Turning decision procedures into disprovers
André Rognes
No Institute Given
Abstract. We describe a method for devising first-order disprovers. One
needs a decision procedure for some first-order theory, to devise a disprover according to the method. We argue that each resulting disprover
works by a natural generalisation of finite model search, for a substantial
fragment of first-order language. Finite generalised models may present
infinite structures. When this is the case, finite model search is strictly
generalised. A downward Löwenheim-Skolem type of result is shown in
that satisfiable purely relational prenex sentences each have a finite generalised model, in the form of a particular kind of finite many-sorted
polyadic set algebra. 1
1
Introduction
In this setting a disprover is a procedure that only recognises first-order sentences that are not recognised by a sound refutation procedure complete for full
first-order logic. The purpose being to use the disprover in conjunction with a
refutation procedure in attempts at deciding whether given sentences are satisfiable.
A general method for devising disprovers is described. 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 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
in the given algebra and increasingly 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 this is an example;
∀x ∃y Rxy∧ ∀x ¬Rxx∧ ∀x ∀y ∀z Rxy ∧ Ryz → Rxz
Satisfiable sentences having this infinity axiom as a logical consequence are also
infinity axioms. Disprovers that to some extent recognise these infinity axioms
have been implemented 2 . Very short instances of these infinity axioms can be
found for which state of the art first-order reasoning procedures appear not to
terminate.
1
2
Parts of this paper appeared in earlier form at DISPROVING06 under the title
“automated relative consistency proving”
http://flipper.berlios.de
The class of sentences recognised by each disprover turns out to be nonrecursive. This is a consequence of a theorem of Trakhtenbrot [Tra50] together
with the ability to recognise any finitely satisfiable sentence of a substantial
fragment of first-order language. This implies that no decidable class of sentences
covers the class 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 the recognised classes either.
As any reasoning procedure can be repaired in a most ad hoc way so as to
recognise any given infinity axiom, the following naturalness property is shown.
The class of sentences recognised by each disprover is closed under logical equivalence, within the aforementioned fragment. This property is not shared with
procedures repaired by first checking whether the input is syntactically equal to
one of a finite set of satisfiable sentences and then going 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 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. 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 terms of 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, [Büc62]. 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 [KMW62], which turns out to be conservative by [Ber66]
and [GK72]. A proof of this is accessible in the book [BGG97]. In the same book
a proof of Trakhtenbrots theorem is available (note exercise 2.1.36).
The generalisation of model search is done by search for satisfying interpretations in finite many-sorted polyadic set algebras. 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 2. In Section 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 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
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.
1.2
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. Relation symbols are further assumed to be of
arity 3. See chapter 6.5 of [DG79] for how to conservatively reduce formulae 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 an unlimited supply of ternary relation symbols.
Ternary relation symbols are also the only kind of symbol of which there is
an unlimited supply. Rxy and xRy are used as an abbreviation for Rxyy. The
notation L′3 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 L′3 sentences, and the symbol Γ is used for theories.
A theory Γ is complete if for every sentence φ ∈ L′3 it is the case that φ ∈ Γ or
¬φ ∈ Γ . If it is the case that for all φ ∈ L′3 either φ ∈ Γ or ¬φ ∈ Γ , and not
both, then Γ is consistent and complete. The set of L′3 sentences true under a
fixed interpretation is a complete and consistent theory. Complete theories may
sometimes be presented by decision procedures, and these may internally use
any sort of symbol, such as an equality symbol, unlimited supplies of variable
and function symbols, or symbols for higher order constructs. Two L′3 formulae
φ and ψ are said to be Γ -equivalent if the sentence ∀x∀y∀z(φ ↔ ψ) is an element
of Γ .
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.
1.3
Algebras, homomorphisms, and preservation
The present paper is about algebras and homomorphisms of several kinds. 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 1.1. Let A and B be algebras, let f be an operation of A and g the
corresponding 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 x, . . . , y ∈ X it is
the case that f (x, . . .) = y implies g(h(x), . . .) = h(y).
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.
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 such that the operations map elements in X out of X. All in
all X must be 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 , as opposed to algebras of boolean signature, turn out
to be very hard to axiomatise. We therefore dispose of with the usual axioms all
together. By the versatile representation theorem of Stone the following is way
of defining boolean algebras.
Definition 1.2. 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.
Proposition 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.
Proposition 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.
