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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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Cited by 11 (3 self)
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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.
Proof Search and CounterModel Construction for Biintuitionistic Propositional Logic with Labelled Sequents
"... Abstract. Biintuitionistic logic is a conservative extension of intuitionistic logic with a connective dual to implication, called exclusion. We present a sound and complete cutfree labelled sequent calculus for biintuitionistic propositional logic, BiInt, following S. Negri’s general method for ..."
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Cited by 6 (1 self)
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Abstract. Biintuitionistic logic is a conservative extension of intuitionistic logic with a connective dual to implication, called exclusion. We present a sound and complete cutfree labelled sequent calculus for biintuitionistic propositional logic, BiInt, following S. Negri’s general method for devising sequent calculi for normal modal logics. Although it arises as a natural formalization of the Kripke semantics, it is does not directly support proof search. To describe a proof search procedure, we develop a more algorithmic version that also allows for countermodel extraction from a failed proof attempt. 1
Kripke semantics for basic sequent systems
 In Proceedings of the 20th international
"... Abstract. We present a general method for providing Kripke semantics for the family of fullystructural multipleconclusion propositional sequent systems. In particular, many wellknown Kripke semantics for a variety of logics are easily obtained as special cases. This semantics is then used to obta ..."
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Cited by 3 (0 self)
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Abstract. We present a general method for providing Kripke semantics for the family of fullystructural multipleconclusion propositional sequent systems. In particular, many wellknown Kripke semantics for a variety of logics are easily obtained as special cases. This semantics is then used to obtain semantic characterizations of analytic sequent systems of this type, as well as of those admitting cutadmissibility. These characterizations serve as a uniform basis for semantic proofs of analyticity and cutadmissibility in such systems. 1
Countermodels from Sequent Calculi in MultiModal Logics
, 2012
"... A novel countermodelproducing decision procedure that applies to several multimodal logics, both intuitionistic and classical, is presented. Based on backwards search in labeled sequent calculi, the procedure employs a novel termination condition and countermodel construction. Using the procedure, ..."
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Cited by 1 (1 self)
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A novel countermodelproducing decision procedure that applies to several multimodal logics, both intuitionistic and classical, is presented. Based on backwards search in labeled sequent calculi, the procedure employs a novel termination condition and countermodel construction. Using the procedure, it is argued that multimodal variants of several classical and intuitionistic logics including K, T, K4, S4 and their combinations with D are decidable and have the finite model property. At least in the intuitionistic multimodal case, the decidability results are new. It is further shown that the countermodels produced by the procedure, starting from a set of hypotheses and no goals, characterize the atomic formulas provable from the hypotheses. 1
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 Unified Semantic Framework for Fullystructural Propositional
"... We identify a large family of fullystructural propositional sequent systems, which we call basic systems. We present a general uniform method for providing (potentially, nondeterministic) strongly sound and complete Kripkestyle semantics, which is applicable for every system of this family. In ad ..."
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We identify a large family of fullystructural propositional sequent systems, which we call basic systems. We present a general uniform method for providing (potentially, nondeterministic) strongly sound and complete Kripkestyle semantics, which is applicable for every system of this family. In addition, this method can also be applied when (i) some formulas are not allowed to appear in derivations, (ii) some formulas are not allowed to serve as cutformulas, and (iii) some instances of the identity axiom are not allowed to be used. This naturally leads to new semantic characterizations of analyticity (global subformula property), cutadmissibility and axiomexpansion in basic systems. We provide a large variety of examples showing that many soundness and completeness theorems for different sequent systems, as well as analyticity, cutadmissibility and axiomexpansion results, easily follow using the general method of this paper.
Decision procedures · Countermodels generation
"... Intuitionistic propositional logic whose proofs are linearly bounded in the length of the formula to be proved and satisfy the subformula property. We also introduce a sequent calculus RJ for intuitionistic unprovability with the same properties of LSJ. We show that from a refutation of RJ of a sequ ..."
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Intuitionistic propositional logic whose proofs are linearly bounded in the length of the formula to be proved and satisfy the subformula property. We also introduce a sequent calculus RJ for intuitionistic unprovability with the same properties of LSJ. We show that from a refutation of RJ of a sequent σ we can extract a Kripke countermodel for σ. Finally, we provide a procedure that given a sequent σ returns either a proof of σ in LSJ or a refutation in RJ such that the extracted countermodel is of minimal depth.