂(A))) ∈ TRv . Therefore the tuple of numbers which σ̂(A) represents is in
L(WRv , h). By the assumed equality this means that the tuple of numbers which A represents is in L(WRσv , h). Hence δ(0, h(A)) ∈ TRσv wich by definition of A means that
q ∈ TRσv .
4. Existential quantification:
π̂
(If ): Assume W |= ∀q ∈ R[T∃vi φ (q) ↔ ∃q 0 , x[q ∼i q 0 ∧ 0 = π̂i (x) ∧ Tφ (δ(q 0 , x))]].
(T∃vi φ not too big): Assume that there are A0 , . . . , Ak−1 ∈ Mp (n, m) such that
δh∗ (0, A0 _ · · · Ak−1 ) ∈ T∃vi φ . We now wish to display l ≥ k and B0 , . . . , Bl−1 ∈ Mp (n, m)
such that
A0 _ · · · _ Ak−1 _ 0n×m _ · · · _ 0n×m and B0 _ · · · _ Bl−1 are equal except possibly on the
i’th row and such that δh∗ (0, B0 _ · · · _ Bl−1 ) ∈ Tφ . To do this let l = k + 1. By the first of
π̂
the current two assumptions there exists a q 0 such that δh∗ (0, A0 _ · · · _ Ak−1 ) ∼i q 0 . Using
this equivalence we do by lemma 2.6.8 get B0 , . . . , Bk−1 ∈ Mp (n, m) with the property
that that A0 _ · · · _ Ak−1 and B0 _ · · · _ Bk−1 are equal except possibly on the i’th row.
Moreover δh∗ (0, B0 _ · · · _ Bk−1 ) = q 0 . By the firs assumption there also is an x such that
δ(q 0 , x) ∈ Tφ . Since h is an isomorphism we can define Bl−1 = h−1 (x) and we get the
B0 , . . . , Bl−1 we sought to display.
(T∃vi φ big enough): Assume that δh∗ (0, B0 _ · · · _ Bk−1 ) ∈ Tφ . By the definition of onesorted n-tape p-multi-automaton this implies that δh∗ (0, B0 _ · · · _ Bk−1 _ 0n×m ) ∈ Tφ . If
we now let both q and q 0 equal δh∗ (0, B0 _ · · · _ Bk−1 ) and let x = 0 we can from the first
assumption infer (from the right to the left side) that δh∗ (0, B0 _ · · · _ Bk−1 ) ∈ T∃vi φ .
(Only if ): Assume now that we have a PTPS automaton with two sets of terminal states that
incidentally are indexed like T∃vi φ and Tφ . Assume also that (V, δ, T∃vi φ , h) recognises the
set set of tapes A0 _ · · · _ Ak−1 _ 0n×N such that there exists a
B0 _ · · · _ Bl−1 _ 0n×N different from A0 _ · · · _ Ak−1 _ 0n×N in at most the i’th row
and such that δh∗ (0, B0 _ · · · _ Bl−1 ) ∈ Tφ . Note that if l < k + 1 we can extend
B0 _ · · · _ Bl−1 by adding 0n×m ’s at the most significant end obtaining a word whose membership status in Tφ is equivalent. Also if l > k + 1 we use the fact that our automaton is
projectively transitive as in lemma 2.6.7 and shorten B0 _ · · · _ Bl−1 so as to obtain a word
whose membership status in Tφ is equivalent and that differs from A0 _ · · · _ Ak−1 _ 0n×N
in at most the i’th row. Without loss of generality we therefore assume that l = k + 1.
Recall that π̂i ∈ P L(Mp (n, m)) is the projection that replaces every entry on the i’th row
with a 0. Using π̂i we translate the present assumptions to the following equivalence.
For all states of the form δh∗ (0, A0 _ · · · _ Ak−1 ) it is the case that
δh∗ (0, A0 _ · · · _ Ak−1 ) ∈ T∃vi φ iff there exists a state of the form δh∗ (0, B0 _ · · ·
that the conjunction of
(a) π̂i (A0 ) _ · · ·
_
π̂i (Ak−1 ) = π̂i (B0 ) _ · · ·
(b) 0n×m = π̂i (Bk )
71
_
π̂i (Bk−1 )
_
Bk ) such
(c) δh∗ (0, B0 _ · · ·
_
Bk ) ∈ Tφ
holds.
Since for each k the matrices {Ai }i<k are arbitrary we may replace δh∗ (0, A0 _ · · · _ Ak−1 )
with any state q reachable from the initial state in k steps, obtaining an equivalence as true
as the original one. Using lemma 2.6.8 and the introduced variable q, the conjunct a) is
π̂
equivalent to q ∼i δh∗ (0, B0 _ . . . _ Bk−1 ).
Since h is an isomorphism, we may replace the conjunct b) with 0 = π̂i (h(Bk )). Since Bk
is arbitrary, for fixed k, and h is surjective we may introduce a variable x to denote h(Bk ).
Accordingly the conjunct b) is in its context equivalent to 0 = π̂i (x).
With the introduced x we can write the conjunct c) as δ(δh∗ (0, B0 _ · · ·
_
Bk−1 ), x) ∈ Tφ .
If we now replace the conjuncts a),b) and c) with equivalents we are from our assumptions
able to establish the following.
for all q, reachable from the initial state in k steps, it is the case that q ∈ T∃vi φ
iff
there exists a state of the form δh∗ (0, B0 _ · · ·
π̂
q ∼i δh∗ (0, B0 _ · · ·
_
_
Bk−1 ) and an x such that
Bk−1 )
and 0 = π̂i (x)
and δ(δh∗ (0, B0 _ · · ·
_
Bk−1 ), x) ∈ Tφ .
We shall now soundly replace the occurrences of δh∗ (0, B0 _ · · · _ Bk−1 ) with a variable q 0 .
By lemma 2.6.8 we do, in the context where q is reachable from the initial state in k steps,
conclude as follows. The requirement that there exists a q 0 of the form δh∗ (0, B0 _ · · · _ Bk−1 )
π̂
such that q ∼i δh∗ (0, B0 _ · · · _ Bk−1 ) is equivalent to the requirement that there exists a q 0
π̂
such that q ∼i q 0 .
We have now shown that for all q, reachable from the initial state in k steps, it is the case
π̂
that: T∃vi φ (q) ↔ ∃q 0 , x[q ∼i q 0 ∧ 0 = π̂i (x) ∧ Tφ (δ(q 0 , x))]. Since k ∈ N was arbitrary we
are done.
qed
Note that in the above each construction on a PTPS automaton that corresponds to a logical
connective is definable using a first-order equivalence.
Corollary 2.7.3 Let Γ be the set of sub-formulae of a purely relational sentence φ. Let Γ0 be the set of
first-order definitions of constructions on automata for formulae that occur in Γ. Let Γ00 be the first-order
definition of PTPS automata. Then Γ has an automatic model iff Γ00 ∪ Γ0 ∪ {∀q ∈ R[Tφ (q)]} has a
finite model.
Proof:
72
1. If Γ has an automatic model then to each φ ∈ Γ we can assign a finite concrete twosorted n-tape p-(multi-)automaton (Wφ , hφ ) that recognises the definition of φ in the automatic model. Using proposition 2.7.1 we can merge these into one finite two-sorted multiautomaton (W, h) where each component is equivalent to a (Wφ , hφ ). The structure W is
finite and clearly a model for Γ00 ∪ Γ0 ∪ {∀q ∈ R[Tφ (q)]}.
2. If Γ00 ∪ Γ0 ∪ {∀q ∈ R[Tφ (q)]} has a finite model W then W is a multi-automaton. To make
W concrete we use a homomomorphism provided by lemma 2.2.2.
qed
2.8
Concluding remarks
With proposition 2.4.6, corollary 2.4.9 and corollary 2.4.10 we have shown that in finite transitive
automata reachability is first-order, we have shown that the class of transitive automata is finitely
axiomatisable. Moreover we have shown that the finite and concrete representatives of the class are
versatile enough to replace automata with fixed alphabets in J.R. Büchis procedure for deciding the
theory of automatic structures. With proposition 2.6.1, corollary 2.6.4 and corollary 2.6.5 we have
proven the same for PTPS automata.
We have investigated the relationship between a PTPS automaton that recognises the interpretation of a formula with the PTPS automata that recognise the interpretation of its sub-formulae.
With proposition 3.3.2 we have shown that this relationship is in a sense first-order. The proposition relies on first-order definability of various variants of reachability in PTPS automata.
PTPS-automata can be used to construct a semi-desicion procedure that terminates on input
on consistent sentences only, as follows.
1. Let Γ be the set of sub-formulae of the input sentence.
2. Let Γ00 ∪ Γ0 be the definition of the PTPS-automaton in corollary 2.7.3.
3. Search for a finite model for Γ00 ∪ Γ0 ∪ {∀q ∈ R[Tφ (q)]} and terminate if one is found.
In step 3 we use the fact that the class of PTPS-automata is basic elementary to make the procedure
terminate when the input sentence has a model definable in an automatic structure. This procedure
terminates on input on at least those infinity axioms that are true in Presburger arithmetic.
One idea of transforming a first-order sentence into a first-order description of automata and
then to search for a finite model of the transformed sentence has been proposed by N. Peltier
[Pel09]. To define reachability N. Peltiers transformation introduces the element-relation and lets
the carrier-set consist of sets of states. In contrast, the transformation outlined in the present paper
has a carrier-set whose members are symbols, some of which are also used as states. To define
reachability we allow for some flexibility in the set of symbols. N. Peltier says of his transformation
that, “The main interest of the present work is to prove that the translation is feasible from a
theoretical point of view”. The same can be said of the transformation proposed in the present
paper. To make search for satisfiable interpretations in automatic models remotely feasible from a
practical point of view the present author proposes to use the atom-strucures of finite representable
polyadic algebras. How to use PTPS-automata to compute such atom-structures is the subject of
the second paper.
73
Chapter 3
Automata for the computation of finite
representable polyadic algebras
75
3.1
Introduction
This is the second of two papers on a kind of automaton that is suitable for consistency proofs and
for the computation of atom-structures of finite representable polyadic algebras, including some
which have purely infinite spectrum. Finite representable algebras with infinite spectrum are of
interest in regards to Hilberts Entscheidungsproblem as the algebras can be used to construct semidecision procedures that recognise not only finitely satisfiable sentences as consistent but also some
infinity axioms, see A. Rognes [Rog09]. Infinity axioms are consistent first-order sentences that have
infinite models only. Devising reasonably natural semi-decision procedures that terminate on input
of even a single infinity axiom is a challenge. Note, for instance, that an eventually periodic infinite
branch of a semantic tree implies finite satisfiability. Similar effects occur with finitely presented
Herbrand models.
Polyadic algebras were introduced by P. R. Halmos who was inspired by A.Tarskis closely related
cylindric algebras. Finite and representable polyadic algebras are mathematical objects that can be
used much like structures and models when recognising given first-order sentences as consistent.
As with structures we may interpret relational sentences in these algebras and only consistent sentences have satisfying interpretations in representable algebras. Curiously some of the finite and
representable algebras have satisfying interpretations for infinity axioms. In computations involving infinity axioms explicitly presented models are generally unsuitable as arguments to computable
functions by virtue of being infinite. Finite representable algebras with satisfying interpretations for
infinity axioms however, work well by virtue of being finite. The atom-structures of finite algebras
are even more suitable as they are considerably smaller that the algebra it self. An atom-structure
plays the same role for a finite polyadic algebra as does a basis for a vector space or a topology.
The present paper, i.e. part two of two papers, can be read independently from part one if one
is willing to accept a few propositions without proof.
Part one makes no use of algebraic logic. In it we introduce a basic elementary class of multiautomata and see how these can be used for showing consistency of first-order sentences, including
some infinity axioms. Part two requires some knowledge of algebraic logic which we apply to the
multi-automata of part one. In part two we show that in the finite of the multi-automata we can
define atom-structures whose complex algebra are representable polyadic algebras. The class of
atom-structures in question is seen to be definable by a given finite set of first-order sentences in
the language of the multi-automata. The fact that these classes are basic elementary implies that
we are able to recursively enumerate the atom-structures of finite simple representable polyadic
algebras, without computing the the whole polyadic algebra in question. Simple algebras are of
interest as these are the basic building blocks of the class at hand. In the context of algebraic
logic, finite representable algebras are of interest in them selves, see H. Andréka and R.D. Maddux
[AM94]. In the context of Hilberts Entscheidungsproblem the atom-structures of finite simple
algebras whose spectrum is purely infinite, form a key component in the implementation of semidecision procedures that go beyond search for finite models, see A. Rognes[Rog09].
In the present paper we consider, the atom-structures of, n-dimensional one- and many-sorted
variants of polyadic algebras. For each kind of algebra we have a notion of homomorphism, embedding and sub-algebra. In accordance with the pattern of H. Andréka and R.D. Maddux [AM94] we
define the following. A set algebra is an algebra whose carrier set is a subset of P(U n ), the power-set
of U n . An algebra is representable if it is embeddable into a product of set algebras. A representable
76
algebra is simple if it is embeddable into a set algebra. The spectrum of a simple representable algebra