2
Algebras of polyadic signature
Polyadic Algebras were introduced and studied by Halmos. Some papers on
these studies are collected in [Hal62]. A particular class of polyadic algebras
are polyadic set algebras of dimension 3, denoted Ps3 . In the present paper
these serve as abstract or generalised models for L3 sentences. The well known
Lindenbaum algebra, L′3 /Γ , of a consistent and complete L′3 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 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 L′3 /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, L′3 , with one relation symbol (<), which together with the three
variables of L′3 gives rise to a finite number of atomic formulae. It also allows
quantifier-elimination within L′3 . 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, L′3 /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 ∀z Rxyy∧ ∀x ∀y ∀z ¬Rxxx∧ ∀x ∀y ∀z Rxyy ∧ Ryzz → Rxzz.
Interpretation and satisfaction in Ps3 are such that; the infinity axiom is satisfied
in L′3 /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 L′3 have interpretations in L′3 /Γ .
When Γ is decidable, and L′3 /Γ has a finite number of equivalence classes,
it is possible to compute an abstract finite Ps3 isomorphic to L′3 /Γ , by using
indices for the equivalence classes of L′3 /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
mostly follows [Ném91]. One exception is that we use notation from lattice theory
for boolean operations, an other exception is that in [Ném91] more operations
related to variable substitutions are used in the definition of polyadic algebras,
here only 3 are used.
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.
Proposition 2.1. p,s and r generate all mappings in 3 → 3, under composition.
2.2
Algebras of polyadic signature
Definition 2.1. 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 propositional connectives. r, s, p relate to variable substitution. c0 , c1 , c2 correspond to existential
quantifiers, and are called cylindrifications.
Definition 2.2. 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 C that preserves each of r, p, s, c0 , c1 , c2 .
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 are by definition 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 2.3. 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∗ is related to only renaming 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 2.4. L3 = (L3 , ∨, ¬, ⊥, r∗ , p∗ , s∗ , ∃0 , ∃1 , ∃2 )
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. As usual in logic a mapping from the variables
of L3 to some set U is called a valuation, and an interpretation is an assignment
of each formula of L3 to a set of valuations. 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 valuations 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 theorems, 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 and fig. 2.
Definition 2.5. Let U be some set. The full Ps3 over U is an algebra
3
U U
(P(U 3 ), ∪, −, ∅, rU , pU , sU , cU
0 , c1 , c2 ) where (P(U ), ∪, −, ∅) is the boolean algebra of sets of triples of elements of U . Operations related to variable substitutions
are;
rU (X) = {u ∈ U 3 : u ◦ r ∈ X} = {abc ∈ U 3 : cab ∈ X}
pU (X) = {u ∈ U 3 : u ◦ p ∈ X} = {abc ∈ U 3 : bac ∈ X}
sU (X) = {u ∈ U 3 : u ◦ s ∈ X} = {abc ∈ U 3 : bbc ∈ X}
Operations for the existential quantifiers follow;
3
cU
0 (X) = {abc ∈ U : there is a u ∈ U such that ubc ∈ X}
3
U
c1 (X) = {abc ∈ U : there is a u ∈ U such that auc ∈ X}
3
cU
2 (X) = {abc ∈ U : there is a u ∈ U such that abu ∈ X}
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 .
Definition 2.6. A Ps3 is an algebra of polyadic signature that is embeddable
into a full Ps3 .
The next two properties relate the defining equations for r∗ , s∗ , p∗ to those
of rU , sU , pU . The first property is a consequence of proposition 2.1.
Proposition 2.2. Let U be a set, let R be a relation-symbol of L3 and let f, g ∈
3 → 3 then, for each each X ⊆ U the following holds. If R(012 ◦ f ) = R(012 ◦ g)
then {u ∈ U 3 : u ◦ f ∈ X} = {u ∈ U 3 : u ◦ g ∈ X}
For the following the author finds geometric interpretation helpful, see fig. 1 and
fig. 2.
Proposition 2.3. For each i ∈ 3,
Fig. 1. Cylindrification along x-axis, z-axis is orthogonal to the present paper and the
dotted line marks the xy-diagonal plane.
rU (¬X) = ¬(rU (X)).
pU (¬X) = ¬(pU (X)).
sU (¬X) = ¬(sU (X)).
rU (X ∪ Y ) = rU (X) ∪ rU (Y ).
pU (X ∪ Y ) = pU (X) ∪ pU (Y ).
sU (X ∪ Y ) = sU (X) ∪ sU (Y ).
U
rU (cU
i (X)) = cr(i) X.
U
pU (cU
i (X)) = pp(i) X.
