The formalization of abductive reasoning is still an open question: there is no general agreement on the boundary of some basic concepts, such as preference criteria for explanations, and the extension to first order logic has not been settled. Investigating the nature of abduction outside the context of resolution based logic programming still deserves attention, in order to characterize abductive explanations without tailoring them to any fixed method of computation. In fact, resolution is surely not the best tool for facing meta-logical and proof-theoretical questions. In this work the analysis of the concepts involved in abductive reasoning is based on analytical proof systems, i.e. tableaux and Gentzen-type systems. A proof theoretical abduction method for first order classical logic is defined, based on the sequent calculus and a dual one, based on semantic tableaux. The methods are sound and complete and work for full first order logic, without requiring any preliminary reductio...

We characterize provability in intuitionistic predicate logic in terms of derivation skeletons and constraints and study the problem of instantiations of a skeleton to valid derivations. We prove that for two different notions of a skeleton the problem is respectively polynomial and NP-complete. As an application of our technique, we demonstrate PSPACE-completeness of the prenex fragment of intuitionistic logic. We outline some applications of the proposed technique in automated reasoning. y y Copyright c fl 1995, 1996 Andrei Voronkov. This technical report and other technical reports in this series can be obtained at http://www.csd.uu.se/~thomas/reports.html or at ftp.csd.uu.se in the directory pub/papers/reports. Some reports can be updated, check one of these addresses for the latest version. Section 1 Introduction The characterization of provability for classical logic in terms of matrices was given by Kanger [9, 10] and Prawitz [19, 20] and is in a fact a reformulation of the...

Simultaneous rigid E-unification was introduced in 1987 by Gallier, Raatz and Snyder. It is used in the area of automated reasoning with equality in extension procedures, like the tableau method or the connection method. Many articles in this area assumed the existence of an algorithm for the simultaneous rigid E-unification problem. There were several faulty proofs of the decidability of this problem. In this paper we prove that simultaneous rigid E-unification is undecidable. As a consequence, we obtain the undecidability of the 9 -fragment of intuitionistic logic with equality. 1 Introduction Simultaneous rigid E-unification plays a crucial role in automatic proof methods for first order logic with equality based on sequent calculi, such as semantic tableaux [13], the connection method [7] (also known as the mating method [1]), model elimination [25] and a dozen other procedures. All these methods are based on the Herbrand theorem and express the idea that the proof-search can ...

Abstract not found

this paper every formula is equivalent to a formula in negation normal form

We demonstrate how to handle equality in the inverse method using equality elimination. In the equality elimination method, proofs consist of two parts. In the first part we try to solve equations obtaining so called solution clauses. In the second part, we perform the usual sequent proof search by the inverse method. Our method is called equality elimination because we eliminate all occurrences of equality in the first part of the proof. Solution clauses are obtained by using a very strong strategy -- basic superposition. Unlike the previous approach proposed by Maslov, we prove completeness of our method with most general substitutions and with ordering restrictions. We also note that these technique can be adapted to extension procedures, like the connection method. Unlike other approaches, we do not require the use of rigid or mixed E-unification.

In this paper we continue to develop the approach to automated search for theorem proofs started in Kyiv in 1960-1970s. This approach presupposes the development of deductive techniques used for the processing of mathematical texts, written in a formal first-order language, close to the natural language used in mathematical papers. We construct two logical calculi, gS and mS, satisfying the following requirements: the syntactical form of the initial problem should be preserved; the proof search should be goal-oriented; preliminary skolemization is not obligatory; equality handling should be separated from the deduction process. The calculus gS is a machine-oriented sequent-type calculus with "large-block" inference rules for first-order classical logic. The calculus mS is a further development of the calculus gS, enriched with formal analogs of the natural proof search techniques such as definition handling and application of auxiliary propositions. The results on soundness and completeness of gS and mS are given.

Recently it was proved that the problem of simultaneous rigid E-unification (SREU) is undecidable. Here we perform an in-depth investigation of this matter and obtain that one can use SREU to uniformly represent any recursively enumerable set. From the exact form of this representation follows that SREU is undecidable already for 6 rigid equations with ground left hand sides and 2 variables. There is a close correspondence between solvability of SREU problems and provability of the corresponding formulas in intuitionistic first order logic with equality. Due to this correspondence we obtain a new (uniform) representation of the recursively enumerable sets in intuitionistic first order logic with equality with one binary function symbol and a countable set of constants. From this result follows the undecidability of the 99-fragment of intuitionistic logic with equality. This is an improvement of a recent result regarding the undecidability of the 9 -fragment in general. Contents 1 ...

We introduce the equality elimination method which is a new procedure for dealing with Horn clause logic programs with equality. The method combines SLD-resolution with a bottom-up equation solving. By solving equations, we try to transform a logic program with equality to a logic program without equality. The transformation uses basic superposition as the main operation. We prove soundness and completeness of the equality elimination method. We also show that approaches based on complete sets of E-unifiers are not satisfying. In particular, we provide a negative solution to the open problem of completeness of SLDE + -resolution. Contents 1 Introduction 2 2 Preliminaries 4 3 The equality elimination method 7 3.1 Optimizations : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 17 4 Related work 21 Bibliography 23 Section 1 Introduction This paper deals with a new approach to logic programming with equality. By a logic program with equality we underst...

this paper is an attempt at providing a systematic presentation of Quantified Modal Logics (with constant domains and rigid designators). We present a set of modular, uniform, normalizing, sound and complete labelled sequent calculi for all QMLs whose frame properties can be expressed as a finite set of first-order sentences with equality. We first present CQK, a calculus for the logic QK, and then we extend it to any such logic QL. Each calculus, called CQL, is modular (obtained by adding rules to CQK), uniform (each added rule is clearly related to a property of the frame), normalizing (frame reasoning only happens at the top of the proof tree) and Kripke-sound and complete for QL. We improve on the existing literature on the subject (mainly, [21]) by extending the class of logics for which such a presentation is given, and by giving a new proof of soundness and completeness.