is the class of cardinalities κ such that there exists an embedding of the simple algebra to a set algebra with carrier set P(κ). To this we add that a simple representable algebra is said to have purely
infinite spectrum if there is no embedding of the algebra to any set algebra with carrier set P(U n )
for any finite U . The latter class of algebras has satisfying interpretations for infinity axioms.
Note that although the algebras considered here are boolean algebras with operators, shortened
BAO’s, the notion of representability used for BAO’s is to weak for consistency proofs for first-order
language in general. Indeed it follows from a well known result of B.Jónsson and A.Tarski that finite
BAO’s are representable over finite sets and so their spectra necessarily contain finite cardinals, see
B.Jónsson and A.Tarski [JT51] or a survey such as R. Goldblatt [Gol00], Theorem 3.1.
3.1.1
Outline of paper
The rest of section 1 introduces notation and recalls some definitions on directed many-sorted
polyadic algebras of dimension n, or dMsPsn for short, as defined in A. Rognes [Rog09].
In section 2 we define the polyadic atom-structure as known from the literature. We do however
introduce the h-complex algebra of a polyadic atom-structure, where h is a homomorphism of
polyadic atom-structures. The h-complex algebra generalises the complex algebra known from the
literature. A criterion on atom-structures is introduced and we prove that the h-complex algebra of
an atom-structure meeting the criterion is representable for a canonical h.
In section 3 we recall the definition of PTPS-automata as known from part one, i.e., A.Rognes
[Rog11]. We introduce properly partitioned automata which serve as a kind of normal form for
PTPS-automata. We identify a polyadic atom-structure with each properly partitioned automaton
and show that for any finite automaton this atom-structure necessarily meets the criterion of section
2, and therefore has a representable h-complex algebra.
In section 4 we show that every dMsPsn that is generated by a finite set of first-order definable relations over the structure (N, +, |p ) is embeddable in the h-complex algebra of a properly
partitioned finite automaton. Here N are the natural numbers, ’+’ is addition and |p is a binary
relation such that x|p y if y is the greatest power of p such that y divides x. We also show that the
above embeddability criterion does not hold for n-dimensional polyadic algebras, Psn , nor does it
hold for many-sorted algebras, MsPsn , unless they are directed. The situation is unchanged when
we consider the corresponding classes with diagonals, namely dMsPEsn , MsPEsn and PEsn .
In section 5 we provide finite sets of first-order axioms for four variants of properly partitioned
automata. These four axiom sets ensure that the h-complex algebra of (the atom-structure of )
a finite automaton is a finite dMsPsn , dMsPEsn , Psn or PEsn respectively. We prove some
result that are of interest in regards to recursively enumerating representatives for the finite members
of the four classes.
3.1.2
Sets, relations and mappings
Uppercase letters such as A, B, C, R, X, Y and Z are used for sets. X × Y , X ∪ Y , X ∩ Y , X\Y
and X ⊆ Y are as usual. A set R is a binary relation if it is a subset of X × Y . The domain of R is
the first projection of R, making it a, possibly proper, subset of X. R is said to be total if its domain
is X. If A ⊆ X then the image of A under R is {y : x ∈ A, (x, y) ∈ R}. We write R(A) for the
image of A under R. If B ⊆ Y then the inverse image of B under R is {x : y ∈ B, (x, y) ∈ R}.
77
We write R−1 (B) for the inverse image of B under R. The range of R is the image of X under R.
If (x, y) ∈ R and (x, y 0 ) ∈ R implies that y = y 0 then R is said to be a partial function from X
to Y . If a partial function happens to be total we call it a mapping from X to Y , or an operation.
The set of mappings from X to Y is written (X → Y ). The power-set of a set U is written P(U ).
If U is a set then B(U ) = (P(U ), ∪, −) is the full boolean set algebra over U . The closure of a
family F ⊆ P(U ) under finite unions is written Cl∪ (F). The notions signature, homomorphism,
embedding, (direct) product, expansion and reduction are used as is common in model theory.
3.1.3
Finite-dimensional (quasi) polyadic algebras
We consider finite-dimensional polyadic algebras in the present papers and we fixate a dimension
n ∈ N throughout. We also assume that 3 ≤ n, since for 0 ≤ n < 3 representable polyadic
algebras of dimension n are representable over finite sets. We make no distinction between quasi
polyadic algebras and polyadic algebras since the two notions coincide for finite-dimensions. There
is a variation of quasi polyadic algebras due to C. Pinter [Pin73] called quantifier algebras. These
in turn are called substitution-cylindric algebras by I. Németi [Ném91] and H. Andréka and I. Sain
and I. Németi [ASN01].
For cylindric algebras the result on representability over finite sets is due to L. Henkin, see
[MHT85] corollary 3.2.66. For (quasi) polyadic algebras it was proven independently by L.
Henkin, and by H. Andréka and I. Németi, see [MHT85] theorem 5.4.23. For simplified proofs
of both these results and some history see M. Marx and S. Mikulás [MM99].
We shortly define a many-sorted variant of quasi polyadic algebras . We begin with the signature.
Definition The tuple A = (Bn , . . . , B0 , r, p, s, c0 , . . . , cn−1 ) is said to have the signature of an
n-dimensional directed many-sorted polyadic algebra, or to be of dMsPn -signature for short, when
Bn , · · · , B0 are algebras which have the signature of a boolean algebra, i.e. for 0 ≤ i ≤ n we have
that Bi = (Bi , ∨Bi , ¬Bi , ⊥Bi ),
r, p, s : Bn → Bn are operations that are called rotation, permutation and substitution and relate
to variable substitutions,
for 0 ≤ i < n it is the case that ci : Bi+1 → Bi is an operation called cylindrification and it relates
to existential quantification.
Note that this signature differs slightly from that found in the literature in that we use only r, p and
s for the operations that relate to variable substitutions. This difference is not important as long as
the intended interpretation of the symbols suffices to generate all variable substitutions. Example
Let (K, <) be the rational numbers with the usual ordering. Let L3 (<) be the set of three-variable
formulae in the language for (K, <). The set of ternary relations definable using quantifier-free
formulae of L3 (<) generate an A of dMsPn -signature where
A A A
A = (B3 , B2 , B1 , B0 , rA , pA , sA , cA
0 , c1 , c2 , c3 ) is defined as follows.
B3 = (B3A , ∪, −, ∅) is the set algebra where B3 ⊆ P(K3 ) is the set of ternary relations definable
using quantifier-free formulae. This boolean algebra is known to have 13 atoms, which can
be verified by a geometric argument as follows. Take a cube and slice it along the three planes
78
defined by the formulae v0 = v1 , v1 = v2 and v0 = v2 . One then ends up with 6 tetrahedra,
wedged between these there are 6 triangles and finally there is a diagonal where the three
defining planes meet.
B2 = (B2A , ∪, −, ∅) is the set algebra where B2 ⊆ P(K3 ) is the set of ternary relations definable
using sentences of the form ∃v2 φ where φ defines a relation in B3 . This algebra has 3 atoms,
which can be visualised by slicing a cube along the plane defined by v0 = v1 .
B1 = (B1A , ∪, −, ∅) is the set algebra where B1 ⊆ P(K3 ) is the set of ternary relations definable
using sentences of the form ∃v1 φ where φ defines a relation in B2 . This algebra has 2
elements.
B0 = (B0A , ∪, −, ∅) is the set algebra where B0 ⊆ P(K3 ) is the set of ternary relations definable
using sentences of the form ∃v0 φ where φ defines a relation in B1 . This algebra has 2
elements.
When X ⊆ K3 we let
cA
0 (X) = {(x0 , x1 , x2 )|∃y(y ∈ K ∧ (y, x1 , x2 ) ∈ X)},
cA
1 (X) = {(x0 , x1 , x2 )|∃y(y ∈ K ∧ (x0 , y, x2 ) ∈ X)},
cA
2 (X) = {(x0 , x1 , x2 )|∃y(y ∈ K ∧ (x0 , x1 , y) ∈ X)}.
These operations are called cylindrifications for the reason that geometrically they would turn a ball
into a cylinder. In this example there are no balls, but
tetrahedra are turned into prisms,
the diagonal is turned into planes,
prisms are turned into prisms or the cube depending on the axis we cylindrify along,
planes are turned into planes or the cube depending on the axis we cylindrify along,
triangles are turned into prisms or planes depending on the axis we cylindrify along,
the cube and the empty set remain as they are.
When X ⊆ K3 we let
rA (X) = {(x0 , x1 , x2 )|(x2 , x0 , x1 ) ∈ X} geometrically X is rotated around the axis x0 = x1 =
x2 an angle a third of the full circle,
pA (X) = {(x0 , x1 , x2 )|(x1 , x0 , x2 ) ∈ X} geometrically X is mirrored in the xy plane,
sA (X) = {(x0 , x1 , x2 )|(x1 , x1 , x2 ) ∈ X} geometrically the part of X that meets the xy-diagonal
and that is orthogonal to xy plane is cylindrified.
As usual in algebraic logic we now generalise the last example to n dimensions and to as many n-ary
relations as possible on a single set U .
79
Definition Let U be a set. The full n-dimensional directed many-sorted polyadic set algebra over U or
the full dMsPsn over U for short, is the
U = (B(U n ), . . . , B(U n ), rU , pU , sU , cU0 , . . . , cUn−1 ) of dMsPn -signature where
B(U n ) = (P(U n ), ∪, −, ∅) is the boolean algebra of subsets of the set U n .
rU (X) = {(x0 , . . . , xn−1 ) ∈ U n |(xn−1 , x0 , x1 . . . , xn−2 ) ∈ X}, i.e. subtract one from each
index modulo n,
pU (X) = {(x0 , . . . , xn−1 ) ∈ U n |(x1 , x0 , x2 . . . , xn−1 ) ∈ X}, i.e. swap the first and the second
component,
sU (X) = {(x0 , . . . , xn−1 ) ∈ U n |(x1 , x1 , x2 . . . , xn−1 ) ∈ X}, i.e. overwrite the first component
with the second.
cU0 (X) = {(x0 , . . . , xn−1 ) ∈ U n |∃y(y ∈ U ∧ (y, x1 , . . . , xn−1 ) ∈ X}
..
.
cUi (X) = {(x0 , . . . , xn−1 ) ∈ U n |∃y(y ∈ U ∧ (x0 , . . . , xi−1 , y, xi+1 , . . . , xn−1 ) ∈ X}
..
.
cUn−1 (X) = {(x0 , . . . , xn−1 ) ∈ U n |∃y(y ∈ U ∧ (x0 , . . . , xn−2 , y) ∈ X}
Example The full dMsPs3 over K. This algebra has four uncountable sorts, all equal to P(K3 ).
The operations of this algebra are defined as in example 3.1.3. We are particularly interested in
finitely generated sub-algebras of full dMsPsn ’s over infinite sets, as these are suitable as objects of
computation as long as they are abstract, see A. Rognes [Rog09].
Definition Let A = (Bn , . . . , B0 , . . .) and A0 = (Bn0 , . . . , B00 , . . .) be two algebras of dMsPn signature. A dMsPn -homomorphism from A to A0 is a tuple f = (fn , . . . , f0 ) of mappings such
that
fn : Bn → Bn0 is a boolean homomorphism,
..
.
f0 : B0 → B00 is a boolean homomorphism,
fn preserves the operations r, p and s,
for i < n it is the case that fi (ci (x)) = c0i (fi+1 (x)).
The homomorphism f is said to be a dMsPn -embedding if each of f0 , . . . , fn is a boolean embedding.
Using embeddings we now define the objects which our main result is about.
80
Definition An algebra A of dMsPn -signature is said to be an n-dimensional many-sorted polyadic
set algebra, or a dMsPsn for short, if there exists a full dMsPsn , say U, and a dMsPn -embedding
from A to U.
Note that dMsPsn ’s are not required to be subsets of full algebras, only isomorphic to such. Example The algebra of dMsP3 -signature generated by the ternary relations definable by quantifier
free formulae in the language of (K, <) restricted to three variables is a dMsPs3 , see example
3.1.3. Example The full dMsPs3 over K is a dMsPs3 , see example 3.1.3.
3.1.4
Purely infinite spectrum
This subsection is about the connection between infinity axioms and algebras with a purely infinite
spectrum. It is not strictly necessary for understanding the present paper, but may serve as motivation. H. Andréka and R.D. Maddux [AM94] define spectrum on relation algebras that are both
simple and representable. This carries over to algebras of dMsPn -signature as follows.
Definition An algebra A of dMsPn -signature is said to be representable if there exists a possibly
infinite sequence of algebras A00 , . . . , A0κ ∈ dMsPsn and a dMsPn -embedding from A to A00 ×
. . . × A0κ .
Definition An algebra A = (Bn , . . . , B0 , . . . , c0 , . . . , cn−1 ) of dMsPn -signature is said to be
simple if
1. for each i < n we have that ci (x) = ⊥ iff x = ⊥,
2. for each i < n it is the case that Cl∪,− (ci (Bi+1 )) = Bi , i.e., Bi is the closure of ci (Bi+1 )
under the boolean operations,
3. c0 · · · cn−1 (Bn ) = {⊥, >}.
The last property ensures that a representable and simple algebra is embeddable into a single set algebra rather than a product. The simple and representable algebras of dMsPsn -signature therefore
are contained in the class dMsPsn .
Definition Let A be simple and representable. Then spec(A) is the class of cardinal numbers κ
such that there exists an embedding from A to the full dMsPsn over κ.