U U
U
s (c0 (X)) = c0 (X).
U U
sU (cU
1 (X)) = c0 (p (X).
U U
U U
s (c2 (X)) = c2 (s (X).
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 proposition 2.2.
Proposition 2.4. 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.
Fig. 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.
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, an interpretation for each formula of L3 is uniquely
determined.
Definition 2.7. 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.
Proposition 2.5. Let X ⊆ Y ⊆ L3 be such that X is sub-formula-closed Let A
be a Ps3 . Each 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 proposition 2.3
which provides semantics for r∗ , s∗ , p∗ .
Proposition 2.6. 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]
(valuations are called sequences there).
Proposition 2.7. 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.
2.6
Additivity
The following proposition is not central for understanding why the presented
disprovers work. It does however contribute to actually fitting the presented
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 2.8. An operation f on a boolean algebra is called additive if f (⊥) =
⊥ and f (x ∨ y) = f (x) ∨ f (y).
Additive operations go by the name of hemimorphisms in [Hal62]. The following
property can to some extent be inspected geometrically, see fig. 1 and fig. 2.
Proposition 2.8. The appropriate reduct of a Ps3 is a boolean algebra and each
of r, p, s, c0 , c1 , c2 is additive.
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 standard 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. Well-orderedness ensures that the
following defines a mapping.
Definition 2.9. µΓ ∈ L′3 → L′3 is defined by µΓ (φ) = ψ where ψ is the minimal L′3 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 2.10. L′3 /Γ = ([L′3 ], ∨′ , ¬′ , ⊥′ , r′ , p′ , s′ , ∃′0 , ∃′1 , ∃′2 ), where for i ∈ 3
[L′3 ] is the µΓ -image of L′3
[φ] ∨′ [ψ] = [φ ∨ ψ]
¬′ [φ] = [¬φ]
r′ [φ] = [r∗ φ]
p′ [φ] = [p∗ φ]
s′ [φ] = [s∗ φ]
∃′i [φ] = [∃i φ]
Note that by this definition, µΓ is a polyadic homomorphism from L′3 to
Moreover it is a satisfying interpretation for every logical consequence of
L′3 /Γ .
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 L′3 -formulae as relations over U , provides a homomorphism h
from L′3 to the full Ps3 over U . The restriction of h to the carrier set of L′3 /Γ is
a homomorphism from L′3 /Γ to the full Ps3 over U . Completeness ensures that
this homomorphism is an embedding.
Proposition 2.9. If Γ is consistent and complete then L′3 /Γ is a Ps3
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 L′3 is a language with a finite number of
relation-symbols and if Γ is a complete and consistent theory that has quantifierelimination within L′3 , then L′3 /Γ has a finite number of equivalence classes. By
the particular representation chosen, this L′3 /Γ is a finite object. Given a decision
procedure for Γ , L′3 /Γ can be computed and put in the form of tables for the
operations on the carrier set of L′3 /Γ . Given such tables one can by exhaustive
search decide whether there exists a satisfying interpretation from L3 to L′3 /Γ
for any L3 sentence φ. This is because one only needs to enumerate mappings
f in {R0 , ...Rm } × 012 → L′3 /Γ , where {R0 , ...Rm } are the relation symbols of
φ, to decide whether there exists an interpretation f ∗ from L3 to L′3 /Γ that is
satisfying for φ.
If a satisfying interpretation f ∗ for φ in L′3 /Γ is found then a satisfying
interpretation for φ in the tarskian sense is obtained by h ◦ f ∗ . Here h is an
embedding from L′3 /Γ to P(U 3 ) provided by proposition 2.9. 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 ⊤.
3
Algebras of directed many-sorted polyadic signature
We have seen how to do exhaustive search for satisfying interpretations in an
L′3 /Γ when Γ has quantifier-elimination within L′3 . Theories do not in general
have quantifier-elimination like this however. Let, for example, L′3 be the language whose only relation-symbol is S, and let Γ be the set of L′3 sentences
that are true when S is interpreted as the successor relation over the natural
numbers. Consider the following sequence of formulae:
φ0 (x) = ∀y ¬Syx
φ1 (x) = ∀y (¬(φ0 (y)) ∨ Syx)
φ2 (x) = ∀y (¬(φ1 (y)) ∨ Syx)
..
.
