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35
A Resolution Theorem Prover for Intuitionistic Logic
 Proceedings of CADE13
, 1996
"... We use the general scheme of building resolution calculi (also called the inverse method) originating from S.Maslov and G.Mints to design and implement a resolution theorem prover for intuitionistic logic. A number of search strategies is introduced and proved complete. The resolution method is show ..."
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Cited by 42 (4 self)
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We use the general scheme of building resolution calculi (also called the inverse method) originating from S.Maslov and G.Mints to design and implement a resolution theorem prover for intuitionistic logic. A number of search strategies is introduced and proved complete. The resolution method is shown to be a decision procedure for a new syntactically described decidable class of intuitionistic logic. We compare the search strategies suitable for the resolution method with strategies suitable for the tableau method. The performance of our prover is compared with the performance of a tableau prover for intuitionistic logic presented in [17].
A Proof Procedure for the Logic of Hereditary Harrop Formulas
 JOURNAL OF AUTOMATED REASONING
, 1993
"... A proof procedure is presented for a class of formulas in intuitionistic logic. These formulas are the socalled goal formulas in the theory of hereditary Harrop formulas. Proof search inintuitionistic logic is complicated by the nonexistence of a Herbrandlike theorem for this logic: formulas cann ..."
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Cited by 30 (12 self)
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A proof procedure is presented for a class of formulas in intuitionistic logic. These formulas are the socalled goal formulas in the theory of hereditary Harrop formulas. Proof search inintuitionistic logic is complicated by the nonexistence of a Herbrandlike theorem for this logic: formulas cannot in general be preprocessed into a form such as the clausal form and the construction of a proof is often sensitive to the order in which the connectives and quantifiers are analyzed. An interesting aspect of the formulas we consider here is that this analysis can be carried out in a relatively controlled manner in their context. In particular, the task of finding a proof can be reduced to one of demonstrating that a formula follows from a set of assumptions with the next step in this process being determined by the structure of the conclusion formula. An acceptable implementation of this observation must utilize unification. However, since our formulas may contain universal and existential quantifiers in mixed order, care must be exercised to ensure the correctness of unification. One way of realizing this requirement involves labelling constants and variables and then using these labels to constrain unification. This form of unification is presented and used in a proof procedure for goal formulas in a firstorder version of hereditary Harrop formulas. Modifications to this procedure for the relevant formulas in a higherorder logic are also described. The proof procedure that we present has a practical value in that it provides the basis for an implementation of the logic programming language lambdaProlog.
Proof Search in Firstorder Linear Logic and Other Cutfree Sequent Calculi
 In LICS
, 1994
"... We present a general framework for proof search in firstorder cutfree sequent calculi and apply it to the specific case of linear logic. In this framework, Herbrand functions are used to encode universal quantification, and unification is used to instantiate existential quantifiers so that the eig ..."
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Cited by 27 (2 self)
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We present a general framework for proof search in firstorder cutfree sequent calculi and apply it to the specific case of linear logic. In this framework, Herbrand functions are used to encode universal quantification, and unification is used to instantiate existential quantifiers so that the eigenvariable conditions are respected. We present an optimization of this procedure that exploits the permutabilities of the subject logic. We prove the soundness and completeness of several related proof search procedures. This proof search framework is used to show that provability for firstorder MALL is in nexptime, and firstorder MLL is in np. Performance comparisons based on Prolog implementations of the procedures are also given. The optimization of the quantifier steps in proof search can be combined effectively with a number of other optimizations that are also based on permutability. 1 Introduction Since proofs contain more information than the theorems they prove, the main challen...
On Proof Normalization in Linear Logic
 Theoretical Computer Science
, 1994
"... We present a prooftheoretic foundation for automated deduction in linear logic. At first, we systematically study the permutability properties of the inference rules in this logical framework and exploit these to introduce an appropriate notion of forward and backward movement of an inference in a ..."