Definition Let A be simple and representable. Then A is said to have purely infinite spectrum if no
finite cardinal number is an element of spec(A).
The following is a proposition to make clear why algebras of purely infinite spectrum are of interest
to us.
Proposition 3.1.1 Let A be finite, simple and representable. Then A has purely infinite spectrum if
and only if there exists an infinity axiom with a satisfying interpretation in A.
81
Proof: We turn a fragment of first-order language into an algebra Lcrc
n of dMsPn -signature as
follows. The sort with index n consist of boolean combinations of atomic formulae built from
a countably infinite supply of n-ary relation symbols. The sort with index i consist of boolean
combinations of formulae of the form ∃xi φ, where φ is of the sort with index i + 1. See see
A. Rognes [Rog09] for further details. Interpretations of Lcrc
n -formulae are nothing more than
crc
dMsPn -homomorphisms from Ln to other algebras of dMsPn -signature. An interpretation f
is satisfying for a sentence φ if f0 (φ) = >.
(⇒): Assume that A = (Bn , . . .) has purely infinite spectrum. We construct an infinity axiom
φ of Lcrc
n , that has a satisfying interpretation f in A, by describing A up to isomorphism. For
this construction we need one n-ary relation symbol, R, for each element of Bn . The satisfying
interpretation of φ is uniquely determined by mapping the atomic formula Ra (x0 , . . . , xn−1 ) to
the element a ∈ Bn . Since A is simple there is for each element a ∈ Bi a formula ψ such that
fi (ψ) = a. This allows us to describe, in Lcrc
n , what the result of any operation of A is, on any
arguments. Since A is finite, the conjunction of these descriptions is again a sentence in Lcrc
n . The
conjunction is clearly satisfied by A and if it had a model over a finite set U then A would be
embeddable into the full dMsPsn over U . But A was assumed to have purely infinite spectrum
so the conjunction is an infinity axiom.
(⇐): Assume now that φ is an infinity axiom of Lcrc
n and let f be a satisfying interpretation, i.e.,
crc
a homomorphism from Ln to A with the property that f0 (φ) = >. Assume, for the purpose of
arriving at a contradiction, that there is a finite cardinal number in spec(A). This means that there
is a finite set U and a homomorphism f 0 from A into the full dMsPsn over U . The composition
of f with f 0 now is an interpretation of φ in the full dMsPsn over U . The composition maps φ
to > since homomorphisms preserve >. But then φ has a model whose carrier set is the finite set
U . This contradicts the assumption that φ is an infinity axiom.
qed
3.2
The polyadic atom-structure and the h-complex algebra
We define the polyadic analog of atom-structure, following the terminology of R. Hirsch and I. M.
Hodkinson, [Hod97] [HH09], who work with relation and cylindric algebras rather than polyadic
algebras. The polyadic version of atom-structures are used in the definition of complex algebra in
the book of L.Henkin, J.D.Monk and A.Tarski [MHT85], but called relational structure. Finite
polyadic atom-structures correspond to (finite many-sorted) polyadic algebras but atom-structures
are preferable as objects of computation as an atom-structure with k elements has a corresponding
algebra with 2k elements. Since n is fixed throughout, we simply write atom-structure to mean
n-dimensional polyadic atom-structure in this section.
3.2.1
Atom-structures and n-homomorphisms
Here the atom-structure is formally defined. Moreover we define a special kind of homomorphism
designed to correspond to dMsPn -embeddings. The homomorphisms are called n-homomorphisms
and are believed to be new.
Definition Let n ∈ N. An n-dimensional polyadic atom-structure is a tuple
(H, /r , /p , /s , E0 , . . . , En−1 ), where
82
H is a set of elements thought of as atoms of a boolean algebra.
/r , /p , /s are binary relations, which correspond to substitution of variables for variables.
E0 , . . . , En−1 are binary relations, which correspond to existential quantifiers.
By the following we associate an atom-structure with each atomic dMsPsn . The associated atomstructure plays the same role as does a basis associated with a vector space. Example Let A =
(Bn , . . . , B0 , r, p, s, c0 , . . . , cn−1 ) be a dMsPsn such that Bn , . . . , B0 are atomic. Then the atomstructure of A is the n-dimensional polyadic atom-structure (H, /r , /p , /s , E0 , . . . , En−1 ), defined
by the following.
H = At(Bn ) ∪ . . . ∪ At(B0 ), i.e. H is the union of the atoms of the boolean algebras Bn , . . . , B0 ,
for σ ∈ {r, p, s} it is the case that x /σ y iff σ(x) is defined and y ≤ σ(x), here ≤ is the usual
ordering of the boolean algebra Bn
for i ≤ n it is the case that xEi y iff ci (x) is defined and y ≤ ci (x).
The following is an explicit definition of the atom-structure of a full dMsPsn .
Definition Let U be a set. The n-dimensional set polyadic atom-structure over U is the tuple
U
(U n , /Ur , /Up , /Us , E0U , . . . , En−1
), where
U n is the set of n-tuples of elements of U .
/Ur ⊆ U n × U n is defined by /Ur = {(u, u0 )|(u0 , u1 , u2 . . . , un−1 ) = (u0n−1 , u00 , u01 , . . . , u0n−2 )},
i.e. subtract one from each index modulo n.
/Up ⊆ U n × U n is defined by /Up = {(u, u0 )|(u0 , u1 , u2 . . . , un−1 ) = (u01 , u00 , u02 , . . . , u0n−1 )}, i.e.
swap the first and the second component.
/Us ⊆ U n × U n is defined by /Us = {(u, u0 )|(u0 , u1 , u2 . . . , un−1 ) = (u01 , u01 , u02 , . . . , u0n−1 )}, i.e.
overwrite the first component with the second.
E0U = {(u, u0 )|(u1 , . . . , un−1 ) = (u01 , . . . , u0n−1 )}
..
.
EiU = {(u, u0 )|(u0 , . . . , ui−1 ) = (u00 , . . . , u0i−1 ) ∧ (ui+1 , . . . , un−1 ) = (u0i+1 , . . . , u0n−1 )}
..
.
U
En−1
= {(u, u0 )|(u0 , . . . , un−2 ) = (u00 , . . . , u0n−2 )}
Definition Let {r, p, s} be a set of formal symbols. Let H and H0 be atom-structures. An nhomomorphism from H to H0 is an n + 1 tuple of mappings h = (hn , . . . , h0 ) such that
for σ ∈ {r, p, s}, if q /σ q 0 then hn (q) /0σ hn (q 0 ).
for i < n, if qEi q 0 then hi+1 (q)Ei0 hi (q 0 ).
83
3.2.2
The complex algebra tailored for many sorts
We introduce a, believed to be new, generalisation of the complex algebra of an atom-structure
which is suitable for many-sorted polyadic algebras. The definition depends on an n-homomorphism, h, on an atom-structure H and is denoted Hh+ . If each component of h = (hn , . . . , h0 )
is the identity mapping on H then Hh+ is nothing more than the complex algebra H+ know form
the literature, see eg. R.Goldblatt [Gol00].
In the following definition we use the fact that a mapping h on a set H induces a partition of
H. A partition is a family of pairwise disjoin and nonempty subsets of H whose union is all of H.
The elements of the partition are called parts and the part that a given q ∈ H belongs to is written
[q]h .
Definition Let {r, p, s} be a set of formal symbols. Let H = (H, /r , /p , /s , E0 , . . . , En−1 ) and
0
H0 = (H 0 , /0r , /0p , /0s , E00 , . . . , En−1
) be polyadic atom-structures. Let h be an n-homomorphism
0
from H to H . Then the h-complex algebra of H , denoted Hh+ , is the algebra
+
(Bn , . . . , B0 , r+ , p+ , s+ , c+
0 , . . . , cn−1 ) of dMsPn -signature defined as follows.
1. For i < n let Bi = (Bi , ∪, −, ∅), where Bi = Cl∪ ({h−i
i hi q|q ∈ H}). Here Bi is the
closure under unions of the partition induced by hi . This makes Bi the boolean set algebra
generated by the partition induced by hi .
2. For σ ∈ {r, p, s} each σ + : Bn → Bn is defined first on parts, then on unions of parts as
follows
(a) for q ∈ H we let σ + ([q]hn ) = /σ ([q]hn ), i.e. the image of [q]hn under /σ .
(b) for Q ⊆ H let σ + ( {[q]hn |q ∈ Q}) = {σ + ([q]hn )|q ∈ Q}, here the last occurrence
of σ + is applied to parts on which σ + is already defined.
S
S
3. for i < n each c+
i : Bi+1 → Bi is defined first on parts, then on unions of parts as follows
(a) for q ∈ H we let c+
i ([q]hi+1 ) = Ei ([q]hi+1 ), i.e. the image of [q]hi+1 under Ei .
+
(b) for Q ⊆ H let c+
i ( {[q]hi+1 |q ∈ Q}) = {ci ([q]hi+1 ))|q ∈ Q}.
S
S
Definition Let H = (H, . . .) be an n-dimensional polyadic atom-structure. Let id : H → H be
the identity mapping. Let h = (id, . . . , id) be the identity n-homomorphism on H. Then the
complex algebra of H, denoted H+ , is the algebra Hh+ .
Lemma 3.2.1 Let U be a set. Let U be the n-dimensional set polyadic atom-structure over U . Then
U + is the full dMsPsn over U .
Proof: Left to the reader.
qed
We now give a criterion for a h-complex algebra to be a dMsPsn , and therefore representable,
rather than any odd algebra of dMsPn -signature. The criterion depends on a possibly infinite
external set polyadic atom-structure. The existence of such an infinite external set polyadic atomstructure may sometimes be inferred from intrinsic criteria. We consider one such intrinsic criterion
later in the present paper where we apply the following proposition to automata.
84
Proposition 3.2.2 Let {r, p, s} be a set of formal symbols. Let U be a set.
Let U = (U n , /Ur , . . . , E0U , . . .) be the n-dimensional set polyadic atom-structure over U . Let H =
(H, /r , . . . , E0 , . . .) and H0 = (H 0 , /0r , . . . , E00 , . . .) be polyadic atom-structures such that there exists
an n-homomorphism g from U to H, and an n-homomorphism h from H to H0 with the following two
properties.
1. For σ ∈ {r, p, s} and q ∈ H we have gn−1 (/σ ([q]hn )) = /σ (gn−1 ([q]hn )).
−1
2. For i < n, q ∈ H we have gi−1 (Ei ([q]hi+1 )) = Ei (gi+1
([q]hi+1 )).
Then the h-complex algebra Hh+ is representable (by an embedding into U + ).
Proof: We intend to show that Hh+ ∈ dMsPsn by displaying an embedding f from Hh+ to U + .
This implies representability. So define f = (fn , . . . , f0 ) from Hh+ to U + as follows. For each
i ≤ n we define fi : Cl∪ {[q]hi |q ∈ H} → Cl∪ {gi−1 ([q]hi )|q ∈ H} first on parts then on unions
of parts.
1. For q ∈ H let fi [q]hi = gi−1 [q]hi .
2. For Q ⊆ H let fi ( {[q]hi |q ∈ Q}) = {gi−1 [q]hi |q ∈ Q}.
S
S
The fact that each fi is a boolean embedding is easy to see, so we skip the formal proof. For
σ ∈ {r, p, s} we now prove that fn (σ(X)) = σ(fn (X)), first for parts then for unions of parts.
The first occurrence of σ is defined on Hh+ and the second on U + .
1. Let q ∈ H. Now
u ∈ fn σ[q]hn
iff u ∈ gn−1 σ[q]hn
by definition of f
iff u ∈ gn−1 (/σ [q]hn )
by definition of σ on parts of Hh+
iff u ∈ /Uσ (gn−1 [q]hn )
by the first of the assumed properties of g and h
iff u ∈ σgn−1 [q]hn
σ is defined in the full dMsPsn over U , see lemma 3.2.1
iff u ∈ σfn [q]hn .
From this we conclude that fn (σ(X)) = σ(fn (X)) when X is a part.
2. Let Q ⊆ H. Now
S
fn (σ(
q∈Q [q]hn ))
σ([q]hn ))
by definition of σ on unions of parts of Hh+
q∈Q
fn (σ([q]hn ))
fn was just seen to be boolean embedding
q∈Q
σ(fn ([q]hn ))
fn and σ were just seen to commute on parts
fn ([q]hn ))
by definition of σ on unions of parts of U + , recall lemma 3.2.1
q∈Q [q]hn )).
fn was just seen to be a boolean embedding
S
= fn (
=
S
=
S
S
= σ(
q∈Q
q∈Q
= σ(fn (
S
From this we conclude that fn (σ(X)) = σ(fn (X)) when X is a union of parts.
For each i ∈ N we can prove that fi+1 (ci (X)) = ci (fi (X)) for parts and unions of parts, X, in
exactly the same way.
qed
85
3.3
The h-complex algebra of a multi-automaton
We define PTPS-automata as introduced in A. Rognes [Rog11] and define operations on them
so as to make them a polyadic atom-structure. This allows us to define the h-complex algebra
of a PTPS-automaton. We use proposition 3.2.2 to show that the h-complex algebra of a finite
PTPS-automaton necessarily is representable.
3.3.1
Concrete PTPS-automata defined
We recall definitions and some basic facts on PTPS-automata here, see A.Rognes [Rog11] for further details. PTPS-automata use matrices over finite fields both as alphabet and as states. We write
Fp for the unique finite field whose order is the prime power p. Moreover Mp (n, m) is the set of n
times m matrices with entries form Fp . We let 0n×m ∈ Mp (n, m) be the matrix whose entries are
all 0 ∈ Fp . Likewise 0n×N is the matrix with n infinite rows whose entries are all 0 ∈ Fp .
S
By definition the set Mp (n, m)∗ = k∈N Mp (n, k · m). The concatenation of the two matrices
A, A0 ∈ Mp (n, m)∗ is written A _ A0 and is an element of Mp (n, m)∗ . We write Mp (n, m)∗ _ {0n×N }
for the set of n times N matrices with finite support and call them n-tapes.
The set Mp (n, m) is a vector-space over Fp when the operations are defined component-wise
and the tuple Mp (n, m) stands for this vector-space. We let P L(Mp (n, m)) stand for the set of
linear-transformations on Mp (n, m) that replace every entry on a given set of rows with 0’s. The
following is an example of an element of P L(Mp (n, m)) which is of special interest to us.
Definition The mapping π : Mp (n, m) → Mp (n, m) is defined by




