This report provides a detailed description of the TPTP Problem Library for automated theorem proving systems. The library is available via Internet, and forms a common basis for development of and experimentation with automated theorem provers. This report provides: ffl the motivations for building the library; ffl a discussion of the inadequacies of previous problem collections, and how these have been resolved in the TPTP; ffl a description of the library structure, including overview information; ffl descriptions of supplementary utility programs; ffl guidelines for obtaining and using the library; Contents 1 Introduction 2 1.1 Previous Problem Collections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 What is Required? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Inside the TPTP 6 2.1 The TPTP Domain Structure . . . . . . . . . . . . . . . . . . . . . ...

In this paper we focus on two powerful techniques to obtain compact clause normal forms: Renaming of formulae and refined Skolemization methods. We illustrate their effect on various examples. By an exhaustive experiment of all first-order TPTP problems, it shows that our clause normal form transformation yields fewer clauses and fewer literals than the methods known and used so far. This often allows for exponentially shorter proofs and, in some cases, it makes it even possible for a theorem prover to find a proof where it was unable to do so with more standard clause normal form transformations. 1

. "prove((E,F),A,B,C,D) :- !, prove(E,[F---A],B,C,D). prove((E;F),A,B,C,D) :- !, prove(E,A,B,C,D), prove(F,A,B,C,D). prove(all(H,I),A,B,C,D) :- !, "+length(C,D), copyterm((H,I,C),(G,F,C)), append(A,[all(H,I)],E), prove(F,E,B,[G---C],D). prove(A,,[C---D],,) :- ((A= -(B); -(A)=B)) -? (unify(B,C); prove(A,[],D,,)). prove(A,[E---F],B,C,D) :- prove(E,F,[A---B],C,D)." implements a first-order theorem prover based on free-variable semantic tableaux. It is complete, sound, and efficient. 1 Introduction The Prolog program listed in the abstract implements a complete and sound theorem prover for first-order logic; it is based on free-variable semantic tableaux (Fitting, 1990). We call this lean deduction: the idea is to achieve maximal efficiency from minimal means. We will see that the above program is indeed very efficient---not although but because it is extremely short and compact. Our approach surely does not lead to a deduction system which is superior to highly sophisticated systems li...

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t represents the sort restrictions on the variables. There are two extra inference rules which transform the sort constraint into solved form: Sort resolution and empty sort. These rules 1 The name is the result of a lunch break, FLOTTER means "faster", in German. 2 Synergetic Prover Augmenting Superposition with Sorts, SPASS means "fun", in German. implement a specific strategy of the sorted unification algorithm [15] on the sort constraint literals. In addition to these inference rules, SPASS includes a splitting rule. The splitting rule is a variant of the usual fi-rule of tableau. If SPASS splits a clause into two different cases, the two parts will not share variables, i.e. these parts can independently be refuted. For SPASS we implemented powerful reduction rules: tautology deletion, subsumption, condensing, an efficient variant of contextual rewriting,

In his well-known paper "How computer should think" ([Be77b]) Belnap argues that four valued semantics is a very suitable setting for computerized reasoning. In this paper we vindicate this thesis by showing that the logical role that the four-valued structure has among Ginsberg's well-known bilattices is similar to the role that the two-valued algebra has among Boolean algebras. Specifically, we provide several theorems that show that the most useful bilatticevalued logics can actually be characterized as four-valued inference relations. In addition, we compare the use of three-valued logics with the use of four-valued logics, and show that at least for the task of handling inconsistent or uncertain information, the comparison is in favor of the latter. Keyworkds: Bilattices, Paraconsistency, Multiple-valued systems, Preferential logics, Reasoning. 1 Introduction In [Be77a, Be77b] Belnap introduced a logic intended to deal in a useful way with inconsistent and incomplete information....

A lot of methods have been proposed (and sometimes implemented) for proof search in the propositional modal logics K, KT, and S4. It is difficult to compare the usefulness of these methods in practice, since mostly only the execution times for a few simple formulas have been published. We try to improve this unsatisfactory situation by presenting a set of benchmark formulas. Note that we do not just list formulas, but give a method that allows to compare different provers today and in the future. As a starting point we give the results we obtained when we applied this benchmark method to the Logics Workbench (LWB). Moreover we hope that the discussion of postulates concerning benchmark tests for automated theorem provers help to obtain improved benchmark methods for other logics, too.

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A logic for specification and verification is derived from the axioms of Zermelo-Fraenkel set theory. The proofs are performed using the proof assistant Isabelle. Isabelle is generic, supporting several different logics. Isabelle has the flexibility to adapt to variants of set theory. Its higher-order syntax supports the definition of new binding operators. Unknowns in subgoals can be instantiated incrementally. The paper describes the derivation of rules for descriptions, relations and functions, and discusses interactive proofs of Cantor’s Theorem, the Composition of Homomorphisms challenge [9], and Ramsey’s Theorem [5]. A generic proof assistant can stand up against provers dedicated to particular logics. Key words. Isabelle, set theory, generic theorem proving, Ramsey’s Theorem,

Abstract: A generic tableau prover has been implemented and integrated with Isabelle [Paulson, 1994]. Compared with classical rst-order logic provers, it has numerous extensions that allow it to reason with any supplied set of tableau rules. It has a higherorder syntax in order to support user-de ned binding operators, such as those of set theory. The uni cation algorithm is rst-order instead of higher-order, but it includes modi cations to handle bound variables. The proof, when found, is returned to Isabelle as a list of tactics. Because Isabelle veri es the proof, the prover can cut corners for e ciency's sake without compromising soundness. For example, the prover can use type information to guide the search without storing type information in full. Categories: F.4, I.1