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15
Efficient LoopCheck for Backward Proof Search in Some NonClassical Propositional Logics
, 1996
"... . We consider the modal logics KT and S4, the tense logic K t , and the fragment IPC (^;!) of intuitionistic logic. For these logics backward proof search in the standard sequent or tableau calculi does not terminate in general. In terms of the respective Kripke semantics, there are several kinds of ..."
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Cited by 33 (1 self)
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. We consider the modal logics KT and S4, the tense logic K t , and the fragment IPC (^;!) of intuitionistic logic. For these logics backward proof search in the standard sequent or tableau calculi does not terminate in general. In terms of the respective Kripke semantics, there are several kinds of nontermination: loops inside a world (KT), innite resp. looping branches (S4, IPC (^;!) ), and innite branching degree (K t ). We give uniform sequentbased calculi that contain specically tailored loopchecks such that the eciency of proof search is not deteriorated. Moreover all these loopchecks are easy to implement and can be combined with optimizations. Note that our calculus for S4 is not a pure contractionfree sequent calculus, but this (theoretical) defect does not hinder its application in practice. 1 Introduction For many nonclassical propositional logics, backward proof search in the usual sequent calculi does not terminate in general. For all the logics we consider in th...
On an interpretation of second order quantification in first order intuitionistic propositional logic
 JOURNAL OF SYMBOLIC LOGIC, 57: 33 { 52
, 1992
"... We prove the following surprising property of Heyting's intuitionistic propositional calculus, IpC. Consider the collection of formulas, , built up from propositional variables (p; q; r; : ::) and falsity (?) using conjunction (^), disjunction (_) and implication (!). Write ` to indicate that such a ..."
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Cited by 23 (0 self)
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We prove the following surprising property of Heyting's intuitionistic propositional calculus, IpC. Consider the collection of formulas, , built up from propositional variables (p; q; r; : ::) and falsity (?) using conjunction (^), disjunction (_) and implication (!). Write ` to indicate that such a formula is intuitionistically valid. We show that for each variable p and formula there exists a formula Ap (e ectively computable from), containing only variables not equal to p which occur in, and such that for all formulas not involving p, ` ! Ap if and only if ` !. Consequently quanti cation over propositional variables can be modelled in IpC, and there is an interpretation of the second order propositional calculus, IpC2, in IpC which restricts to the identity on rst order propositions. An immediate corollary is the strengthening of the usual Interpolation Theorem for IpC to the statement that there are least and greatest interpolant formulas for any given pair of formulas. The result also has a number of interesting consequences for the algebraic counterpart of IpC, the theory of Heyting algebras. In particular we show that a model of IpC² can be constructed whose algebra of truthvalues is equal to any given Heyting algebra.
Connection Methods in Linear Logic and Proof Nets Construction
 Theoretical Computer Science
, 1999
"... Linear logic (LL) is the logical foundation of some typetheoretic languages and also of environments for specification and theorem proving. In this paper, we analyse the relationships between the proof net notion of LL and the connection notion used for efficient proofsearch in different logics. A ..."
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Cited by 12 (2 self)
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Linear logic (LL) is the logical foundation of some typetheoretic languages and also of environments for specification and theorem proving. In this paper, we analyse the relationships between the proof net notion of LL and the connection notion used for efficient proofsearch in different logics. Aiming at using proof nets as a tool for automated deduction in linear logic, we define a connectionbased characterization of provability in Multiplicative Linear Logic (MLL). We show that an algorithm for proof net construction can be seen as a proofsearch connection method. This central result is illustrated with a specific algorithm that is able to construct, for a provable MLL sequent, a set of connections, a proof net and a sequent proof. From these results we expect to extend to other LL fragments, we analyse what happens with the additive connectives of LL by tackling the additive fragment in a similar way.
Admissibility of Structural Rules for ContractionFree Systems of Intuitionistic Logic
 Journal of Symbolic Logic
, 2000
"... We give a direct proof of admissibility of cut and contraction for the contractionfree sequent calculus G4ip for intuitionistic propositional logic and for a corresponding multisuccedent calculus; this proof extends easily in the presence of quantifiers, in contrast to other, indirect, proofs, i.e. ..."
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Cited by 11 (4 self)
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We give a direct proof of admissibility of cut and contraction for the contractionfree sequent calculus G4ip for intuitionistic propositional logic and for a corresponding multisuccedent calculus; this proof extends easily in the presence of quantifiers, in contrast to other, indirect, proofs, i.e. those which use induction on sequent weight or appeal to admissibility of rules in other calculi.