π(


A0
A1
..
.
An−1





)








=
A0
01×m
..
.
01×m







We let SL(Mp (n, m)) stand for the set of linear-transformations on Mp (n, m) that swaps or
overwrites rows with other rows according to some mapping σ on row-indices. The following is an
example of an element of SL(Mp (n, m)) which also is of special interest to us.
Definition The mapping r : Mp (n, m) → Mp (n, m) is defined by




r(


A0
A1
..
.
An−1




)




A1

.. 



. 
=



An−1 

A0
Using these two definition we now define a class of special vector-spaces which serve as alphabet
(and states) of PTPS-automata.
Definition A concrete n-fold vector-space is a tuple
Vp,n,m = (Mp (n, m), π, r) where Mp (n, m), π, r are defined as above.
86
We consider p, n, m as fixed throughout so we mostly write V rather than Vp,n,m .
We write On×N for the concrete one-element n-fold vector-space over Fp whose sole element is
the infinite n-tape whose entries are all 0. What follows are the objects that PTPS-automata devour.
It is the set of infinite n-tapes with operations that allow us to compare and overwrite tracks on the
tape, i.e. rows of matrices with finite support.
∗
_
Definition Vp,n,m
On×N is the concrete n − f old vector-space whose carrier set is
Mp (n, m)∗ _ {0n×N }, where the vector-space operations are defined component-wise and where
π(A0 _ · · ·
_
Ak−1 _ 0n×N ) = π(A0 ) _ · · ·
_
π(Ak−1 ) _ 0n×N
and
r(A0 _ · · ·
_
Ak−1 _ 0n×N ) = r(A0 ) _ · · ·
_
r(Ak−1 ) _ 0n×N
Each track holds the p-nary expansion of a natural number. Thus there is a natural one to one
∗
_
correspondence between Vp,n,m
On×N and n-tuples of natural numbers.
We shall define PTPS-automata in two steps. First we define its signature then after having
defined some notions on the signature we introduce the axioms in a second step.
Definition Let J be a finite set. A concrete automaton of PTPS-signature is a
W = (V, δ, {Tj }j∈J ) s.t.
V = (V, +, . . .) is a concrete n-fold vector-space thought of as both alphabet and state set,
δ : V × V → V is the transition function,
for each j ∈ J the set Tj ⊆ V is a set of terminal states.
We mention that the zero-vector of V is used as the initial state. We will use variables such as
q, q 0 , q 00 . . . for elements of the vector-space when we think of them as states. When we think
of them as symbols we use A, A0 , B, C, A0 , . . . instead. Automata recognise sets of n-tapes, and
therefore sets of n-tuples of numbers, by means of the following.
Definition Let W = (V, δ, {Tj }j∈J ) be of PTPS-signature. Let V = (V, . . .). Then:
1. δ ∗ : V × V ∗ → V is defined by δ ∗ (q, A _ A0 ) = δ(δ ∗ (q, A), A0 ),
2. Wj = (V, δ, Tj ) and is called the j-th projection of W.
3. Wj recognises A ∈ Mp (n, m)∗ if δ ∗ (0, A) ∈ Tj .
We now define two relations on automata that relate to the existential quantifier.
Definition Let π̂i ∈ P L(V) be the linear transformation that replaces every entry on the i-th row
with a 0. Let t ∈ P L(V) be arbitrary. Then,
t
t
1. ∼⊆ V × V is defined by q ∼ q 0 if there exist A, A0 ∈ V such that t(A) = t(A0 ) and
δ(0, A) = q and δ(0, A0 ) = q 0 ,
π̂
2. Ei ⊆ V × V is defined by qEi q 0 if δ(q, 0) ∼i q 0 and q can be reached from the initial state
in one step, i.e., there exists an A ∈ V such that q = δ(0, A).
87
Similarly we define two relations on automata that relate to substitution of variables for variables.
Definition Let σ̂ ∈ SL(V), i.e., σ̂ is a linear transformation that swaps and overwrites columns
according to some mapping on track indices. Then,
1. /σ̂ ⊆ V × V is defined by q /σ̂ q 0 is there exists an A ∈ V such that δ(0, σ̂(A)) = q and
δ(0, A) = q 0 ,
2. σ̂ ⊆ V × V is defined by q σ̂ q 0 if there exists a q 00 ∈ V such that q /σ̂ q 00 and q 0 /σ̂ q 00 .
As was seen in A.Rognes [Rog11] the elements of P L(V) and SL(V) are definable using the
operations of automata of PTPS-signature, more specifically the operations π, r and +. It follows
t
that for each t and σ the relations ∼, Ei , σ̂ , /σ̂ are definable as well. We use this observation to
shorten the definition of PTPS-automata now.
Definition A concrete PTPS-automaton is a W = (V, δ, {Tj }j∈J ) of concrete PTPS-signature such
that, For each j ∈ J, each t ∈ P L(V) and σ̂ ∈ SL(V) the following axioms hold:
1. W |= Tj (q) ↔ Tj (δ(q, 0)), to ensure well definedness of relations recognised by automata,
2. W |= ∀qAB∃C(δ(δ(q, t(A)), t(B)) = δ(q, t(C))), i.e., a state reachable by a tape of the
form t(A) _ t(B) is also reachable by a tape of the form t(C),
t
t
3. W |= ∀qq 0 AB(q ∼ q 0 ∧ t(A) = t(B) → δ(q, A) ∼ δ(q 0 , B)), e.g, if t = π̂i and A _ A0 and
π̂
B _ B 0 are tapes that are the same except on the i-th row then δ ∗ (0, A _ A0 ) ∼i δ ∗ (0, B _ B 0 ),
t
t
4. W |= ∀qq 0 A(δ(q, A) ∼ q 0 → ∃q 00 B(q ∼ q 00 ∧ t(A) = t(B) ∧ δ(q 00 , B) = q 0 )), e.g., if
π̂
t = π̂i and δ ∗ (0, A _ A0 ) ∼i q 0 then there exists a tape B _ B 0 that is the same as A _ A0
except possibly on the i-th row such that δ ∗ (0, B _ B 0 ) = q 0 see A.Rognes [Rog11],
5. W |= ∀qq 0 A(q 0 /σ̂ q → δ(q 0 , σ̂(A)) /σ̂ δ(q, A)), this is similar to 3,
6. W |= ∀qq 0 A(q 0 /σ̂ δ(q, A) → ∃q 00 (q 00 /σ̂ q ∧ δ(q 00 , σ̂(A)) = q 0 )) this is similar to 4.
The first axiom schema here is to ensure that we can extend the definition of recognition from finite
tapes to infinite tapes with finite support, and thus tuples of natural numbers, as follows.
∗
_
Definition For A _ 0n×N ∈ Vp,n,m
On×N we say that Wj = (V, δ, Tj ) recognises A _ 0n×N if
δ ∗ (0, A) ∈ Tj .
Note that the definition works regardless of which of the many possible p-nary expansions A for
a given tuple of numbers we have chosen, i.e., regardless of how many 0’s there are at the most
significant end of A.
It is known that with each definable relation of an automatic structure, we can associate a
classical finite n-tape p-automaton that recognises exactly the tapes that represent tuples of the
definable relation, see, J.R. Büchi [B6̈0]. We recall one of the main results from A.Rognes [Rog11],
namely Proposition 7.2. The result allows us to merge finite sets of classical automata into one
finite concrete PTPS-automaton.
88
Proposition 3.3.1 There is an effective mapping from the set of finite sets X of classical n-tape pautomata to the set of finite concrete PTPS-automata W = (V, δ, {Tj }j∈J ), such that for each W 0 ∈ X
there exists a j ∈ J such that Wj recognises the same relation as does W 0 .
Proof: We outline a proof, for a full proof see A.Rognes [Rog11]. The mapping is defined by
constructing a PTPS-automaton as follows. We take the product of the automata in X and obtain
a two-sorted multi-automaton Π(X). The alphabet of Π(X) is Mp (n, 1). We then replace the the
alphabet of Π(X) with one of the form Mp (n, m) where m is chosen so as to make the resulting
automaton a two-sorted PTPS-automaton, W. Finally we imbed the states of W into the alphabet
so as to obtain a one-sorted PTPS-automaton.
qed
We also recall proposition 7.3.3 in A.Rognes [Rog11]. The proposition tells us how the components
of a PTPS-automaton, that recognise interpretations for first-order formulae and immediate subformulae are related. Note that q 0 /σ̂ q here means the same as q .σ̂ q 0 in A.Rognes [Rog11]. We
use a set of formulae Γ as index set for the sole purpose of increasing readability.
Proposition 3.3.2 Let W = (V, δ, {Tφ }φ∈Γ ) be a concrete PTPS automaton. Let R be the set of
states reachable from the initial state, which since W is projectively transitive is the set of states, q,
π̂i
/σ̂ and σ̂ as defined in the
defined by the formula ∃x[δ(0, x) = q]. Consider the operations ∼,
language of PTPS automata.
1. Negation: Let {φ, ¬φ} ⊆ Γ. Then
W |= ∀q ∈ R[T¬φ (q) ↔ ¬Tφ (q)]
iff
W¬φ recognises the complement of the relation recognised by Wφ .
2. Disjunction: Let {φ, ψ, φ ∨ ψ} ⊆ Γ. Then
W |= ∀q ∈ R[Tφ∨ψ (q) ↔ (Tφ (q) ∨ Tψ (q))]
iff
Wφ∧ψ recognises the union of the relations recognised by Wφ and Wψ .
3. Substituting variables for variables:
Let σ : n → n and let σ̂ ∈ SL(Mp (n, m)) be the operation that swaps and overwrites rows
according to the mapping σ. Let {R(v0 , . . . , vn−1 ), R(vσ(0) , . . . , vσ(n−1) )} ⊆ Γ then
W |= ∀q ∈ R[TR(vσ(0) ,...,vσ(n−1) ) (q) ↔ ∃q 0 [q 0 /σ̂ q ∧ TR(v0 ,...,vn−1 ) (q 0 )]]
∧∀q 0 q 00 ∈ R[q 0 σ̂ q 00 → (TR(v0 ,...,vn−1 ) (q 0 ) ↔ TR(v0 ,...,vn−1 ) (q 00 ))]
iff
WR(vσ(0) ,...,vσ(n−1) ) recognises {(x0 , . . . , xn−1 ) : (xσ(0) , . . . , xσ(n−1) ) ∈ L(WR(v0 ,...,vn−1 ) )}
where L(WR(v0 ,...,vn−1 ) ) is the set recognised by WR(vσ(0) ,...,vσ(n−1) ) .
4. Existential quantification:
Let i ≤ n. Let {φ, ∃vi φ} ⊆ Γ. Then
π̂
W |= ∀q ∈ R[(T∃vi φ (q) ↔ ∃q 0 , x[q ∼i q 0 ∧ 0 = π̂i (x) ∧ Tφ (δ(q 0 , x))]]
iff
W∃vi φ recognises the set of (x0 , . . . , xn−1 ) ∈ Nn s.t. there exists a
(y0 , . . . , yn−1 ) ∈ Nn that is equal to (x0 , . . . , xn−1 ) except possibly in the i’th component and
s.t. Wφ recognises (y0 , . . . , yn−1 ).
89
3.3.2
Properly partitioned automata
We introduce the notion of a properly partitioned automaton. The notion can be thought of as
a normal form for PTPS-automata. Automata of this form can be turned into polyadic atomstructures fairly directly.
The second axiom schema, in the definition of PTPS-automata, allows us to give a first order
definition of reachability. We continue using the symbol R for the set of reachable states, for which,
in the case of PTPS-automata, we can give a first-order definition as follows. The set R ⊆ V
consists of the states q such that ∃A(δ(0, A) = q).
Definition Let W = (V, δ, {Tj }j∈J ) be a concrete PTPS-automaton. Then W is said to be
properly partitioned if
1. {Tj }j∈J is a partition of R,
2. For each j ∈ J and σ̂ ∈ SL(V) we have W |= ∀q 0 q 00 ∈ R(q 0 σ̂ q 00 → (Tj (q 0 ) ↔ Tj (q 00 ))).
We simply say properly partitioned automaton to mean a properly partitioned PTPS automaton. The
reason that we may think of properly partitioned automata as being in normal form is the following
proposition.
Proposition 3.3.3 Let W = (V, δ, {Tj }j∈J ) be a PTPS-automaton such that:
1. J is finite,
2. For each j ∈ J there exists a k ∈ J such that Tj ∪ Tk = R
3. For each σ : n → n and Wj that recognises the relation R(v0 , . . . , vn−1 ) there exists a k ∈ J
such that Wk recognises R(vσ(0) , . . . , vσ(n−1) ),
Then there is a properly partitioned automaton W 0 = (V, δ, {Tj0 }j∈J 0 ) such that {Tj0 }j∈J 0 generates
the same boolean algebra as does {Tj }j∈J .
Proof: The proposition follows when we let {Tj0 }j∈J 0 consist of the atoms of the boolean algebra
T
that {Tj }j∈J generates. Then each Tj00 = {Tj }j∈K for some K ⊆ J. Now clearly condition 1 of
being properly partitioned holds. If we now assume that condition 2 of being properly partitioned
does not hold then for some σ̂ ∈ SL(V) and q 0 , q 00 ∈ R it is the case that q 0 σ̂ q 00 and q 0 ∈ Tj00
whilst q 00 ∈
/ Tj00 . Therefore for some j ∈ K it must be the case that q 0 ∈ Tj whilst q 00 ∈
/ Tj . If
we let R(v0 , . . . , vn−1 ) denote the relation that Aj recognises then by assumption 3 of the present
proposition there is an automaton Ak that recognises R(vσ(0) , . . . , vσ(n−1) ). But this is in violation
with proposition 3.3.2, which states that if Wk recognises R(vσ(0) , . . . , vσ(n−1) ) and Wj recognises
R(v0 , . . . , vn−1 ) then Tj is closed under σ̂ .
qed
3.3.3
Automata as atom-structures
We carefully select the binary relations that occur in the signature of an n-dimensional polyadic
atom-structure amongst the relations definable on a properly partitioned automaton.
Definition The three mappings r̂, ŝ, p̂ ∈ SL(Vp,n,m ) are defined as follows.
90




