TeMP: A Temporal Monodic Prover

We present TeMP—the first experimental system for testing validity of monodic temporal logic formulae. The prover implements fine-grained temporal resolution. The core operations required by the procedure are performed by an efficient resolution-based prover for classical first-order logic. 1 Monodic First-Order Temporal Logic First-Order Temporal Logic, FOTL, is an extension of classical first-order logic by temporal operators for a discrete linear model of time (isomorphic to N, that is, the most commonly used model of time). Formulae of this logic are interpreted over structures that associate with each element n of N, representing a moment in time, a first-order structure (Dn; In) with its own non-empty domain Dn. In this paper we make the expanding domain assumption, that is, Dn Dm if n < m. The set of valid formulae of this logic is not recursively enumerable. However, the set of valid monodic formulae is known to be finitely axiomatisable [13]. A formula φ in a FOTL language without equality and function symbols (constants are allowed) is called monodic if any subformulae of φ of the form gψ, ψ, ψ, ψ1Uψ2 or ψ1Wψ2 contains at most one free variable. For example, the formulae 8x 9yP(x;y) and 8x P(x;c) are monodic, while 8x;y(P(x;y)) P(x;y)) is not monodic. The monodic fragment has a wide range of novel applications, for example in spatio-temporal logics [14, 5] and temporal description logics [1]. In this abstract we describe TeMP, the first automatic theorem prover for the monodic fragment of FOTL. 2 Monodic fine-grained temporal resolution Our temporal prover is based on fine-grained temporal resolution [9] which we briefly describe in this section. Every monodic temporal formula can be translated in a satisfiability equivalence preserving way into a clausal form. The calculus operates on four kinds of temporal clauses, called initial, universal, step, and eventuality clauses. Essentially, initial clauses hold only in the initial moment in time, all other kinds of clause hold in every moment in time. Initial and universal are ordinary first-order clauses, containing no temporal operators. Step clauses in the clausal form of monodic temporal ? Work supported by EPSRC grant GR/L87491. ?? On leave from Steklov Institute of Mathematics at St.Petersburg formulae are of the form p ) gq, where p and q are propositions, or of the form P(x)) gQ(x), where P and Q are unary predicate symbols and x a variable. During a derivation more general step clauses can be derived, which are of the form C ) gD, where C is a conjunction of propositions, atoms of the form P(x) and ground formulae of the form P(c), where P is a unary predicate symbol and c is a constant such that c occurs in the input formula, D is a disjunction of arbitrary literals, such that C and D have at most one free variable in common. The eventuality clauses are of the form L(x), where L(x) is a literal having at most one free variable. Monodic fine-grained temporal resolution consists of the eventuality resolution rule: 8x(A1(x)) g(B1(x))) : : : 8x(An(x)) g(Bn(x))) L(x) 8xVni=1:Ai(x) ( Ures) ; where U is the current set of all universal clauses, 8x(Ai(x) ) gBi(x)) are complex combinations of step clauses, called full merged step clauses [9], such that for all i 2 f1; : : : ;ng, the loop side conditions 8x(U^Bi(x)):L(x)) and 8x(U^Bi(x)) Wn j=1(A j(x))) are both valid; and the following five rules of fine-grained step resolution 1. First-order resolution between two universal clauses and factoring on a universal clause. The result is a universal clause. 2. First-order resolution between an initial and a universal clause, between two initial clauses, and factoring on an initial clause. The result is again an initial clause. 3. Fine-grained (restricted) step resolution. C1 ) g(D1_L) C2 ) g(D2_:M) (C1^C2)σ ) g(D1_D2)σ C1 ) g(D1_L) D2_:M C1σ) g(D1_D2)σ 4. (Step) factoring. C1 ) g(D1_L_M) C1σ) g(D1_L)σ (C1^L^M)) gD1 (C1^L)σ ) gD1σ 5. Clause conversion. A step clause of the form C ) gfalse is rewritten into the universal clause :C. In rules 1 to 5, we assume that different premises and conclusions of the deduction rules have no variables in common; variables may be renamed if necessary. In rules 3 and 4, σ is a most general unifier of the literals L and M such that σ does not map variables from C1 or C2 into a constant or a functional term. The input formula is unsatisfiable over expanding domains if and only if finegrained temporal resolution derives the empty clause (see [9], Theorem 8).