We introduce a modal language L which is obtained from standard modal logic by adding the Boolean operators on accessibility relations, the identity relation, and the converse of relations. It is proved that L has the same expressive power as the two-variable fragment F O2 of first-order logic, but speaks less succinctly about relational structures: if the number of relations is bounded, then L-satisfiability is ExpTimecomplete but F O2 satisfiability is NExpTime-complete. We indicate that the relation between L and F O2 provides a general framework for comparing modal and temporal languages with first-order languages.

. We give a finer analysis of the difficulty of proof search in classical first-order logic, other than just saying that it is undecidable. To do this, we identify several measures of difficulty of theorems, which we use as resource bounds to prune infinite proof search trees. In classical first-order logic without interpreted symbols, we prove that for all these measures, the search for a proof of bounded difficulty (i.e, for a simple proof) is \Sigma p 2 -complete. We also show that the same problem when the initial formula is a set of Horn clauses is only NP-complete, and examine the case of first-order logic modulo an equational theory. These results allow us not only to give estimations of the inherent difficulty of automated theorem proving problems, but to gain some insight into the computational relevance of several automated theorem proving methods. Topics: computational complexity, logics, computational issues in AI (automated theorem proving). 1 Introduction First-order ...

1. INTRODUCTION. For geometers, Hilbert’s influential work on the foundations of geometry is important. For analysts, Hilbert’s theory of integral equations is just as important. But the address “Mathematische Probleme ” [37] that David Hilbert (1862– 1943) delivered at the second International Congress of Mathematicians (ICM) in Paris has tremendous importance for all mathematicians. Moreover, a substantial part of

We improve upon a number of recent undecidability results related to the so-called Herbrand Skeleton Problem, the Simultaneous Rigid E-Unification Problem and the prenex fragment of intuitionistic logic with equality. Partially supported by grants from NSF, ONR and the Faculty of Science and Technology of Uppsala University. 1 Introduction We study classical first-order logic with equality but without any other relation symbols. The letters ' and / are reserved for quantifier-free formulas. The signature of a syntactic object S (a term, a set of terms, a formula, etc.) is the collection of function symbols in S augmented, in the case when S contains no constants, with a constant c. The language of S is the language of the signature of S. Any syntactic object is ground if it contains no variables. A substitution is ground if its range is ground, and it is said to be in a given language if the terms in its range are in that language. A set of substitutions is ground if each membe...

We prove that the two-variable fragment of first-order intuitionistic logic is undecidable, even without constants and equality. We also show that the two-variable fragment of a quantified modal logic L with expanding first-order domains is undecidable whenever there is a Kripke frame for L with a point having infinitely many successors (such are, in particular, the first-order extensions of practically all standard modal logics like K, K4, GL, S4, S5, K4.1, S4.2, GL.3, etc.). For many quantified modal logics, including those in the standard nomenclature above, even the monadic two-variable fragments turn out to be undecidable.

Abstract. In this paper we study the decidability of various fragments of monodic first-order temporal logic by temporal resolution. We focus on two resolution calculi, namely, monodic temporal resolution and finegrained temporal resolution. For the first, we state a very general decidability result, which is independent of the particular decision procedure used to decide the first-order part of the logic. For the second, we introduce refinements using orderings and selection functions. This allows us to transfer existing results on decidability by resolution for first-order fragments to monodic first-order temporal logic and obtain new decision procedures. The latter is of immediate practical value, due to the availability of TeMP, an implementation of fine-grained temporal resolution. 1

Abstract. In this paper, we show how the clausal temporal resolution technique developed for temporal logic provides an effective method for searching for invariants, and so is suitable for mechanising a wide class of temporal problems. We demonstrate that this scheme of searching for invariants can be also applied to a class of multi-predicate induction problems represented by mutually recursive definitions. Completeness of the approach, examples of the application of the scheme, and overview of the implementation are described. 1

Until recently, first-order temporal logic has been little understood.

In this paper we consider the specification and verification of infinite-state systems using temporal logic. In particular, we describe parameterised systems using a new variety of first-order temporal logic that is both powerful enough for this form of specification and tractable enough for practical deductive verification. Importantly, the power of the temporal language allows us to describe (and verify) asynchronous systems, communication delays and more complex liveness and fairness properties. These aspects appear difficult for many other approaches to infinite-state verification. 1.

it has less than 2 vertices. Finally, one adds a 6th, nullary operation, of type o: the constant 0, standing for the empty graph with no vertices. ⊓⊔ A Equivalences of logical formulas In this appendix, we discuss some equivalences and transformations of logical formulas which can be used to give upper bounds for the index of congruences considered in this paper, and to complete the proof of the effectiveness of certain notions (e.g. quantifier-free definition schemes). More specifically, we make precise in what sense we can state, as we do in the body of the paper, that the set of first-order (resp. monadic second-order) formulas over finite sets of relations, constants and free variables, and with a bounded quantification depth, can be considered as finite. Moreover, explicit upper bounds on the size of these finite sets are derived, which can be used to justify the termination of some of our algorithms, and in evaluating their complexity. That these upper bounds have unbounded levels of exponentiation is not unexpected. A.1 Boolean formulas Let p1,..., pn be Boolean variables and let Bn be the set of Boolean formulas written with these variables. It is well known that Bn is finite up to logical equivalence. For further reference, we record the following more precise statement. Proposition A.1 There exists a subset Bred n of Bn, of cardinality 22n such that every formula in Bn can be effectively reduced to an equivalent formula in Bred n. be the set of Boolean formulas in disjunctive normal form, where in each disjunct, variables occur at most once and in increasing order, no two disjuncts are equal, and disjuncts are ordered lexicographically. These constraints guarantee the announced cardinality of Bred n; the rest of the proof is classical. ⊓⊔ Proof. We let B red n Of course, the formula in B red