r̂(






ŝ(






p̂(


A0
A1
..
.
An−1
A0
A1
..
.
An−1
A0
A1
..
.
An−1





)








=





)








=





)








=
An−1
A0
..
.
An−2
A1
A1
..
.
An−1
A1
A0
..
.
An−1




 i.e.


each row is shifted one down,




 i.e.


the second row is copied to the first row,







i.e. the first and second rows are swapped.
For the following recall the definitions of /σ̂ and Ei in section 3.3.1.
Definition Let W = (V, δ, {Tj }j∈J ) be a PTPS-automaton. Then (R, /r̂ , /ŝ , /p̂ , {Ei }i∈n ) is
called the atom-structure of W.
Note that the atom-structure of a PTPS-automaton coincides with the atom-structure of the properly partitioned automaton of a PTPS-automaton since the definitions of R , /σ̂ and Ei depend
only on V and δ. Since we have a fairly canonical way of turning PTPS-automata into properly
partitioned automata we will be concerned with the latter from now on.
3.3.4
The n-homomorphism induced by an automaton
We define an n-homomorphism, h, that allows us to show that the h-complex algebra of the atomstructure of a properly partitioned automaton is representable. The n-homomorphism is defined
by means of the following sequence of partitions.
Definition Let W = (V, δ, {Tj }j∈J ) be a properly partitioned automaton, with V = (V, . . .). For
each i ≤ n we define a partition of R, where the part containing a given q ∈ R is written [q]i .
1. When i = n we define [q]i as follows. If q ∈ Tj then [q]n = Tj .
2. When i < n we let [q]i = {q 0 |q 0 ∈ R ∧ δ(q 0 , 0) = δ(q, 0)}.
Note that for i, i0 < n the partitions are the same, i.e., [q]i = [q]i0 . If for each i ≤ n we now define
a mapping hi : R → P(R) by hi (q) = [q]i we may trivially turn P(R), i.e., the power-set of the
reachable states, into a polyadic atom-structure such that h is an n-homomorphism.
Definition Let W = (V, δ, {Tj }j∈J ) be a properly partitioned automaton. Then the sequence
(hn , . . . , h0 ) where for i ≤ n and q ∈ R the mapping hi (q) = [q]i is called the n-homomorphism
induced by W.
Definition The h-complex algebra of a properly partitioned automaton W is the h-complex algebra
of W where h is the n-homomorphism induced by W.
91
3.3.5
Representability of the h-complex algebra of an automaton
Using proposition 3.2.2, we show that if h is the n-homomorphism induced by an automaton W,
then the h-complex algebra of W is representable. To do this we use the following three items. We
∗
_
use Vp,n,m
On×N which, by virtue of a by now obvious isomorphism, we think of as the atomstructure of the full n-dimensional polyadic set algebra over the natural numbers. It corresponds
to U in proposition 3.2.2. We use the n-homomorphism induced by W, which corresponds to
h in proposition 3.2.2. Finally we use a mapping γ which we define now. It corresponds to g in
proposition 3.2.2.
Definition Let W = (V, δ, {Tj }j∈J ) be a properly partitioned automaton, where V = (V, . . .).
Then γ : V ∗ _ {0n×N } → V is defined by
1. γ(0n×N ) = δ(0, 0),
2. If A ∈ V ∗ has length greater than 0 then γ(A _ 0n×N ) = δ(δ ∗ (0, A), 0).
Note that γ provides a mapping from representatives of Nn to states that is independent of which
particular representative we choose, i.e. how many 0’s there are at the most significant end.
Here is a definition that shortens the upcoming proofs.
Definition For σ̂ ∈ SL(Vp,n,m ) we define σ̂ ∗ : Mp (n, m)∗ → Mp (n, m)∗ as follows. If A0 _ · · ·
Mp (n, m)k then σ̂ ∗ (A0 _ · · · _ Ak−1 ) = σ̂(A0 ) _ · · · _ σ̂(Ak−1 ).
_
Now a lemma which we shortly use to prove the main result of the present paper. In the following
the the expression σ̂ ∗ (A) _ 0n×N /Uσ̂ A _ 0n×N means that the n-tuples of natural numbers that
σ̂ ∗ (A) and A represent are in /Uσ̂ relation, see the definition of set polyadic atom-structure.
Lemma 3.3.4 Let W = (V, δ, {Tj }j∈J ) be a properly partitioned automaton. Let σ̂ ∈ {r̂, ŝ, p̂}. Let
V = (V, . . .) and let A ∈ V ∗ . Then the following two are equivalent.
1. There exists a q 0 such that q 0 ∈ [q]n and such that q 0 /σ̂ γ(A _ 0n×N ).
2. γ(σ̂ ∗ (A) _ 0n×N ) ∈ [q]n and σ̂ ∗ (A) _ 0n×N /Uσ̂ A _ 0n×N .
Proof: (1 implies 2): Let q 0 ∈ [q]n and q 0 /σ̂ γ(A _ 0n×N ). By definition of γ this means
that q 0 /σ̂ δ(δ ∗ (0, A), 0). By point 5 in the definition of PTPS-automaton we also have that
δ(δ ∗ (0, σ̂ ∗ (A)), 0) /σ̂ δ(δ ∗ (0, A), 0). Therefore by the definition of σ̂ it is the case that q 0 σ̂
δ(δ ∗ (0, σ̂ ∗ (A)), 0). Since W is properly partitioned, [q]n = Tj for some j ∈ J which is closed
under σ̂ , we have δ(δ ∗ (0, σ̂ ∗ (A)), 0) ∈ [q]n . By definition of γ we now have γ(σ̂ ∗ (A) _ 0n×N ) ∈
[q]n . The rest of this direction follows from the definition of /Uσ̂ .
(2 implies 1): Immediate when letting q 0 = γ(σ̂ ∗ (A) _ 0n×N ).
qed
We now show a result which together with proposition 3.2.2 means that the h-complex algebra of
a properly partitioned automaton is representable.
Proposition 3.3.5 Let W = (V, δ, {Tj }j∈J ) be a properly partitioned automaton. Let V = (V, . . .)
and q ∈ R. Then
92
Ak ∈
1. for σ̂ ∈ {r̂, ŝ, p̂} we have γ −1 (/σ̂ ([q]n )) = /Uσ̂ (γ −1 ([q]n ))
2. and for i < n we have γ −1 (Ei ([q]i+1 )) = EiU (γ −1 ([q]i+1 ))
Proof:
1. Dropping the superscript U, we show that for A ∈ V ∗ we have that
A _ 0n×N ∈ γ −1 (/σ̂ ([q]n )) iff A _ 0n×N ∈ /σ̂ (γ −1 ([q]n )). So assume:
A _ 0n×N ∈ γ −1 (/σ̂ ([q]n ))
iff γ(A _ 0n×N ) ∈ /σ̂ ([q]n )
by definition of inverse image
iff there exists a q 0 s.t. q 0 ∈ [q]n and q 0 /σ̂ γ(A _ 0n×N ) this is what lying in the /σ̂ -image of
[q]n means
iff γ(σ̂ ∗ (A) _ 0n×N ) ∈ [q]n and σ̂ ∗ (A) _ 0n×N /Uσ̂ A _ 0n×N
iff γ(σ̂ ∗ (A) _ 0n×N ) ∈ [q]n
by lemma 3.3.4
right conjunct follows from the definition of /Uσ̂
iff σ̂ ∗ (A) _ 0n×N ∈ γ −1 ([q]n )
by definition of inverse image
iff A _ 0n×N ∈ /σ̂ (γ −1 ([q]n ))
compare the definitions of σ̂ and /Uσ̂
2. This time we show that for A ∈ V k where k ∈ N and i < n we have that A _ 0n×N ∈
γ −1 (Ei (δ(q, 0))) iff A _ 0n×N ∈ Ei (γ −1 (δ(q, 0))). The theorem now follows for those [q]i
where i < n, since then [q]i = {q 0 |q 0 ∈ R ∧ δ(q 0 , 0) = δ(q, 0)}. The theorem also follows
for [q]n since [q]n = [δ(q, 0)]n by the first axiom schema in the definition of PTPS-automata.
We introduce the following notation for this proof, if A = A0 _ · · ·
π̂i (A0 ) _ · · · _ π̂i (Ak−1 ).
_
Ak−1 then πi∗ (A) =
We prove the equivalence in two separate directions. For the first direction assume:
A _ 0n×N ∈ γ −1 (Ei (δ(q, 0)))
then γ(A _ 0n×N ) ∈ Ei (δ(q, 0))
then δ(q, 0)Ei γ(A _ 0n×N )
by the definition of image of a binary relation
then δ(q, 0)Ei δ(δ ∗ (0, A), 0)
by definition of γ
π̂
π̂
then δ(δ(q, 0), 0) ∼i δ(δ ∗ (0, A), 0)
since Ei is defined by means of ∼i like this
π̂
since δ(δ(q, 0), 0) = δ(q, 0), by the definition of
π̂
by the definition of δ ∗
then δ(q, 0) ∼i δ(δ ∗ (0, A), 0)
PTPS-automaton schema 2
then δ(q, 0) ∼i δ ∗ (0, A _ 0)
then there exists a B ∈ V k+1 s.t. πi∗ (A _ 0) = πi∗ (B) and s.t. δ ∗ (0, B) = δ(q, 0) by lemma
6.8 in A. Rognes [Rog11]
93
then there exists a B ∈ V k+1 s.t. πi∗ (A _ 0) = πi∗ (B) and s.t. δ(δ ∗ (0, B), 0) = δ(q, 0) since
δ(δ(q, 0), 0) = δ(q, 0) by the definition of PTPS-automaton schema 2
then there exists a B ∈ V k+1 s.t. πi∗ (A _ 0) = πi∗ (B) and s.t. γ(B _ 0n×N ) = δ(q, 0)
definition of γ
by
then there exists a B ∈ V k+1 s.t. B _ 0n×N EiU A _ 0n×N and γ(B _ 0n×N ) = δ(q, 0)
definition of EiU
by
then there exists a B ∈ V k+1 s.t. A _ 0n×N ∈ EiU (B _ 0n×N ) and γ(B _ 0n×N ) = δ(q, 0)
then A _ 0n×N ∈ EiU (γ −1 (δ(q, 0)))
Now we prove the other direction of the equivalence. So assume
A _ 0n×N ∈ EiU (γ −1 (δ(q, 0)))
then there exists a B ∈ V ∗ s.t. πi∗ (A) _ 0n×N = πi∗ (B) _ 0n×N and s.t. γ(B _ 0n×N ) =
δ(q, 0)
by definition of EiU
then there exists a B ∈ V k+1 s.t. πi∗ (A) _ 0n×N = πi∗ (B) _ 0n×N and s.t. γ(B _ 0n×N ) =
δ(q, 0)
by lemma 6.7 in A.Rognes [Rog11]
then there exists a B ∈ V k+1 s.t. πi∗ (A _ 0) = πi∗ (B) and s.t. δ(δ ∗ (0, B), 0) = δ(q, 0)
definition of γ
by
then there exists a B ∈ V k+1 s.t. πi∗ (A _ 0) = πi∗ (B) and s.t. δ ∗ (0, B _ 0) = δ(q, 0)
definition of δ ∗
by
then there exists a B ∈ V k+1 s.t. πi∗ (A _ 0 _ 0) = πi∗ (B _ 0) and s.t. δ ∗ (0, B _ 0) = δ(q, 0)
by definition of πi∗
π̂
by lemma 6.8 in A.Rognes [Rog11]
π̂
since δ(δ(q, 0), 0) = δ(q, 0) by the definition of
then δ ∗ (0, A _ 0 _ 0) ∼i δ(q, 0)
then δ ∗ (0, A _ 0 _ 0) ∼i δ(δ(q, 0), 0)
PTPS-automaton schema 2
π̂
π̂
then δ(δ(q, 0), 0) ∼i δ ∗ (0, A _ 0 _ 0)
since ∼i is symmetric by definition
then δ(q, 0)Ei δ ∗ (0, A _ 0 _ 0)
by definition of Ei
then δ(q, 0)Ei δ(δ ∗ (0, A _ 0), 0)
by definition of δ ∗
then δ(q, 0)Ei γ(A _ 0n×N )
by definition of γ
then γ(A _ 0n×N ) ∈ Ei (δ(q, 0))
by definition of image of a relation
then A _ 0n×N ∈ γ −1 (Ei (δ(q, 0)))
qed
94
3.3.6
The main result
Combining the last proposition with proposition 3.2.2 we get the following which we regard as the
main result.
