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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 19 (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 10 (3 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 the ..."
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Cited by 5 (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)’, IEEE Computer
, 1996
"... ..."
Strategies in RigidVariable Methods
"... We study complexity of methods using rigid variables, like the method of matings or the tableau method, on a decidable class of predicate calculus with equality. We show some intrinsic complications introduced by rigid variables. We also consider strategies for increasing multiplicity in rigidvaria ..."
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We study complexity of methods using rigid variables, like the method of matings or the tableau method, on a decidable class of predicate calculus with equality. We show some intrinsic complications introduced by rigid variables. We also consider strategies for increasing multiplicity in rigidvariable methods, and formally show that the use of intelligent strategies can result in an essential gain in efficiency. 1
CHAPTER 1 EQUALITY AND OTHER THEORIES
"... Theory reasoning is an important technique for increasing the efficiency of automated deduction systems. The knowledge from a given domain (or theory) is made use of by applying efficient methods for reasoning in that domain. The general purpose foreground reasoner calls a special purpose background ..."
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Theory reasoning is an important technique for increasing the efficiency of automated deduction systems. The knowledge from a given domain (or theory) is made use of by applying efficient methods for reasoning in that domain. The general purpose foreground reasoner calls a special purpose background reasoner to
Rigid Variables Considered Harmful
, 1997
"... We study complexity of methods using rigid variables, like the method of matings or the tableau method, on a decidable class of predicate calculus with equality. We show some intrinsic complications introduced by rigid variables. We also consider strategies for increasing multiplicity in rigidvaria ..."
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We study complexity of methods using rigid variables, like the method of matings or the tableau method, on a decidable class of predicate calculus with equality. We show some intrinsic complications introduced by rigid variables. We also consider strategies for increasing multiplicity in rigidvariable methods, and formally show that the use of intelligent strategies can result in an essential gain in efficiency. 1 Section 1. Introduction Section 1 Introduction Automated reasoning methods for firstorder classical logic can generally be divided in two classes. Methods of the first class use universal variables (resolution [34], the inverse method [27]). Variables in these methods are local to a clause (formula, sequent) and can be considered as universally quantified in this clause (respectively formula or sequent). [29, 28] characterized these methods as local methods (see also [30]). Methods of the second class use rigid variables (the tableau method [4], the mating or the connecti...