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Natural Deduction as HigherOrder Resolution
 Journal of Logic Programming
, 1986
"... An interactive theorem prover, Isabelle, is under development. In LCF, each inference rule is represented by one function for forwards proof and another (a tactic) for backwards proof. In Isabelle, each inference rule is represented by a Horn clause. ..."
Abstract

Cited by 54 (8 self)
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An interactive theorem prover, Isabelle, is under development. In LCF, each inference rule is represented by one function for forwards proof and another (a tactic) for backwards proof. In Isabelle, each inference rule is represented by a Horn clause.
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...
ProofTerms for Classical and Intuitionistic Resolution (Extended Abstract)
, 1996
"... We exploit a system of realizers for classical logic, and a translation from resolution into the sequent calculus, to assess the intuitionistic force of classical resolution for a fragment of intuitionistic logic. This approach is in contrast to formulating locally intuitionistically sound resol ..."
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Cited by 12 (3 self)
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We exploit a system of realizers for classical logic, and a translation from resolution into the sequent calculus, to assess the intuitionistic force of classical resolution for a fragment of intuitionistic logic. This approach is in contrast to formulating locally intuitionistically sound resolution rules. The techniques use the fflcalculus, a development of Parigot's calculus.
Hintikka Multiplicities in Matrix Decision Methods for Some Propositional Modal Logics
, 1997
"... . This work is a study of the intertranslatability of two closely related proof methods, i.e. tableau (or sequent) and connection based, in the case of the propositional modal logics K, K4, T, S4, paying particular attention to the relation between matrix multiplicity and multiple use of 0formula ..."
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Cited by 3 (0 self)
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. This work is a study of the intertranslatability of two closely related proof methods, i.e. tableau (or sequent) and connection based, in the case of the propositional modal logics K, K4, T, S4, paying particular attention to the relation between matrix multiplicity and multiple use of 0formulae (contractions) in tableaux/sequent proofs. The motivation of the work is the following. Since the role of a multiplicity in matrix methods is the encoding of the number of copies of a given formula that are needed in order to prove a valid formula, it is important to find upper bounds for multiplicities in order to reduce as much as possible the search space for proofs. Moreover, it is obviously a crucial issue if the matrix method is to be used as a decision method. We exploit previous results establishing upper bounds on the number of contractions in tableau/sequent proofs [4], in order to establish upper bounds for multiplicities in matrix systems. We obtain two kinds of upper bounds: i...
On the Intuitionistic Force of Classical Search (Extended Abstract)
 PROC. TABLEAUX '96, LNCS 1071
, 1996
"... The combinatorics of proofsearch in classical propositional logic lies at the heart of most efficient proof procedures because the logic admits leastcommitment search. The key to extending such methods to quantifiers and nonclassical connectives is the problem of recovering this leastcommitm ..."
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Cited by 2 (2 self)
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The combinatorics of proofsearch in classical propositional logic lies at the heart of most efficient proof procedures because the logic admits leastcommitment search. The key to extending such methods to quantifiers and nonclassical connectives is the problem of recovering this leastcommitment principle in the context of the nonclassical /nonpropositional logic; i.e., characterizing when a leastcommitment (classical) search yields sufficient evidence for provability in the (nonclassical) logic. In this paper, we present such a characterization for the (oe; ) fragment of intuitionistic logic using the calculus: a system of realizers for classical free deduction (cf. natural deduction) due to Parigot. We show how this characterization can be used to define a notion of uniform proof, and a corresponding proof procedure, which extends that of Miller et al. to multipleconclusioned sequent systems. The procedure is sound and complete for the fragment of intuitionisti...