For each i ∈ ω it is the case that φi defines the natural number i. Thus Γ has
a model where one can define ω distinct relations. Thus Γ has ω formulae no
distinct pair of which are Γ -equivalent. This implies that L′3 /Γ has ω distinct
equivalence classes. The equivalence classes of L′3 /Γ with open formulae in them,
form a finite sub-boolean algebra of L′3 /Γ , as this is a finitely generated boolean
algebra. So an infinite number of the φi are not Γ -equivalent to an open formula.
We conclude that Γ does not have quantifier-elimination within L′3 .
In this section a particular kind of many-sorted polyadic set algebra is introduced that can be computed by means of a decision procedure for some L′3 -theory
Γ , regardless of whether Γ has quantifier-elimination within L′3 . In algebras of
this kind, exhaustive search for satisfying interpretations can be done for any
conjunction of prenex sentences of L3 .
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 3.1. An algebra of directed many-sorted polyadic signature is a manysorted 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 .
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 3.2. 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.
There also is a many-sorted counterpart to the notion of sub-algebra.
Definition 3.3. 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 .
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 3.4. 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 }
Proposition 3.1. L33 ∪ L32 ∪ L31 ∪ L30 is a sub-formula-closed sub-set of L3 .
Proof. L33 ∪ L32 ∪ L31 ∪ L30 are generated by means of logical connectives beginning with the atomic formulae.
⊓
⊔
The above four algebras of boolean signature are now interconnected with
operations related to variable substitution and existential quantifiers.
= (L33 , L32 , L31 , L30 , r∗ |, p∗ |, s∗ |, ∃0 |, ∃1 |, ∃2 |) where
Definition 3.5. Lcrc
3
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
is a conservative reduction class.
Proposition 3.2. The sort L30 of Lcrc
3
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 conservative reduction class of Kahr
More and Wang. 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 .
⊓
⊔
The following is virtually the same as above but works for the reduction class of
Büchi.
has every conjunction of prenex senProposition 3.3. The sort L30 of Lcrc
3
tences 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 lemma then follows since L30 has boolean signature and thus has
conjunctions.
⊓
⊔
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 3.6. 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 .
3.4
Interpretation
We show that one can interpret conjunctions of prenex L3 sentences in dMsPs3 ’s
much as in Ps3 ’s, and that this form of interpretation coincides with tarskian
interpretation.
Proposition 3.4. Let (h3 , h2 , h1 , h0 ) be a directed many-sorted polyadic homomorphism from Lcrc
to an A in Ps3 . Then h3 ∪ h2 ∪ h1 ∪ h0 is a mapping from
3
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 proposition 2.5, 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 proposition
2.6.
⊓
⊔
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.
Proposition 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 many-sorted polyadic homomorphism (f3∗ , f2∗ , f1∗ , f0∗ )
to A.
from Lcrc
3
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 extraboolean operations of Lcrc
are preserved and that this is a unique directed
3
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.
Proposition 2.6 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 property 2.4, the two are the same on atomic
formulae. Letting X be the atomic formulae and Y = L33 ∪ L32 ∪ L31 ∪ L30 ,
in property 2.5 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.
⊓
⊔
As before a homomorphism is satisfying for a sentence if it is mapped to ⊤.
Definition 3.7. Let L30 be the sort of Lcrc
consisting of sentences. A directed
3
to an A in dMsPs3 is satismany-sorted polyadic homomorphism h from Lcrc
3
fying for a sentence φ ∈ L30 if h(φ) = ⊤.
The following allows us to use homomorphisms as representatives of tarskian
interpretations.
to an A in dMsPs3 correProposition 3.6. Each homomorphism from Lcrc
3
sponds to a tarskian interpretation of L3 as relations over some set U . Moreover
the 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 from Lcrc
3 to A with an embedding f from A to a full Ps3 . The
definition of dMsPs3 provides such an f . Secondly we use proposition 3.4 and
extend f ◦ h to a homomorphism (f ◦ h)∗ from L3 to the full Ps3 . By proposition
2.7 this is an interpretation in the usual sense.
If h(φ) = ⊤ then (f ◦ h)∗ is a satisfying interpretation since h and f are
homomorphisms, since homomorphisms are closed under composition and since
extension does not alter that which is extended. 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.
⊓
⊔
By the above we may refer to homomorphisms from Lcrc
to dMsPs3 ’s as
3
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 3.7. Let A be a (finite) dMsPs3 , let φ be a sentence of Lcrc
and
3
h an interpretation in A such that h is satisfying for φ. If ψ is a sentence of Lcrc
3
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.
