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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.
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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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
Deep inference in Biintuitionistic logic
 In Int Workshop on Logic, Language, Information and Computation, WoLLIC 2009, LNAI 5514
, 2009
"... Abstract. Biintuitionistic logic is the extension of intuitionistic logic with exclusion, a connective dual to implication. Cutelimination in biintuitionistic logic is complicated due to the interaction between these two connectives, and various extended sequent calculi, including a display calcu ..."
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Abstract. Biintuitionistic logic is the extension of intuitionistic logic with exclusion, a connective dual to implication. Cutelimination in biintuitionistic logic is complicated due to the interaction between these two connectives, and various extended sequent calculi, including a display calculus, have been proposed to address this problem. In this paper, we present a new extended sequent calculus DBiInt for biintuitionistic logic which uses nested sequents and “deep inference”, i.e., inference rules can be applied at any level in the nested sequent. We show that DBiInt can simulate our previous “shallow ” sequent calculus LBiInt. In particular, we show that deep inference can simulate the residuation rules in the displaylike shallow calculus LBiInt. We also consider proof search and give a simple restriction of DBiInt which allows terminating proof search. Thus our work is another step towards addressing the broader problem of proof search in display logic. 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
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
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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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
An Interactive Prover for Biintuitionistic Logic
"... Abstract. In this paper we present an interactive prover for deciding formulas in propositional biintuitionistic logic (BiInt). This tool is based on a recent connectionbased characterization of biintuitionistic validity through biintuitionistic resource graphs (biRG). After giving the main conc ..."
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Abstract. In this paper we present an interactive prover for deciding formulas in propositional biintuitionistic logic (BiInt). This tool is based on a recent connectionbased characterization of biintuitionistic validity through biintuitionistic resource graphs (biRG). After giving the main concepts and principles we illustrate how to use this interactive proof or countermodel building assistant and emphasize the interest of biintuitionistic resource graphs for proving or refuting BiInt formulas. 1 Biintuitionistic Propositional Logic Biintuitionistic logic (BiInt) is a conservative extension of intuitionistic logic that introduces a connective �, called coimplication, which acts as a dual to implication. It was first studied by Rauszer who gives a Hilbert calculus with Kripke and algebraic semantics [5] and more recently by Crolard as a way to define proof systems that work as programming languages in which values and continuations are handled in a symmetric way [1]. Cutfree calculi for BiInt have been proposed using deep inference, nested
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.