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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
Proof search and countermodel construction for biintuitionistic propositional logic with labelled sequents
 In TABLEAUX
, 2009
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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.
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
AnnotationFree Sequent Calculi for Full Intuitionistic Linear Logic ∗
"... Full Intuitionistic Linear Logic (FILL) is multiplicative intuitionistic linear logic extended with par. Its proof theory has been notoriously difficult to get right, and existing sequent calculi all involve inference rules with complex annotations to guarantee soundness and cutelimination. We give ..."
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Full Intuitionistic Linear Logic (FILL) is multiplicative intuitionistic linear logic extended with par. Its proof theory has been notoriously difficult to get right, and existing sequent calculi all involve inference rules with complex annotations to guarantee soundness and cutelimination. We give a simple and annotationfree display calculus for FILL which satisfies Belnap’s generic cutelimination theorem. To do so, our display calculus actually handles an extension of FILL, called BiIntuitionistic Linear Logic (BiILL), with an ‘exclusion ’ connective defined via an adjunction with par. We refine our display calculus for BiILL into a cutfree nested sequent calculus with deep inference in which the explicit structural rules of the display calculus become admissible. A separation property guarantees that proofs of FILL formulae in the deep inference calculus contain no trace of exclusion. Each such rule is sound for the semantics of FILL, thus our deep inference calculus and display calculus are conservative over FILL. The deep inference calculus also enjoys the subformula property and terminating backward proof search, which gives the NPcompleteness of BiILL and FILL.
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
G.L.: Nested Sequents Calculi for Normal Conditional Logics
 Journal of Logic and Computation
"... Nested sequent calculi are a useful generalization of ordinary sequent calculi, where sequents are allowed to occur within sequents. Nested sequent calculi have been profitably employed in the area of (multi)modal logic to obtain analytic and modular proof systems for these logics. In this work, we ..."
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Nested sequent calculi are a useful generalization of ordinary sequent calculi, where sequents are allowed to occur within sequents. Nested sequent calculi have been profitably employed in the area of (multi)modal logic to obtain analytic and modular proof systems for these logics. In this work, we extend the realm of nested sequents by providing nested sequent calculi for the basic conditional logic CK and some of its significant extensions. We provide also a calculus for Kraus Lehman Magidor cumulative logic C. The calculi are internal (a sequent can be directly translated into a formula), cutfree and analytic. Moreover, they can be used to design (sometimes optimal) decision procedures for the respective logics, and to obtain complexity upper bounds. Our calculi are an argument in favour of nested sequent calculi for modal logics and alike, showing their versatility and power. 1
Tableau Development for a BiIntuitionistic Tense Logic ⋆
"... Abstract. The paper introduces a biintuitionistic logic with two modal operators and their tense versions. The semantics is defined by Kripke models in which the set of worlds carries a preorder relation as well as an accessibility relation, and the two relations are linked by a stability conditio ..."
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Abstract. The paper introduces a biintuitionistic logic with two modal operators and their tense versions. The semantics is defined by Kripke models in which the set of worlds carries a preorder relation as well as an accessibility relation, and the two relations are linked by a stability condition. A special case of these models arises from graphs in which the worlds are interpreted as nodes and edges of graphs, and formulae represent subgraphs. The preorder is the incidence structure of the graphs. These examples provide an account of time including both time points and intervals, with the accessibility relation providing the order on the time structure. The logic we present is decidable and has the effective finite model property. We present a tableau calculus for the logic which is sound, complete and terminating. The MetTel system has been used to generate a prover from this tableau calculus. 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