⊓
⊔
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 O3′ denote the open formulae of L′3 , and O3′ /Γ the appropriate subboolean algebra of L′3 /Γ . Recall that O3′ is closed under the operations r∗ , p∗ , s∗ .
Definition 3.8. This defines the directed many-sorted polyadic closure of the
atomic formulae of L′3 in L′3 /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
B2
B1
B0
=
=
=
=
the
the
the
the
boolean algebra O3′ /Γ
sub-boolean-algebra of L′3 /Γ generated by the ∃′2 -image of B3
sub-boolean-algebra of L′3 /Γ generated by the ∃′1 -image of B2
sub-boolean-algebra of L′3 /Γ generated by the ∃′0 -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 }
Proposition 3.8. Let Γ be a consistent and complete L′3 theory. As long as the
number of atomic formulae of L′3 is finite, their directed many-sorted polyadic
closure in L′3 /Γ is finite.
Proof. The initial boolean algebra B3 is O3′ /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.
⊓
⊔
Note that in the above proposition, Γ needs only be complete enough to contain
formulae in the closure. Since the closure is finite there is to each satisfiable L′3
sentence a finite Γ sufficient for the closure to be defined. There is no general way
of computing consistent completions 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 satisfiable first order
theory has a completion.
Corollary 3.1. A finite conjunction of prenex L3 sentences, is satisfiable iff it
is satisfiable in a finite dMsPs3 .
Proof. Let L′3 denote the language whose relation symbols are those of the given
conjunction. View such a conjunction as an L′3 theory by itself, and let Γ denote
one of its consistent completions. The sentence is then satisfied in L′3 /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 L′3 /Γ . For the
other direction; a satisfying interpretation in a finite dMsPs3 does by proposition 3.4 extend to a satisfying interpretation in the Ps3 of which the finite
dMsPs3 is a directed many-sorted sub-algebra.
⊓
⊔
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 n-th order language, or any other language where
checking whether an axiom holds in a finite algebra is recursive.
Corollary 3.2. The class of finite dMsPs3 ’s is not recursively enumerable.
Proof. For the purpose of arriving at a contradiction assume that there is a way
of recursively enumerating finite dMsPs3 ’s. One could then write a procedure
that recognised only satisfiable and all satisfiable first order sentence that worked
as follows. This procedure would first transform an input to an equi-satisfiable
conjunction of prenex L3 sentence by the reduction of Büchi. Then the procedure would enumerate finite dMsPs3 ’s, and do exhaustive search for satisfying
interpretations in each of them. If a satisfying interpretation were found it would
terminate. Since every satisfiable conjunction of prenex sentences has a satisfying interpretation in a finite dMsPs3 (corollary 3.1), this procedure terminates
for every satisfiable sentence. It only terminates for satisfiable sentences as satisfying interpretations in dMsPs3 ’s extend to interpretations in the usual sense
(proposition 3.4). Assuming the existence of such a procedure was shown to lead
to a contradiction by Church and Turing.
⊓
⊔
3.6
The closure as an algorithm
Assume that Γ is a complete and consistent theory in a language L′3 , 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 L′3 in L′3 /Γ
can then be seen as an algorithm. First of all, the definition of the closure has the
overall structure of an algorithm. 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 O3′ /Γ , and the process terminates because the number
of equivalence classes of O3′ /Γ is finite. One can also show that all of O3′ /Γ is
generated like this by induction on the well-ordering used in the definition of
L′3 /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 L′3 /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 (proposition
2.8), all one needs to store, is information about how the operations behave on
the atoms of the sub-boolean algebras. Let, for example, [φ] and [ψ] be atoms
of O3′ /Γ . Note that φ and ψ need not be atomic formulae. Then r′ ([φ ∨ ψ]) for
instance is determined by r′ ([φ]) ∨′ r′ ([ψ]). By proposition 1.2 one may represent
O3′ /Γ by an algebra with 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 r′ ([φ]) ∨′ r′ ([ψ]) is then determined by
the bit-wise join of the representatives of r′ ([φ]) and r′ ([ψ]). This allows for a
logarithmic reduction of the size of the tables for the corresponding algebra. For
instance, O3′ /Γ of the theory of a dense order without endpoints has 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 cubic and reasonably
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. 3. One
ought then end up with 6 pieces that are 3 dimensional, and thus visible. Wedged
between these pieces there are 6 triangular 2-dimensional pieces, and finally there
is a 13th, 1-dimensional piece where the 3 defining planes meet. These 13 pieces
correspond to the atoms of O3′ /Γ by the embedding obtained by restricting the
usual interpretation of O3′ , as ternary relations over the dense order (R, <), to
the image of µΓ (definition 2.9).
