. "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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"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 based on free-variable semantic tableaux [6]. We call this lean theorem proving : the idea is to achieve maximal efficiency from minimal means. We will see that the above program is indeed very efficient---not...

We analyze the search efficiency of a number of common refutational theorem proving strategies for first-order logic. Search efficiency is concerned with the total number of proofs and partial proofs generated, rather than with the sizes of the proofs. We show that most common strategies produce search spaces of exponential size even on simple sets of clauses, or else are not sensitive to the goal. However, clause linking, which uses a reduction to propositional calculus, has behavior that is more favorable in some respects, a property that it shares with methods that cache subgoals. A strategy which is of interest for term-rewriting based theorem proving is the A-ordering strategy, and we discuss it in some detail. We show some advantages of A-ordering over other strategies, which may help to explain its efficiency in practice. We also point out some of its combinatorial inefficiencies, especially in relation to goal-sensitivity and irrelevant clauses. In addition, SLD-reso...

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Cut-elimination is the most prominent form of proof transformation in logic. The elimination of cuts in formal proofs corresponds to the removal of intermediate statements (lemmas) in mathematical proofs. The cut-elimination method CERES (cut-elimination by resolution) works by constructing a set of clauses from a proof with cuts. Any resolution refutation of this set then serves as a skeleton of an LK-proof with only atomic cuts. In this paper we present an extension of CERES to a calculus LKDe which is stronger than the Gentzen calculus LK (it contains rules for introduction of definitions and equality rules). This extension makes it much easier to formalize mathematical proofs and increases the performance of the cut-elimination method. The system CERES already proved efficient in handling very large proofs.

The distinction between analytic and synthetic proofs is a very old and important one: An analytic proof uses only notions occurring in the proved statement while a synthetic proof uses additional ones. This distinction has been made precise by Gentzen’s famous cut-elimination theorem stating that synthetic proofs can be transformed into analytic ones. CERES (cut-elimination by resolution) is a cut-elimination method that has the advantage of considering the original proof in its full generality which allows the extraction of different analytic arguments from it. In this paper we will use an implementation of CERES to analyze Fürstenberg’s topological proof of the infinity of primes. We will show that Euclid’s original proof can be obtained as one of the analytic arguments from Fürstenberg’s proof. This constitutes a proof-of-concept example for a semi-automated analysis of realistic mathematical proofs providing new information about them.

Modal nonmonotonic logics constitute a well-known family of knowledge -representation formalisms capturing ideally rational agents reasoning about their own beliefs. Although these formalisms are extensively studied from a theoretical point of view, most of these approaches lack generally available solvers thus far. In this paper, we show how variants of Moore's autoepistemic logic can be axiomatised by means of quantified Boolean formulas (QBFs). More specifically, we provide polynomial reductions of the basic reasoning tasks associated with these logics into the evaluation problem of QBFs. Since there are now efficient QBF-solvers, this reduction technique yields a practicably relevant approach to build prototype reasoning systems for these formalisms. We incorporated our encodings within the system QUIP and tested their performance on a class of benchmark problems using different underlying QBF-solvers.

. 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 ...

In this paper, we show how an approach to belief revision and belief contraction can be axiomatised by means of quantified Boolean formulas. Specifically, we consider the approach of belief change scenarios, a general framework that has been introduced for expressing different forms of belief change. The essential idea is that for a belief change scenario (K, R, C), the set of formulas K, representing the knowledge base, is modified so that the sets of formulas R and C are respectively true in, and consistent with the result. By restricting the form of a belief change scenario, one obtains specific belief change operators including belief revision, contraction, update, and merging. For both the general approach and for specific operators, we give a quantified Boolean formula such that satisfying truth assignments to the free variables correspond to belief change extensions in the original approach. Hence, we reduce the problem of determining the results of a belief change operation to that of satisfiability. This approach has several benefits. First, it furnishes an axiomatic specification of belief change with respect to belief change scenarios. This then leads to further insight into the belief change framework.