Corollary 3.3.6 Let W = (V, δ, {Tj }j∈J ) be a properly partitioned automaton and H the atomstructure of W. Then Hh+ is representable (by suitable restrictions of γ −1 : P(V ) → P(U )).
3.4
Diversity of the h-complex algebras of finite automata
We have seen that the complex algebra of (the atom-structure of ) a properly partitioned automaton
is a dMsPsn , but so far we have no results on the diversity of dMsPsn ’s that may occur as
the complex algebra of properly partitioned automata. To prove a result on diversity we use the
structure (N, +, |p ) where N are the natural numbers + is addition and |p (x, y) is the binary
relation that is true if y is the greatest power of p such that y divides x. We shall in particular
use the Büchi-Bruyère theorem, which states that an n-ary relation is first-order definable over
(N, +, |p ) if and only if the relation is recognised by a finite n-tape p-automaton, see V. Bruyère
et.al. [BHMV94].
The result we prove on diversity is that every dMsPsn , generated by a finite set of first-order
definable relations over (N, +, |p ), is embeddable in the h-complex algebra of some properly partitioned finite automaton.
We also recall different well known variants of finite-dimensional polyadic algebras, and show
that the result on diversity rarely carries over to these. The algebras we recall are; the undirected
many-sorted MsPsn ’s, the one-sorted Psn ’s and polyadic equality variants of the above namely,
dMsPEsn ’s, MsPEsn ’s and PEsn ’s.
3.4.1 dMsPsn ’s
Proposition 3.4.1 Every dMsPsn generated by a finite set of first-order definable relations over
(N, +, |p ) can be embedded in the h-complex algebra of a properly partitioned finite automaton, where
h is the induced n-homomorphism.
Proof: Let A = (Bn , . . . , B0 , r, p, s, c0 , . . . , cn−1 ) be a dMsPsn that is generated by a finite set
of first-order definable relations over (N, +, |p ). We now define the properly partitioned automaton
the h-complex algebra of which we eventually embed A into. Let Γ be a finite set of sentences in
the language of (N, +, |p ) that define a set of generators for A. By the Büchi-Bruyère theorem
there is a finite set Y of n-tape p-automata that recognises each of the generators for A. Let X be
the minimal set of automata such that
1. If Wφ ∈ Y then Wφ ∈ X,
2. If Wφ ∈ X then W¬φ ∈ X where W¬φ recognises the complement of the relation recognised
by Wφ ,
3. If Wφ ∈ X and Wφ recognises R(v0 , . . . , vn−1 ) and σ : n → n and Wσφ recognises
R(vσ(0) , . . . , vσ(n−1) ) then Wσφ ∈ X.
95
By well known results from automata theory we can, given a finite set Y of automata, effectively construct such an X. By proposition 3.3.1 there is one finite PTPS-automaton W =
(V, δ, {Tj }j∈J ) that recognises each relation recognised by the elements of X. Now X was carefully designed so as to make W meet the criteria of proposition 3.3.3. Therefore there is a properly
partitioned automaton, W 0 = (V, δ, {Tj0 }j∈J 0 ), such that {Tj0 }j∈J 0 generates the same boolean
algebra as does {Tj }j∈J . We see that W 0 is finite since it has the same carrier set as W.
Let now H = (R, . . .) be the atom-structure of W 0 , let h = (hn , . . . , h0 ) be the n-homomorphism induced by W 0 and let Hh+ = (Bn0 , . . . , B00 , r0 , p0 , s0 , c00 , . . . , c0n−1 ) be the h-complex
algebra of H. By proposition 3.3.5 we may define an embedding f = (fn , . . . , f0 ) from Hh+ to the
full dMsPsn over N as in the proof of proposition 3.2.2, i.e.
for i ≤ n and q ∈ R we let fi [q]hi = γ −1 [q]hi ,
for Q ⊆ R let fi ( {[q]hi |q ∈ Q}) = {γ −1 [q]hi |q ∈ Q}.
S
S
We now show that the given A is a sub-dMsPsn of the image of Hh+ under f , the embedding
needed to prove the present proposition is then obtained by restricting f −1 to A. Since {Tj0 }j∈J 0
generates the same boolean algebra as does {Tj }j∈J and since each set recognised by an automaton
of Y is of the form γ −1 (Tj ) we see that the sort Bn is a sub-boolean algebra of the image of Bn0
under fn . Therefore the finitely generated A is a sub-dMsPsn of the image of Hh+ under f . qed
3.4.2 Psn ’s
Here we show that in the case of one-sorted polyadic algebras we are not generally able to obtain an
embedding as in the case of dMsPsn . We use a trick and define one-sorted n-dimensional polyadic
algebras as a sub-class of dMsPsn . It is left to the reader to verify that this definition coincides with
n-dimensional (quasi) polyadic algebra as found in the literature, see e.g. L. Henkin, J.D. Monk and
A. Tarski [MHT85] or I. Németi [Ném91]. These may also be compared to the quantifier algebras
due to C.Pinter [Pin73], which are called substitution-cylindric algebras by I. Németi [Ném91] and
H. Andréka and I. Sain and I. Németi [ASN01].
Definition An n-dimensional polyadic set algebra, or Psn for short, is an
A = (B, r, p, s, c0 , . . . , cn−1 ) such that A0 = (B, . . . , B, r, p, s, c0 , . . . , cn−1 ) is a dMsPsn .
Moreover a mapping f from A to some Psn is a Pn -embedding if (f, . . . , f ) is a dMsPn embbeding from A0 to some dMsPsn .
Example Let (N, <) be the natural numbers with the usual ordering. Let L3 (<) be the set of
three-variable formulae in the language for (N, <). The set of ternary relations definable using
A A
quantifier-free formulae of L3 (<) generate an A ∈ Psn where A = (B, rA , pA , sA , cA
0 , c1 , c2 ) is
defined as follows.
B = (B A , ∪, −) is the set algebra where B3 ⊆ P(N3 ) is the set of ternary relations definable
using L3 (<) formulae.
A A
The operations rA , pA , sA , cA
0 , c1 , c2 are defined as in example 3.1.3
Proposition 3.4.2 There exists a Psn generated by a finite set of first-order definable relations over
(N, +, |p ) that can not be embedded in the h-complex algebra of a properly partitioned finite automaton
for any n-homomorphism h.
96
Proof: It is easy to see that the h-complex algebra of a properly partitioned finite automaton
is finite. So to prove the proposition it suffices to display n, p ∈ N and an infinite Psn that is
generated by a finite set of first-order definable relations over (N, +, |p ).
Let n = 3, p = 2 and consider the A ∈ Ps3 with the following generator
X = {(x0 , x1 , x2 )|x1 < x0 }. The generator is definable by the formula ¬(∃v2 (v0 + v2 = v1 )),
which says that it is not the case that v0 ≤ v1 . We will now by induction show that for each a ∈ N
it is the case that the set {(x0 , x1 , x2 )|a < x0 } lies in A. Let N denote the full dMsPs3 over N.
To begin with {(x0 , x1 , x2 )|0 < x0 } lies in A, since cN
1 (X) = {(x0 , x1 , x2 )|∃y(x0 , y, x2 ) ∈
X)}. This set is exactly the set of triples such that the second component is greater that 0.
To proceed the induction, assume that A = {(x0 , x1 , x2 )|a < x0 } lies in A. Consider
c1 (pN (A) ∩ X). Here pN (A) = {(x0 , x1 , x2 )|a < x1 } therefore pN (A) ∩ X = {(x0 , x1 , x2 )|a <
x1 ∧ x1 < x0 } furthermore c1 (pN (A) ∩ X) = {(x0 , x1 , x2 )|∃y(y ∈ N ∧ a < y ∧ y < x0 )} which
of course is {(x0 , x1 , x2 )|a + 1 < x0 }.
qed
3.4.3 MsPsn ’s
−
Definition Let A = (Bn , . . . , B0 , r, p, s, c−
0 , . . . , cn−1 ) be a dMsPsn . An n-dimensional many+
sorted polyadic set algebra, or MsPsn for short, is an A0 = (A, c+
0 , . . . , cn−1 ) such that for each
i < n the mapping c+
i : Bi → Bi+1 has the property that for any dMsPn -embedding f from
A to a full U ∈ dMsPsn and any i < n it is the case that c+
i is a boolean embedding and
+ −
U
that f (ci (ci (x))) = c (f (x)). Moreover an n-homomorphism f from A0 to some MsPsn
is a MsPn -embedding if f is an embedding from A to some dMsPsn that preserves each of
+
c+
0 , . . . , cn−1 .
Corollary 3.4.3 There exists a MsPsn generated by a finite set of first-order definable relations over
(N, +, |p ) that can not be embedded in the h-complex algebra of a properly partitioned finite automaton
for any n-homomorphism h.
Proof: As in the case of Psn consider the Ps3 generated by the set X = {(x0 , x1 , x2 )|x1 < x0 }.
+
Let A = (Bn , . . . , B0 , . . . , c+
0 , . . . , cn−1 ) be the MsPsn generated by the set X. For each i < n
+
the boolean algebra Bi can be embedded in Bn using c+
0 , . . . cn−1 . Therefore the Ps3 generated by
X is definable over the boolean algebra Bn . Combining this proposition 3.4.2 we conclude that
Bn is infinite.
qed
3.4.4
Polyadic equality algebras
Here we will see that the situation is the same for polyadic equality algebras as for polyadic algebras
using one or many sorts. Polyadic equality algebras were considered already by P. R. Halmos. In
the finite-dimensional case they coincide with C.C. Pinters quantifier algebras with equality, see
[Pin73]. The cylindric algebras of A. Tarski and the relation algebras of A. De Morgan and C.
Peirce are reducts of polyadic equality algebras.
Definition Let A = (Bn , . . .) be either a dMsPsn , a Psn or a MsPsn . Let for i, j < n the
ij-diagonal dij ∈ Bn be defined by dij = {(x0 , . . . , xn−1 )|xi = xj }. Let A0 = (A, {dij }i,j<n ).
Then
97