Fast Tacticbased Theorem Proving
 TPHOLs 2000, LNCS 1869
, 2000
"... Theorem provers for higherorder logics often use tactics to implement automated proof search. Tactics use a generalpurpose metalanguage to implement both generalpurpose reasoning and computationally intensive domainspecific proof procedures. The generality of tactic provers has a performance pe ..."
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Cited by 9 (4 self)
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Theorem provers for higherorder logics often use tactics to implement automated proof search. Tactics use a generalpurpose metalanguage to implement both generalpurpose reasoning and computationally intensive domainspecific proof procedures. The generality of tactic provers has a performance penalty; the speed of proof search lags far behind specialpurpose provers. We present a new modular proving architecture that significantly increases the speed of the core logic engine.
STRIP: Structural sharing for efficient proofsearch
 Asian Computing Science Conference, ASIAN'99, LNCS 1742
, 1999
"... The STRIP system is a theorem prover for intuitionistic propositional logic with two main characteristics: it deals with the duplication of formulae during proofsearch from a fine and explicit management of formulae (as resources) based on a structural sharing and it builds, for a given formula, e ..."
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Cited by 7 (2 self)
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The STRIP system is a theorem prover for intuitionistic propositional logic with two main characteristics: it deals with the duplication of formulae during proofsearch from a fine and explicit management of formulae (as resources) based on a structural sharing and it builds, for a given formula, either a proof or a countermodel.
Semantic Labelled Tableaux for Propositional BI
 Journal of Logic and Computation
, 2003
"... In this paper, we study semantic labelled tableaux for the propositional Bunched Implications logic (BI) that freely combines intuitionistic logic (IL) and multiplicative intuitionistic linear logic (MILL). ..."
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Cited by 4 (0 self)
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In this paper, we study semantic labelled tableaux for the propositional Bunched Implications logic (BI) that freely combines intuitionistic logic (IL) and multiplicative intuitionistic linear logic (MILL).
Admissibility of Structural Rules for Extensions of Contractionfree Sequent Calculi
"... The contractionfree sequent calculus G4 for intuitionistic logic is extended by rules following a general rulescheme for nonlogical axioms. Admissibility of structural rules for these extensions is proved in a direct way by induction on derivations. This method permits the representation of variou ..."
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Cited by 4 (1 self)
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The contractionfree sequent calculus G4 for intuitionistic logic is extended by rules following a general rulescheme for nonlogical axioms. Admissibility of structural rules for these extensions is proved in a direct way by induction on derivations. This method permits the representation of various applied logics as complete, contraction and cutfree sequent calculus systems with some restrictions on the nature of the derivations. As specic examples, intuitionistic theories of apartness and order and (Robinsonstyle) arithmetic are treated. Keywords: applied sequent calculus, contractionfree, apartness, conservativity, cutelimination 1 Introduction Extension of a cutfree sequent calculus with new rules can easily destroy the cutfree nature of the calculus. In other words, if two calculi are equivalent in the sense of deriving the same sequents, the addition of some new rules to each calculus can generate nonequivalent calculi. In this paper we address a particular example wh...
Decidability Extracted: Synthesizing ``CorrectbyConstruction'' Decision Procedures from Constructive Proofs
, 1998
"... The topic of this thesis is the extraction of efficient and readable programs from formal constructive proofs of decidability. The proof methods employed to generate the efficient code are new and result in clean and readable Nuprl extracts for two nontrivial programs. They are based on the use of ..."
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Cited by 3 (0 self)
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The topic of this thesis is the extraction of efficient and readable programs from formal constructive proofs of decidability. The proof methods employed to generate the efficient code are new and result in clean and readable Nuprl extracts for two nontrivial programs. They are based on the use of Nuprl's set type and techniques for extracting efficient programs from induction principles. The constructive formal theories required to express the decidability theorems are of independent interest. They formally circumscribe the mathematical knowledge needed to understand the derived algorithms. The formal theories express concepts that are taught at the senior college level. The decidability proofs themselves, depending on this material, are of interest and are presented in some detail. The proof of decidability of classical propositional logic is relative to a semantics based on Kleene's strong threevalued logic. The constructive proof of intuitionistic decidability presented here is the first machine formalization of this proof. The exposition reveals aspects of the Nuprl tactic collection relevant to the creation of readable proofs; clear extracts and efficient code are illustrated in the discussion of the proofs.