First-Order Theorem Proving in Practice
M. C. Fernández-Gago, U. Hustadt, C. Dixon, M. Fisher and B. Konev
Department of Computer Science
The University of Liverpool, Liverpool L69 7ZF, UK
{mcarmen,ullrich,clare,michael,konev}@csc.liv.ac.uk
http://www.csc.liv.ac.uk/research/logics/
Introduction
First-order temporal logic (FOTL), the extension of first-order logic with operators dealing with time, is a
powerful and expressive formalism with many potential applications. This expressive logic can be viewed
as a framework in which we can investigate problems specified in other logics. The monodic fragment of
first-order temporal logic [6] is a useful fragment which possesses good computational properties such as
completeness and sometimes even decidability.
Temporal logics of knowledge, KL(n) , (see for example [4]) are useful for dealing with situations
where the knowledge of agents in a system is involved. Here we use a translation from temporal logics
of knowledge into the monodic fragment of first-order temporal logic. Essentially the epistemic part is
translated into a fragment of classical first-order logic. We can then use a theorem-prover for monodic
first-order temporal logic to prove properties of the translated formulae. This allows problems specified
in temporal logics of knowledge to be verified automatically without needing a specialised theorem prover
for temporal logics of knowledge.
We assume the usual syntax and semantics for first-order temporal logic [6] and for the temporal
logic of knowledge [4]. A FOTL formula φ is called monodic if any subformulae of the form T φ, where
, g (or φ1 T φ2 , where T is one of U or W ) contains at most one free variable.
T is one of ♦,
Translating KL(n) into FOTL
We want to translate KL(n) formulae into the monodic fragment of first-order temporal logic. Without
loss of generality we can assume that formulae are already in SNFK normal form (a normal form for
KL(n) that removes many temporal operators and requires formulae to be of a particular form, see [3]).
Let ∪j Tj be a set of clauses, written in the normal form SNFK . Then ∪j Tj can be translated into firstorder temporal logic by applying the transformations π0 and π1 . In the following, p is a literal, φ and ψ
are formulae in KL(n) , Q is a new predicate symbol introduced in order to define the beginning of time,
st is a constant representing the initial moment in time, QKi p is a new predicate uniquely associated
with Ki p and Ri is the accessibility relation for the modal operator Ki :
^
π0 [∪j Tj ] = Q(st) ∧
∀xπ1 (Tj , x).
j
The translation π1 is as given in Figure 1. For each Ki p we add the clauses:
π1 (start , x) = Q(x)
π1 (true , x) = true
π1 (false , x) = false
π1 (p, x) = P (x)
π1 (¬p, x) = ¬P (x)
π1 (φ ∨ ψ, x) = π1 (φ, x) ∨ π1 (ψ, x)
π1 (φ ∧ ψ, x)
π1 (φ ⇒ ψ, x)
π1 ( gφ, x)
π1 (♦φ, x)
π1 (Ki p, x)
π1 (¬Ki p, x)
Figure 1: π1 translation
=
=
=
=
=
=
π1 (φ, x) ∧ π1 (ψ, x)
π1 (φ, x) ⇒ π1 (ψ, x)
gπ1 (φ, x)
♦π1 (φ, x)
QKi p (x)
Q¬Ki p (x)
(QKi p (x) ⇒ (∀y.Ri (x, y) ⇒ QKi p (y)))
(QKi p (x) ⇒ (∀y.Ri (x, y) ⇒ P (y)))
and for each ¬Ki p we add the clause:
(Q¬Ki p (x) ⇒ (∃y.Ri (x, y) ∧ P (y))).
For every modal operator, Ki , we will also add reflexivity and symmetry axioms to the translation.
∀x.Ri (x, x)
Reflexivity
∀x, y.(Ri (x, y) ⇒ Ri (y, x))
Symmetry
Note, the translation of the epistemic part of the logic is non-standard. In particular, transitivity is dealt
with differently, as compared to reflexivity and symmetry, to ensure termination when the resolution
rules are applied. This translation is based on the axiomatic translation principle given in [9]. The
translation has been shown to preserve satisfiability in [5].
Experimental Results
We have applied the above translation to problems specified in KL(n) . The resultant formulae have
been sent to TeMP [7], a resolution theorem prover for monodic FOTL, based on the calculus presented
in [8]. The implementation of TeMP makes use of the fact that inference steps in this calculus can be
simulated by inference steps in a first-order ordered resolution calculus.
We have proved properties of case studies and examples using this approach including a specification of
the board game Cluedo Re [2], the well known muddy children problem (see for example [3]), and security
protocols [1], each originally expressed in KL(n) . Experimental results are given in [5]. Comparisons
with the standard translation are also given.
Acknowledgements The full version of this abstract appears in [5]. This work was partially supported by the EPSRC project: Analysis and Mechanisation of Decidable First-Order Temporal Logics
(GR/R45376/01).
References
[1] C. Dixon and M.-C. Fernández Gago and M. Fisher and W. van der Hoek. Using Temporal Logics
of Knowledge in the Formal Verification of Security Protocols. In Proceedings of TIME 2004 the
Eleventh International Symposium on Temporal Representation and Reasoning, Tatihou, Normandie,
France, July 2004. IEEE Computer Society Press.
[2] C. Dixon. Miss Scarlett in the Ballroom with the Lead Piping. In 16th European Conference on
Artificial Intelligence (ECAI 2004), pages 995–996, Valencia, Spain, August 2004. IOS Press.
[3] C. Dixon, M. Fisher, and M. Wooldridge. Resolution for Temporal Logics of Knowledge. Journal of
Logic and Computation, 8(3):345–372, 1998.
[4] R. Fagin, J. Y. Halpern, Y. Moses, and M. Y. Vardi. Reasoning About Knowledge. MIT Press, 1995.
[5] M.C. Fernández Gago, U. Hustadt, C. Dixon, M. Fisher, and B. Konev. First-Order Verification in
Practice. Journal of Automated Reasoning, 2005. (To appear).
[6] I. Hodkinson, F. Wolter, and M. Zakharyaschev. Decidable Fragments of First-Order Temporal
Logics. Annals of Pure Applied Logic, 106(1-3):85–134, 2000.
[7] U. Hustadt, B. Konev, A. Riazanov, and A. Voronkov. TeMP: A temporal monodic prover. In
David A. Basin and Michaël Rusinowitch, editors, Proceedings of the Second International Joint
Conference on Automated Reasoning (IJCAR), volume 3097 of LNAI, pages 326–330. Springer, 2004.
[8] B. Konev, A. Degtyarev, C. Dixon, M. Fisher, and U. Hustadt. Mechanising First-Order Temporal
Resolution. Information and Computation, 2005. In press.
[9] R. A. Schmidt and U. Hustadt. A principle for incorporating axioms into the first-order translation
of modal formulae. In Automated Deduction—CADE-19, volume 2741 of Lecture Notes in Artificial
Intelligence, pages 412–426. Springer, 2003.
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