A0 is an n-dimensional polyadic equality set algebra or a PEsn for short if A is a Psn , likewise
A0 is a MsPEsn if A is a MsPsn ,
A0 is a dMsPEsn if A is a dMsPsn .
Moreover a mapping f from A0 to a dMsPEsn , a PEsn or a MsPEsn respectively, is a dMsPEn , PEn - or a MsPEn -embedding if f is a dMsPn -, Pn - or a MsPn embedding from A and if f
preserves each of {dij }i,j<n .
Corollary 3.4.4 Every dMsPEsn generated by a finite set of first-order definable relations over (N, +, |p )
can be embedded in the h-complex algebra of a properly partitioned finite automaton, where h is the
induced n-homomorphism.
Proof: This proof is carried out as the equality-free case, i.e. proposition 3.4.1, by including
{dij }i,j<n as generators.
qed
Definition An n-dimensional polyadic equality set algebra, or PEsn for short, is an A = (B, r, . . . , {dij }i,j<n )
such that A0 = (B, . . . , B, r, . . . , {dij }i,j<n ) is a dMsPEsn . Moreover a mapping f from A to
some PEsn is an embedding if (f, . . . , f ) is an embedding from A0 to some dMsPEsn .
Corollary 3.4.5 There exists a PEsn and a MsPEsn generated by a finite set of first-order definable
relations over (N, +, |p ) that can not be embedded in the h-complex algebra of a properly partitioned
finite automaton for any n-homomorphism h.
Proof: Again this proof is carried out as in the equality-free case by treating each of {dij }i,j<n as
a generator.
qed
3.5
Axiomatisation by a finite set of first order sentences
For each n ∈ N we now provide axioms for four basic elementary classes of properly partitioned
n-tape automata. These four axiom sets ensure that the h-complex algebra of (the atom-structure
of ) a finite automaton is a finite dMsPsn , dMsPEsn , Psn or PEsn respectively. We prove some
result that are of interest in regards to recursively enumerating representatives for the finite members
of the four classes. The cases of MsPsn and MsPEsn are left out, since the finite algebras whose
sorts are all equal can be enumerated in the same way as their one-sorted counterparts.
Each of the four classes consists of properly partitioned automata whose alphabet are matrices
of height n. To axiomatise matrices of height n we have chosen a formalism involving abstract
vector-spaces. The reader may use his or her own formalism as long as for each n ∈ N the axioms
are finite in number and have the property that for all m ∈ N the structure (Mp (n, m), π, r) is a
model and as long as each finite model of the axioms can be thought of as some (Mp (n, m), π, r).
See A. Rognes [Rog11] for details on the example used here.
For the first two classes of automata we use one equivalence relation T to partition the reachable
states, rather than a family of disjoint sets. Recall that in the signature of (Mp (n, m), π, r) we have
expressions for each of the elements of P L(V) and SL(V). We are also able to define the reachable
t
states R. Moreover for t ∈ P L(V), i < n and σ̂ ∈ SL(V) it is the case that ∼, Ei , /σ̂ and σ̂ are
definable.
98
Definition Let n, p ∈ N where p is a prime power. A dMsPn -automaton is
a W = (V, π, r, δ, T ) where,
V = (V, . . .) is a vector-space over Fp ,
π, r : V → V are called projection and rotation,
δ : V × V → V is called the transition function,
T ⊆ V × V is an equivalence relation called the partitioning equivalence.
Ax1 If V is finite then (V, π, r) is isomorphic to (Mp (n, m), π 0 , r0 ) for some m ∈ N depending
on the size of V. See A.Rognes [Rog11] for an example of how this can be axiomatised with
a first-order sentence.
Ax2 W is a PTPS-automaton using the abstract vector-space, V, as alphabet.
Ax3 W is properly partitioned. Since we are using an equivalence relation this can be expressed by
adding the following axiom for each σ̂ ∈ SL(V). W |= ∀q 0 q 00 ∈ R(q 0 σ̂ q 00 → T (q 0 , q 00 ))
For the second class of automata we add equality, i.e., for each i, i0 < n there is a subset of states,
Dii0 , consisting of those states that we run through with some tape whose row i and row i0 are
equal.
Definition Let n, p ∈ N where p is a prime power. A dMsPEn -automaton is a W = (W 0 , {Dii0 }i,i0 <n )
where
Ax1-3 W 0 = (V, π, r, δ, T ) is a dMsPn -automaton,
AxD1 for i, i0 < n and the σ̂ ∈ SL(V) that overwrites row i with the content of row i0 the
diagonal Dii0 comprises those equivalence classes of T that contain reachable states q 0 , such
that q 0 /σ̂ q 0 , i.e., W |= ∀q ∈ R[(Dii0 (q) ↔ ∃q 0 ∈ R[T (q, q 0 ) ∧ q 0 /σ̂ q 0 ])].
Proposition 3.5.1 The classes of dMsPn -automata and dMsPEn -automata are basic elementary.
Moreover each of the relations R, /r̂ , /p̂ , /ŝ E0 , . . . , En−1 , needed to define the atom-structure of W,
is definable by a finite set of first-order sentences in the language of dMsPn -automata and dMsPEn automata respectively.
Proof: This is a matter of going through the definitions of each of the notions.
qed
Proposition 3.5.2 Every dMsPsn and every dMsPEsn generated by a finite set of first-order definable relations over (N, +, |p ) can be embedded in the h-complex algebra of a finite dMsPn -automaton,
where h is the induced n-homomorphism.
Proof: This is a minor variation of proposition 3.4.1.
99
qed
We now define a class of automata that can be used to define the atom-structures of certain
Psn ’s and MsPsn ’s. In this definition we abandon the equivalence relations T and revert to
partitioning by an indexed family {Tj }j∈J where J is a finite set. To explain the reason for this
let Bn be the boolean algebra generated by our partition. To make an atom-structure of a Psn
or MsPsn we need to ensure that the image of Bn under each cylindrification is a sub-boolean
algebra of Bn . The latter appears to require an infinite union or some other higher-order construct,
see Ax4 below.
Definition Let n, p ∈ N where p is a prime power and let J be a finite set. A J-indexed Pn automaton is a W = (V, π, r, δ, {Tj }j∈J ) where
V = (V, . . .) is a vector-space over Fp ,
π, r : V → V are called projection and rotation,
δ : V × V → V is called the transition function,
Tj ⊆ V for each j ∈ J.
Ax1 If V is finite then (V, π, r) is isomorphic to (Mp (n, m), π, r) for some m ∈ N depending
on the size of V.
Ax2 W is a PTPS-automaton using the abstract vector-space, V, as alphabet.
Ax3b W is properly partitioned.
Ax4 For each i < n and j ∈ J there exists a X ⊆ J such that Ei (Tj ) = {Tj 0 |j 0 ∈ X}.
S
Again we add a variant with equality, i.e., one that for each i, i0 < n has a subset of states, Dii0 ,
consisting of those states that we run through with some tape whose row i and row i0 are equal.
Definition Let n, p ∈ N where p is a prime power and let J be a finite set. A J-indexed PEn automaton is a W = (W 0 , {Dii0 }i,i0 <n ) where
Ax1-4 W 0 = (V, π, r, δ, {Tj }j∈J ) is a Pn -automaton,
AxD1b for i, i0 < n and j ∈ J and the σ̂ ∈ SL(V) that overwrites row i with the content of row
i0 the diagonal Dii0 comprises those Tj that contain reachable states q 0 such that q 0 /σ̂ q 0 , i.e.,
W |= ∀q ∈ R[Tj (q) → (Dii0 (q) ↔ ∃q 0 ∈ R[Tj (q 0 ) ∧ q 0 /σ̂ q 0 ])].
Proposition 3.5.3 For fixed J, the classes of J-indexed Pn -automata and PEn -automata are basic
elementary. Moreover each of the relations R, /r̂ , /p̂ , /ŝ E0 , . . . , En−1 , needed to define the atomstructure of W, is definable by a finite set of first-order sentences in the language of J-indexed Pn automata and PEn -automata respectively.
Proof: Axiom 4 can be expressed by a disjunction indexed over subsets of J. Since J is fixed and
finite this can be done in first-order language. It is easy to check that the rest of the axioms for
J-indexed Pn -automata are first-order and finite in number, thus the class is basic elementary. As
before the definitions of R, /r̂ , /p̂ , /ŝ E0 , . . . , En−1 are finite and first-order in the language of the
signature of (V, π, r, δ).
qed
100
The following corresponds to proposition 3.5.2. This time however we have to assume that the
finitely generated algebra is finite since by proposition 3.4.2 a finitely generated algebra is not finite
in general.
Proposition 3.5.4 Let J be a finite set and let k ∈ N be the number of elements of J, then every
Psn or PEsn generated by k disjoint first-order definable relations over (N, +, |p ) and that has 2k
elements can be embedded in the h-complex algebra of a J-indexed Pn -automaton, where h is the
induced n-homomorphism.
Proof: This is also a variation of proposition 3.4.1. As we have seen, finitely generated Psn ’s or
PEsn ’s aren’t guaranteed to be finite so we restrict our selves to Psn ’s and PEsn ’ whose boolean
reduct is the boolean algebra generated by the partition {Tj }j∈J . This algebra has 2k elements
when J has k elements.
qed
3.6
Concluding remarks
In section 3.3.4 we introduced the h-complex algebra of a properly partitioned automaton. With