Fig. 3. Three planes of a cube defined by ¬(x < y ∨ y < x), ¬(x < z ∨ z < x) and
¬(y < z ∨ z < y).
3.7
Exhaustive search for satisfying interpretations in a dMsPs3
L′3
Let
be a language with a finite number of relation-symbols and Γ a finite
consistent L′3 theory that is complete or sufficiently complete for a directed
many-sorted polyadic closure to be defined. If Γ is known then the closure can
be computed and put in the form of tables for operations over a finite set of
formulae.
Given 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 L′3 in L′3 /Γ . If a satisfying interpretation
is found, that interpretation represents a satisfying interpretation in the usual
sense by proposition 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 theory
turned out to have 13 atoms. For full Ps3 ’s over finite sets U the atoms are the
one-element subsets of U 3 . Therefore, the full Ps3 over 2 has 8 atoms and 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.
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 a component of a disprover. 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 4.1. 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) as follows.
r(t)(abc) = r(t(abc ◦ r)),
p(t)(abc) = p(t(abc ◦ p)),
s(t)(abc) = s(t(abc
◦ s)),
W
c0 (t)(abc) = Wu∈2 c0 (t(ubc)),
c1 (t)(abc) = Wu∈2 c1 (t(auc)),
c2 (t)(abc) = u∈2 c2 (t(abu)).
By abuse of notation let 23 → A denote the algebra just defined. Two propositions about the construction follow.
Proposition 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. 4.
⊓
⊔
Proposition 4.2. Any finitely satisfiable sentence is satisfied in an iterate of
the above construction beginning with an arbitrary Ps3 .
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 finite set, with 2n elements for a suitable n ∈ ω.
The proposition follows since any finitely satisfiable equality-free sentence is
satisfiable in a structure with 2n elements, for some n ∈ ω.
⊓
⊔
Fig. 4. Cylindrification along x-axis of an element of 23 → P([0, 1i3 ). Here [0, 1i is a
half-open interval of the reals and the element represents a relation in [0, 2i3 visualised
by suitably stacking 8 copies of [0, 1i3 .
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. The author has used an implemented decision procedure for Presburger
arithmetic for this, [KMS02]. For instance one can, over night, compute the
closure of the atomic formulae of L′3 in L′3 /Γ , where L′3 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. The inputs of the disprover are sentences φ ∈ L3 ,
whose relation symbols may be disjoint from, and many more than, those of L′3 .
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.
..
.
Progress is made as each step can be carried out by exhaustive search. A brief
analysis, in terms of size of search-space follows. By Trakhtenbrots theorem
[Tra50] and by proposition 4.2 the sentences recognised by this procedure form
a non-recursive class. In the example, the initial sort B3 of A, turns out to have
13 atoms, which can be verified by a geometric argument similar to the one
depicted in fig. 3. 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 property 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
according to the presented method, and are publicly available with source code
included. 3 .
5
Related constructions and algebras
The construction of definition 4.1 is rather similar to what is called the cardinal
multiple of theories in [FV59] (Section 4.7), when bearing in mind that polyadic
set algebras correspond to complete and consistent theories. The constructions of
[FV59] have been generalised and carried over to the setting of polyadic algebras
in [Dai63], where the main construction is called the tensor product of polyadic
algebras. The latter is mainly about infinite, not necessarily atomic nor complete,
polyadic algebras so the definition goes via the Stone-space of the boolean reducts
of the algebras. 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 finitedimensional 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 ’∨’ and ’∧’ instead of the ring-operations + and · respectively. 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 A and B 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 proposition 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 [Ber59], more are mentioned in [Ném91], including category theory
approaches. 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
3
http://flipper.berlios.de
partial or many-sorted algebras are not generally such that each arity respecting
mapping of non-logical symbols into an A extends naturally to an interpretation
in A of each sub-formula of each sentence of an entire reduction class. 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.
6
Concluding remarks
The author believes that the identification of the class of dMsPs3 ’s, is a contribution. The contribution being that they are are directed. Directedness enables
the downward Löwenheim-Skolem like property (corollary 3.1), for each sentence of a 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 4.1 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 3.7. The disprover works by a generalisation
of finite model search since, by proposition 4.2, 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.
6.1
Acknowledgement
The author is thankful to Prof. S. O. Aanderaa for ideas and guidance regarding
precursors to the disprovers described here.
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