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Integrating Gandalf and HOL
 Theorem Proving in Higher Order Logics: TPHOLs ’99, LNCS 1690
, 1999
"... Gandalf is a firstorder resolution theoremprover, optimized for speed and specializing in manipulations of large clauses. In this paper I describe GANDALF TAC, a HOL tactic that proves goals by calling Gandalf and mirroring the resulting proofs in HOL. This call can occur over a network, and a ..."
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Cited by 43 (2 self)
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Gandalf is a firstorder resolution theoremprover, optimized for speed and specializing in manipulations of large clauses. In this paper I describe GANDALF TAC, a HOL tactic that proves goals by calling Gandalf and mirroring the resulting proofs in HOL. This call can occur over a network, and a Gandalf server may be set up servicing multiple HOL clients. In addition, the translation of the Gandalf proof into HOL fits in with the LCF model and guarantees logical consistency.
Focusing the inverse method for linear logic
 Proceedings of CSL 2005
, 2005
"... 1.1 Quantification and the subformula property.................. 3 1.2 Ground forward sequent calculus......................... 5 1.3 Lifting to free variables............................... 10 ..."
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Cited by 39 (11 self)
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1.1 Quantification and the subformula property.................. 3 1.2 Ground forward sequent calculus......................... 5 1.3 Lifting to free variables............................... 10
Tabled HigherOrder Logic Programming
 In 20th International Conference on Automated Deduction
, 2003
"... Elf is a general metalanguage for the specification and implementation of logical systems in the style of the logical framework LF. Based on a logic programming interpretation, it supports executing logical systems and reasoning with and about them, thereby reducing the effort required for each ..."
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Elf is a general metalanguage for the specification and implementation of logical systems in the style of the logical framework LF. Based on a logic programming interpretation, it supports executing logical systems and reasoning with and about them, thereby reducing the effort required for each particular logical system. The traditional logic programming paradigm is extended by replacing firstorder terms with dependently typed terms and allowing implication and universal quantification in the bodies of clauses. These higherorder features allow us to model concisely and elegantly conditions on variables and the discharge of assumptions which are prevalent in many logical systems. However, many specifications are not executable under the traditional logic programming semantics and performance may be hampered by redundant computation. To address these problems, I propose a tabled higherorder logic programming interpretation for Elf. Some redundant computation is eliminated by memoizing subcomputation and reusing its result later. If we do not distinguish different proofs for a property, then search based on tabled logic programming is complete and terminates for programs with bounded recursion. In this proposal, I present a prooftheoretical characterization for tabled higherorder logic programming. It is the basis of the implemented prototype for tabled logic programming interpreter for Elf. Preliminary experiments indicate that many more logical specifications are executable under the tabled semantics. In addition, tabled computation leads to more efficient execution of programs. The goal of the thesis is to demonstrate that tabled logic programming allows us to efficiently automate reasoning with and about logical systems in the logical f...
ProofSearch in Intuitionistic Logic Based on Constraint Satisfaction
 Theorem Proving with Analytic Tableaux and Related Methods. 5th International Workshop, TABLEAUX '96, volume 1071 of Lecture Notes in Artificial Intelligence
, 1996
"... 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 NPcomplete. As ..."
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Cited by 20 (7 self)
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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 NPcomplete. As an application of our technique, we demonstrate PSPACEcompleteness 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...
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...
The ILTP problem library for intuitionistic logic, release v1.1
 Journal of Automated Reasoning
"... Abstract. The Intuitionistic Logic Theorem Proving (ILTP) library provides a platform for testing and benchmarking automated theorem proving (ATP) systems for propositional and firstorder intuitionistic logic. It includes about 2800 problems in a standardized syntax from 24 problem domains collecte ..."
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Abstract. The Intuitionistic Logic Theorem Proving (ILTP) library provides a platform for testing and benchmarking automated theorem proving (ATP) systems for propositional and firstorder intuitionistic logic. It includes about 2800 problems in a standardized syntax from 24 problem domains collected from various sources that are appropriate for intuitionistic logic. For each problem intuitionistic status and difficulty rating were obtained by running comprehensive tests of currently available intuitionistic ATP systems on all problems in the library. Thus for the first time the testing and evaluation of intuitionistic ATP systems is put onto a firm basis. Keywords: ILTP, problem library, benchmarking, experimental evaluation, ATP, intuitionistic logic
ileanTAP: An Intuitionistic Theorem Prover
 Proc. 6th TABLEAUX Conference, LNAI 1227
, 1997
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The Inverse Method
, 2001
"... this paper every formula is equivalent to a formula in negation normal form ..."
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Cited by 13 (1 self)
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this paper every formula is equivalent to a formula in negation normal form
Towards efficient subsumption
 Conference on Automated Deduction
, 1998
"... Abstract. We propose several methods for writing efficient subsumption procedures for nonunit clauses, tested in practice as parts incorporated into the Gandalf family of theorem provers. Versions of Gandalf exist for classical logic, first order intuitionistic logic and type theory. Subsumption is ..."
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Abstract. We propose several methods for writing efficient subsumption procedures for nonunit clauses, tested in practice as parts incorporated into the Gandalf family of theorem provers. Versions of Gandalf exist for classical logic, first order intuitionistic logic and type theory. Subsumption is one of the most important techniques for cutting down search space in resolution theorem proving. However, for many problem categories most of the proof search time is spent on subsumption. While acceptable efficiency has been achieved for subsuming unit clauses (see [7], [2]), the nonunit subsumption tends to slow provers down prohibitively. We propose several methods for writing efficient subsumption procedures for nonunit clauses, succesfully tested in practice as parts built into the Gandalf family of theorem provers: – ordering literals according to a certain subsumption measure – indexing first two literals of each nonunit clause – precomputed properties of terms, literals and clauses – a hierarchy of fast filters for clausetoclause subsumption – combining subsumption with clause simplification – linear search among the strongly reduced number of candidates for back subsumption The presented methods for substitution were among the key techniques enabling the classical version of Gandalf to win the MIX division of the CASC14 prover contest in 1997. The approach of the paper is purely empirical, presenting the methods and bringing some statistical evidence. 1 Gandalf Family of Provers Before continuing with the details of the subsumption methods we will present an overview of the Gandalf family of provers. We use the name Gandalf for the interdependent, codesharing, resolutionbased automated theorem provers we are developing: a resolution prover for firstorder intuitionistic logic Tammet [9], for a fragment of MartinLöf’s type theory Tammet [10] and for firstorder
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 ...