corollary 3.3.6 we showed that the h-complex algebra of a properly partitioned automaton is representable. Proposition 3.4.1 states that every dMsPsn generated by a finite set of first-order definable relations over (N, +, |p ) can be embedded in the h-complex algebra of a properly partitioned
finite automaton.
With proposition 3.5.1 we noted that, for a fixed number of tapes, a variation of properly
partitioned automata called dMsPn -automata, is basic elementary. The dMsPn -automata use
an equivalence relation for partitioning and a variant of proposition 3.4.1 also holds for these.
The latter two proposition also hold for for the class of dMsPEn -automata, which differ from
dMsPn -automata only in that they have diagonals and therefore are suitable for polyadic equality
algebras. By proposition 3.5.1 we also have a way of recursively enumerating the atom-structures
of both finite dMsPn -automata and finite dMsPEn -automata, without explicitly computing the
whole complex algebra. Proposition 3.5.2 measures the extent of the representable algebras whose
atom-structures are enumerated thusly.
We finally introduced the variants of properly partitioned automata called J-indexed Pn automata and J-indexed PEn -automata, designed to have the property that the h-complex algebra
of a finite automaton is a finite polyadic algebra or finite polyadic equality algebra respectively.
In order to make these two classes basic elementary we did not use an equivalence relation but a
partition in the form of one unary relation symbol for each member of an index set J. This has
the somewhat undesirable property that the finite axiomatisation only works for a fixed number
of tapes and a fixed index set J. Since the members of the set J correspond to the atoms of the
associated complex algebra this means that only a finite number of isomorphism classes of polyadic
(equality) algebras may occur as the complex algebra of J-indexed Pn -automata or PEn -automata
for fixed n and J.
We leave it as an open problem whether it is possible to restrict the class of dMsPn -automata
by means of a first-order sentence so as to ensure that the h-complex algebra is a finite Pn whilst
retaining a property similar to proposition 3.4.1.
Regardless of the open problem, by proposition 3.5.3 we have a way of recursively enumerating
the atom-structures of finite J-indexed Pn -automata and J-indexed PEn -automata, for J ranging
101
over finite sets. Again we avoid explicitly computing the whole of the complex algebras in question. Proposition 3.5.4 measures the extent of the representable algebras whose atom-structures are
enumerated thusly.
102
Bibliography
[Aan71]
S.O. Aanderaa. On the decision problem for fomulas in which all disjunctions are
binary. In Proc. 2nd Scandinavian Logic Symp., volume 48, pages 1–18, 1971.
[Aan83]
S.O. Aanderaa. On the solvability of the extended ∀∃ ∧ ∃∀∗ ackermann class with
identity. In Logic and Machines, pages 270–284, 1983.
[AGM+ 98] H. Andréka, S. Givant, S. Mikulás, I. Németi, and A. Simon. Notions of density that
imply representability in algebraic logic. Ann. Pure Appl. Logic, 91(2-3):93–190, 1998.
[AL73]
S.O. Aanderaa and H.R. Lewis. Prefix classes of krom formulas. J. Symb. Log.,
38(4):628–642, 1973.
[AM94]
H. Andréka and R.D. Maddux. Representations for small relation algebras. Notre
Dame Journal of Formal Logic, 35(4):550–562, 1994.
[ANS01]
H. Andreka, I. Nemeti, and I. Sain. Algebraic logic. In D. Gabbay and F. Guenther,
editors, Handbook of Philosophical Logic, volume 2, pages 133–247. 2001.
[ASN01]
H. Andréka, I. Sain, and I. Németi. Algebraic logic. In Handbook of Philosophical
Logic, Vol 2, volume 2, pages 133–247. Kluwer Academic Publishers, 2001.
[B6̈0]
J. R. Büchi. Weak second-order arithmetic and finite automata. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 6:66–92, 1960.
[Ber59]
P. Bernays. Über eine natürliche erweiterung des relationenkalküls. In A. Heyting,
editor, Constuctivity in mathermatics, pages 1–14, Amsterdam, 1959. North-Holland.
[Ber66]
R. Berger. The undecidability of the domino problem. Memoirs American Mathematical Society, 66:1966, 1966.
[BG04]
A. Blumensath and E. Grädel. Finite presentations of infinite structures: Automata
and interpretations. Theory Comput. Syst., 37(6):641–674, 2004.
[BGG97]
E. Börger, E. Grädel, and Y. Gurevich. The Classical Decision Problem. Perspectives in
Mathematical Logic. Springer-Verlag, 1997.
[BHMV94] V. Bruyére, G. Hansel, C. Michaux, and R. Villemaire. Logic and p-recognizable sets
of integers. Bulletin of the Belgian Mathematical Society, 1:191–238, 1994. Available
from http://www.emis.de/
journals/BBMS/Bulletin/bul942/BRU.PDF.
103
[Büc62]
J.R. Büchi. Turing machines and the entscheidungsproblem. Math.Annalen, 148:201–
213, 1962.
[CP00]
R. Caferra and N. Peltier. Combining enumeration and deductive techniques in order
to increase the class of constructible infinite models. J. Symb. Comput., 29(2):177–
211, 2000.
[Dai63]
A. Daigneault. Tensor products of polyadic algebras. J. Symb. Log., 28(3):177–200,
1963.
[DG79]
B. Dreben and W.D. Goldfarb. The Decision Problem: Solvable Classes of Quantificational Formulas. Addison-Wesley Publishing Company, Reading, Massachusetts,
1979.
[EF06]
H-D. Ebbinghaus and J. Flum. Finite model theory. Springer Monographs in Mathematics. Springer-Verlag, Berlin, enlarged edition, 2006.
[FLHT01] C. G. Fermüller, A. Leitsch, U. Hustadt, and T. Tammet. Resolution decision procedures. In Handbook of Automated Reasoning, pages 1791–1849. 2001.
[FS06]
M. Ferenczi and G. Sági. On some developments in the representation theory of
cylindric-like algebras. Algebra Universalis, 55:345–353, 2006.
[FV59]
S. Feferman and R.L. Vaught. The first-order properties of algebraic systems. Fund.
Math., 47:57–103, 1959.
[GK72]
Y. Gurevich and I.O. Koriakov. A remark on berger’s paper on the domino problem.
Siberian Mathematical Journal, 13:319–321, 1972.
[Gol84]
W. D. Goldfarb. The unsolvability of the gödel class with identity. J. Symb. Log.,
49(4):1237–1252, 1984.
[Gol00]
R. Goldblatt. Algebraic polymodal logic: A survey. Logic Journal of the IGPL,
8(4):393–450, 2000.
[Hal62]
P. R. Halmos. Algebraic Logic. Chelsea, New York, 1962.
[Her75]
I. N. Herstein. Topics in Algebra. Wiley, 2nd edition, 1975.
[HH09]
R. Hirsch and I.M. Hodkinson. Strongly representable atom structures of cylindric
algebras. J. Symb. Log., 74(3):811–828, 2009.
[HMA+ 81] L. Henkin, J.D. Monk, H. Andréka, A. Tarski, and I. Németi. Cylindric set algebras.
Lecture notes in mathematics. Springer-Verlag, 1981.
[Hod97]
I.M. Hodkinson. Atom structures of cylindric algebras and relation algebras. Ann.
Pure Appl. Logic, 89(2-3):117–148, 1997.
[Joh69]
J.S. Johnson. Nonfinitizability of classes of representable polyadic algebras. J. Symb.
Log., 34(3):344–352, 1969.
104
[JT51]
B. Jónsson and A. Tarski. Boolean algebras with operators, part i. American Journal of
Mathematics, 73:891–932, 1951.
[KMS02]
N. Klarlund, A. Møller, and M.I. Schwartzbach. Mona implementation secrets. Int.
J. Found. Comput. Sci., 13(4):571–586, 2002.
[KMW62] A. Kahr, E. Moore, and H. Wang. Entscheidungsproblem reduced to the ∀∃∀-case.
In Proc. Nat. Acad. Sci. U.S.A., volume 48, pages 365–377, 1962.
[Men87]
E. Mendelson. Introduction to mathematical logic (3rd ed.). Wadsworth Publ. Co.,
Belmont, CA, USA, 1987.
[MHT85] J.D. Monk, L. Henkin, , and A. Tarski. Cylindric algebras part II. North-Holland,
1985.
[MM99]
M. Marx and S. Mikulás. Decidability of cylindric set algebras of dimension two and
first-order logic with two variables. J. Symb. Log., 64(4):1563–1572, 1999.
[Mon69]
J.D. Monk. Nonfinitizability of classes of representable cylindric algebras. J. Symb.
Log., 34(3):331–343, 1969.
[Ném91]
I. Németi. Algebraizations of quantifier logics: an introductory overview. Studia
Logica, 50:485–570, 1991.
[Ném97]
I. Németi. Algebraizations of quantifier logics, an introductory overview. Alfred Renyi
Institute of Mathematics, 1997. version 12.1.
[Pel97a]
N. Peltier. Increasing model building capabilities by constraint solving on terms with
integer exponents. J. Symb. Comput., 24(1):59–101, 1997.
[Pel97b]
N. Peltier. Tree automata and automated model building. Fundam. Inform., 30(1):59–
81, 1997.
[Pel09]
N. Peltier. Constructing infinite models represented by tree automata. Ann. Math.
Artif. Intell., 56(1):65–85, 2009.
[Pin73]
C.C. Pinter. A simple algebra of first order logic. Notre Dame Journal of Formal Logic,
14(3):361–366, 1973.
[Pin97]
J.E. Pin. Syntactic semigroups. In Handbook of Formal Languages, pages 679–746.
Springer, 1997.
[Rog09]
A. Rognes. Turning decision procedures into disprovers. Math. Log. Q., 55(1):87–104,
2009.
[Rog11]
A. Rognes. Automata for mechanising consistency proofs and for the computation of
finite representable polyadic algebras, part 1, submitted 2011.
[Tho96]
W. Thomas. Languages, automata, and logic. In Handbook of Formal Languages, pages
389–455. Springer, 1996.
105