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Cited by 26 (12 self)
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We present a prooftheoretic foundation for automated deduction in linear logic. At first, we systematically study the permutability properties of the inference rules in this logical framework and exploit these to introduce an appropriate notion of forward and backward movement of an inference in a proof. Then we discuss the naturallyarising question of the redundancy reduction and investigate the possibilities of proof normalization which depend on the proof search strategy and the fragment we consider. Thus, we can define the concept of normal proof that might be the basis of works about automatic proof construction and design of logic programming languages based on linear logic. 1 Introduction Linear logic is a powerful and expressive logic with connections to a variety of topics in computer science. We are mainly interested by the significance it may have in different domains as logic programming or program synthesis through theorem proving. As a matter of fact, classical linear ...
First Order Abduction Via Tableau and Sequent Calculi
 Bulletin of the IGPL
, 1993
"... 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 conte ..."
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Cited by 22 (6 self)
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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 metalogical and prooftheoretical questions. In this work the analysis of the concepts involved in abductive reasoning is based on analytical proof systems, i.e. tableaux and Gentzentype 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...
Foundations of Proof Search Strategies Design in Linear Logic
 In Symposium on Logical Foundations of Computer Science
, 1994
"... In this paper, we investigate automated proof construction in classical linear logic (CLL) by giving logical foundations for the design of proof search strategies. We propose common theoretical foundations for topdown, bottomup and mixed proof search procedures with a systematic formalization of s ..."
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Cited by 20 (11 self)
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In this paper, we investigate automated proof construction in classical linear logic (CLL) by giving logical foundations for the design of proof search strategies. We propose common theoretical foundations for topdown, bottomup and mixed proof search procedures with a systematic formalization of strategies construction using the notions of immediate or chaining composition or decomposition, deduced from permutability properties and inference movements in a proof. Thus, we have logical bases for the design of proof strategies in CLL fragments and then we can propose sketches for their design.
On the Intuitionistic Force of Classical Search
 THEORETICAL COMPUTER SCIENCE
, 1996
"... The combinatorics of classical propositional logic lies at the heart of both local and global methods of proofsearch enabling the achievement of leastcommitment search. Extension of such methods to the predicate calculus, or to nonclassical systems, presents us with the problem of recovering ..."
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Cited by 19 (5 self)
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The combinatorics of classical propositional logic lies at the heart of both local and global methods of proofsearch enabling the achievement of leastcommitment search. Extension of such methods to the predicate calculus, or to nonclassical systems, presents us with the problem of recovering this leastcommitment principle in the context of noninvertible rules. One successful approach is to view the nonclassical logic as a perturbation on search in classical logic and characterize when a leastcommitment (classical) search yields sufficient evidence for provability in the (nonclassical) logic. This technique has been successfully applied to both local and global methods at the cost of subsidiary searches and is the analogue of the standard treatment of quantifiers via skolemization and unification. In this paper, we take a typetheoretic view of this approach for the case in which the nonclassical logic is intuitionistic. We develop a system of realizers (proofobje...
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 18 (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 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...
Linear Logic and Computation: A Survey
 Proof and Computation, Proceedings Marktoberdorf Summer School
, 1993
"... . This is a survey of computational aspects of linear logic related to proof search. Keywords. Linear logic, cut free proof search, logic programming, complexity. 1 Introduction Linear logic, introduced by Girard [14, 36, 32], is a refinement of classical logic. While the central notions of truth ..."
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Cited by 16 (6 self)
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. This is a survey of computational aspects of linear logic related to proof search. Keywords. Linear logic, cut free proof search, logic programming, complexity. 1 Introduction Linear logic, introduced by Girard [14, 36, 32], is a refinement of classical logic. While the central notions of truth (emphasized in classical logic) and proof construction (emphasized in intuitionistic logic) remain important in linear logic, it might be said that the emphasis in linear logic is on state. Linear logic is sometimes described as being resource sensitive because it provides an intrinsic and natural accounting of process states, events, and resources. Linear logic also sheds new light on classical logic and its relationship to intuitionistic logic, see Girard [15, 16] and Danos et al. [11]. An evocative semantic paradigm for linear logic by means of games is proposed by Blass [7] and by Abramsky and Jagadeesan [2]. As an intuitive motivation, let us consider reading logical deductions so tha...