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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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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
Taming displayed tense logics using nested sequents with deep inference
 In Martin Giese and Arild Waaler, editors, Proceedings of TABLEAUX, volume 5607 of Lecture Notes in Computer Science
, 2009
"... Abstract. We consider two sequent calculi for tense logic in which the syntactic judgements are nested sequents, i.e., a tree of traditional onesided sequents built from multisets of formulae. Our first calculus SKt is a variant of Kashima’s calculus for Kt, which can also be seen as a display calcu ..."
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Abstract. We consider two sequent calculi for tense logic in which the syntactic judgements are nested sequents, i.e., a tree of traditional onesided sequents built from multisets of formulae. Our first calculus SKt is a variant of Kashima’s calculus for Kt, which can also be seen as a display calculus, and uses “shallow ” inference whereby inference rules are only applied to the toplevel nodes in the nested structures. The rules of SKt include certain structural rules, called “display postulates”, which are used to bring a node to the top level and thus in effect allow inference rules to be applied to an arbitrary node in a nested sequent. The cut elimination proof for SKt uses a proof substitution technique similar to that used in cut elimination for display logics. We then consider another, more natural, calculus DKt which contains no structural rules (and no display postulates), but which uses deepinference to apply inference rules directly at any node in a nested sequent. This calculus corresponds to Kashima’s S2Kt, but with all structural rules absorbed into logical rules. We show that SKt and DKt are equivalent, that is, any cutfree proof of SKt can be transformed into a cutfree proof of DKt, and vice versa. We consider two extensions of tense logic, Kt.S4 and S5, and show that this equivalence between shallow and deepsequent systems also holds. Since deepsequent systems contain no structural rules, proof search in the calculi is easier than in the shallow calculi. We outline such a procedure for the deepsequent system DKt and its S4 extension. 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 Theorem Prover for Boolean BI
"... While separation logic is acknowledged as an enabling technology for largescale program verification, most of the existing verification tools use only a fragment of separation logic that excludes separating implication. As the first step towards a verification tool using full separation logic, we d ..."
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While separation logic is acknowledged as an enabling technology for largescale program verification, most of the existing verification tools use only a fragment of separation logic that excludes separating implication. As the first step towards a verification tool using full separation logic, we develop a nested sequent calculus for Boolean BI (Bunched Implications), the underlying theory of separation logic, as well as a theorem prover based on it. A salient feature of our nested sequent calculus is that its sequent may have not only smaller child sequents but also multiple parent sequents, thus producing a graph structure of sequents instead of a tree structure. Our theorem prover is based on backward search in a refinement of the nested sequent calculus in which weakening and contraction are built into all the inference rules. We explain the details of designing our theorem prover and provide empirical evidence of its practicality.
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
On the BlokEsakia Theorem
"... Abstract We discuss the celebrated BlokEsakia theorem on the isomorphism between the lattices of extensions of intuitionistic propositional logic and the Grzegorczyk modal system. In particular, we present the original algebraic proof of this theorem found by Blok, and give a brief survey of genera ..."
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Abstract We discuss the celebrated BlokEsakia theorem on the isomorphism between the lattices of extensions of intuitionistic propositional logic and the Grzegorczyk modal system. In particular, we present the original algebraic proof of this theorem found by Blok, and give a brief survey of generalisations of the BlokEsakia theorem to extensions of intuitionistic logic with modal operators and coimplication. In memory of Leo Esakia 1
Grammar Logics in Nested Sequent Calculus: Proof Theory and Decision Procedures
"... A grammar logic refers to an extension of the multimodal logic K in which the modal axioms are generated from a formal grammar. We consider a proof theory, in nested sequent calculus, of grammar logics with converse, i.e., every modal operator [a] comes with a converse [ā]. Extending previous works ..."
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A grammar logic refers to an extension of the multimodal logic K in which the modal axioms are generated from a formal grammar. We consider a proof theory, in nested sequent calculus, of grammar logics with converse, i.e., every modal operator [a] comes with a converse [ā]. Extending previous works on nested sequent systems for tense logics, we show all grammar logics (with or without converse) can be formalised in nested sequent calculi, where the axioms are internalised in the calculi as structural rules. Syntactic cutelimination for these calculi is proved using a procedure similar to that for display logics. If the grammar is contextfree, then one can get rid of all structural rules, in favor of deep inference and additional propagation rules. We give a novel semidecision procedure for contextfree grammar logics, using nested sequent calculus with deep inference, and show that, in the case where the given contextfree grammar is regular, this procedure terminates. Unlike all other existing decision procedures for regular grammar logics in the literature, our procedure does not assume that a finite state automaton encoding the axioms is given. Keywords: Nested sequent calculus, display calculus, modal logics, deep inference.
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.
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