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On ProofSearch in Intuitionistic Logic with Equality, or Back to Simultaneous Rigid EUnification
 Automated Deduction  CADE13
, 1996
"... We characterize provability in intuitionistic logic with equality in terms of a constraint calculus. This characterization uncovers close connections between provability in intuitionistic logic with equality and solutions to simultaneous rigid Eunification. We show that the problem of existence of ..."
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Cited by 17 (9 self)
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We characterize provability in intuitionistic logic with equality in terms of a constraint calculus. This characterization uncovers close connections between provability in intuitionistic logic with equality and solutions to simultaneous rigid Eunification. We show that the problem of existence of a sequent proof with a given skeleton is polynomialtime equivalent to simultaneous rigid Eunifiability. This gives us a proof procedure for intuitionistic logic with equality modulo simultaneous rigid Eunification. We also show that simultaneous rigid Eunifiability is polynomialtime reducible to intuitionistic logic with equality. Thus, any proof procedure for intuitionistic logic with equality can be considered as a procedure for simultaneous rigid Eunifiability. In turn, any procedure for simultaneous rigid Eunifiability gives a procedure for establishing provability in intuitionistic logic with equality. 2 2 Copyright c fl 1995, 1996 Andrei Voronkov. This technical report and ot...
Uniform Representation of Recursively Enumerable Sets with Simultaneous Rigid EUnification
, 1996
"... Recently it was proved that the problem of simultaneous rigid Eunification (SREU) is undecidable. Here we perform an indepth 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 ..."
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Cited by 9 (2 self)
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Recently it was proved that the problem of simultaneous rigid Eunification (SREU) is undecidable. Here we perform an indepth 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 99fragment of intuitionistic logic with equality. This is an improvement of a recent result regarding the undecidability of the 9 fragment in general. Contents 1 ...
Herbrand's Theorem, Automated Reasoning and Semantic Tableaux
 in `Proc. IEEE Conference on Logic in Computer Science (LICS)', IEEE Computer
, 1998
"... We overview recent results related to Herbrand's theorem and tableaulike methods of automated deduction. Based on an analysis and discussion of these results, open problems are posed and new research directions are suggested. Section 1 Introduction Numerous results related to the Herbrand theorem ..."
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Cited by 4 (3 self)
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We overview recent results related to Herbrand's theorem and tableaulike methods of automated deduction. Based on an analysis and discussion of these results, open problems are posed and new research directions are suggested. Section 1 Introduction Numerous results related to the Herbrand theorem and the foundations of semantics tableaux were obtained recently. These results shed a new light on the nature of such methods and computational complexity of several problems related to the Herbrand theorem. The aim of this paper is to overview these results, sketch new research directions and pose some open problems. We also prove several new results. The main questions we ask in the paper are the following. ffl Are tableaux and related methods inherently inefficient? ffl In what cases are these methods efficient? ffl In what cases and how can we increase efficiency of these methods? This paper is structured as follows. In Section 3 we give a short description of the tableau and relate...
Simultaneous rigid EUnification and other decision problems related to the Herbrand theorem
, 1998
"... Recently, a number of results have been published related to simultaneous rigid Eunification and Herbrand's theorem for logic with equality. The aim of this article is to overview these results, fill in some proofs that have only been sketched before, and present some new results. ..."
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Cited by 3 (2 self)
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Recently, a number of results have been published related to simultaneous rigid Eunification and Herbrand's theorem for logic with equality. The aim of this article is to overview these results, fill in some proofs that have only been sketched before, and present some new results.
Simultaneous Rigid EUnification and Related Algorithmic Problems
 in ``Eleventh Annual IEEE Symposium on Logic in Computer Science (LICS'96
, 1996
"... The notion of simultaneous rigid Eunification was introduced in 1987 in the area of automated theorem proving with equality in sequentbased methods, for example the connection method or the tableau method. Recently, simultaneous rigid Eunification was shown undecidable. Despite the importance of ..."
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Cited by 1 (0 self)
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The notion of simultaneous rigid Eunification was introduced in 1987 in the area of automated theorem proving with equality in sequentbased methods, for example the connection method or the tableau method. Recently, simultaneous rigid Eunification was shown undecidable. Despite the importance of this notion, for example in theorem proving in intuitionistic logic, very little is known of its decidable fragments. We prove decidability results for fragments of monadic simultaneous rigid Eunification and show the connections between this notion and some algorithmic problems of logic and computer science. 1 Introduction Simultaneous rigid Eunification plays a crucial role in automatic proof methods for firstorder logic with equality based on sequent calculi, such as semantic tableaux [13], the connection method [6] (also known as the mating method [1]), model elimination [26] and a dozen other procedures. All these methods are based on the Herbrand theorem and express the idea that ...
The Logic in Computer Science Column
 In Bulletin of the European Association for Theoretical Computer Science
, 1996
"... The famous Herbrand's theorem of mathematical logic plays an important role in automated theorem proving. In the first part of this article, we recall the theorem and formulate a number of natural decision problems related to it. Somewhat surprisingly, these problems happen to be equivalent. One of ..."
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The famous Herbrand's theorem of mathematical logic plays an important role in automated theorem proving. In the first part of this article, we recall the theorem and formulate a number of natural decision problems related to it. Somewhat surprisingly, these problems happen to be equivalent. One of these problems is the socalled simultaneous rigid Eunification problem. In the second part, we survey recent result on the simultaneous rigid Eunification problem.
The Logic in Computer Science Column
 In Bulletin of the European Association for Theoretical Computer Science
, 1988
"... The famous Herbrand's theorem of mathematical logic plays an important role in automated theorem proving. In the first part of this article, we recall the theorem and formulate a number of natural decision problems related to it. Somewhat surprisingly, these problems happen to be equivalent. One of ..."
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The famous Herbrand's theorem of mathematical logic plays an important role in automated theorem proving. In the first part of this article, we recall the theorem and formulate a number of natural decision problems related to it. Somewhat surprisingly, these problems happen to be equivalent. One of these problems is the socalled simultaneous rigid Eunification problem. In the second part, we survey recent result on the simultaneous rigid Eunification problem. 1 Problems This simultaneous rigid Eunification : : : It has too procedural a definition. Vaughan Pratt, private communication. We start with some decision problems having nonprocedural definitions. Most of these problems naturally stem from the famous Herbrand theorem. 1.1 Herbrand's theorem and six decision problems The Herbrand theorem [28] asserts that an existential firstorder formula 9¯x'(¯x) is provable (in classical firstorder logic) if and only if a particular disjunction '( ¯ t 1 ) : : : '( ¯ t n ) is provab...
The Logic in Computer Science Column
 In Bulletin of the European Association for Theoretical Computer Science
, 1988
"... The famous Herbrand's theorem of mathematical logic plays an important role in automated theorem proving. In the first part of this article, we recall the theorem and formulate a number of natural decision problems related to it. Somewhat surprisingly, these problems happen to be equivalent. One ..."
Abstract
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The famous Herbrand's theorem of mathematical logic plays an important role in automated theorem proving. In the first part of this article, we recall the theorem and formulate a number of natural decision problems related to it. Somewhat surprisingly, these problems happen to be equivalent. One of these problems is the socalled simultaneous rigid Eunification problem. In the second part, we survey recent result on the simultaneous rigid problem.