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Cutelimination and proofsearch for biintuitionistic logic using nested sequents
, 2008
"... We propose a new sequent calculus for biintuitionistic logic which sits somewhere between display calculi and traditional sequent calculi by using nested sequents. Our calculus enjoys a simple (purely syntactic) cutelimination proof as do display calculi. But it has an easily derivable variant cal ..."
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We propose a new sequent calculus for biintuitionistic logic which sits somewhere between display calculi and traditional sequent calculi by using nested sequents. Our calculus enjoys a simple (purely syntactic) cutelimination proof as do display calculi. But it has an easily derivable variant calculus which is amenable to automated proof search as are (some) traditional sequent calculi. We first present the initial calculus and its cutelimination proof. We then present the derived calculus, and then present a proofsearch strategy which allows it to be used for automated proof search. We prove that this search strategy is terminating and complete by showing how it can be used to mimic derivations obtained from an existing calculus GBiInt for biintuitionistic logic. As far as we know, our new calculus is the first sequent calculus for biintuitionistic logic which uses no semantic additions like labels, which has a purely syntactic cutelimination proof, and which can be used naturally for backwards proofsearch.
Relating Sequent Calculi for Biintuitionistic Propositional Logic
"... Abstract. Biintuitionistic logic is the conservative extension of intuitionistic logic with a connective dual to implication. It is sometimes presented as a symmetric constructive subsystem of classical logic. In this paper, we compare three sequent calculi for biintuitionistic propositional logic ..."
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Abstract. Biintuitionistic logic is the conservative extension of intuitionistic logic with a connective dual to implication. It is sometimes presented as a symmetric constructive subsystem of classical logic. In this paper, we compare three sequent calculi for biintuitionistic propositional logic: (1) a basic standardstyle sequent calculus that restricts the premises of implicationright and exclusionleft inferences to be singleconclusion resp. singleassumption and is incomplete without the cut rule, (2) the calculus with nested sequents by Goré et al., where a complete class of cuts is encapsulated into special “unnest ” rules and (3) a cutfree labelled sequent calculus derived from the Kripke semantics of the logic. We show that these calculi can be translated into each other and discuss the ineliminable cuts of the standardstyle sequent calculus. 1
A Connectionbased Characterization of Biintuitionistic Validity
"... Abstract. We give a connectionbased characterization of validity in propositional biintuitionistic logic in terms of speci c directed graphs called Rgraphs. Such a characterization is wellsuited for deriving labelled proofsystems with countermodel construction facilities. We rst de ne the noti ..."
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Abstract. We give a connectionbased characterization of validity in propositional biintuitionistic logic in terms of speci c directed graphs called Rgraphs. Such a characterization is wellsuited for deriving labelled proofsystems with countermodel construction facilities. We rst de ne the notion of biintuitionistic Rgraph from which we then obtain a connectionbased characterization of propositional biintuitionistic validity and derive a sound and complete freevariable labelled sequent calculus that admits cutelimination and also variable splitting. 1
Nested Sequents for Intuitionistic Logics Melvin Fitting
, 2012
"... Relatively recently nested sequent systems for modal logics have come to be seen as an attractive deep reasoning extension of familiar sequent calculi. In an earlier paper I showed there was a strong connection between modal nested sequents and modal prefixed tableaux. In this paper I show the conne ..."
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Relatively recently nested sequent systems for modal logics have come to be seen as an attractive deep reasoning extension of familiar sequent calculi. In an earlier paper I showed there was a strong connection between modal nested sequents and modal prefixed tableaux. In this paper I show the connection continues to intuitionistic logic, both standard and constant domain, relating nested intuitionistic sequent calculi to intuitionistic prefixed tableaux. Modal nested sequent machinery generalizes one sided sequent calculi—intuitionistic nested sequents similarly generalize two sided sequents. It is noteworthy that the resulting system for constant domain intuitionistic logic is particularly simple. It amounts to a combination of intuitionistic propositional rules and classical quantifier rules, a combination that is known to be inadequate when conventional intuitionistic sequent systems are used.
Relating Sequent Calculi for Biintuitionistic Propositional Logic Luı́s Pinto
"... Biintuitionistic logic is the conservative extension of intuitionistic logic with a connective dual to implication. It is sometimes presented as a symmetric constructive subsystem of classical logic. In this paper, we compare three sequent calculi for biintuitionistic propositional logic: (1) a ba ..."
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Biintuitionistic logic is the conservative extension of intuitionistic logic with a connective dual to implication. It is sometimes presented as a symmetric constructive subsystem of classical logic. In this paper, we compare three sequent calculi for biintuitionistic propositional logic: (1) a basic standardstyle sequent calculus that restricts the premises of implicationright and exclusionleft inferences to be singleconclusion resp. singleassumption and is incomplete without the cut rule, (2) the calculus with nested sequents by Gore ́ et al., where a complete class of cuts is encapsulated into special “unnest ” rules and (3) a cutfree labelled sequent calculus derived from the Kripke semantics of the logic. We show that these calculi can be translated into each other and discuss the ineliminable cuts of the standardstyle sequent calculus. 1