Sequent of Relations Calculi: A Framework for Analytic Deduction in Many-valued Logics

Baaz, Matthias; Ciabattoni, Agata; Fermüller, Christian G.

doi:10.1007/978-3-7908-1769-0_6

Cited by 7 publications

(9 citation statements)

References 21 publications

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“…Among them, several calculi are better suited for proof search than hypersequent calculi. This holds in particular for sequent of relations calculi [17,13], goal-oriented proof procedures [43], the systems recently defined in [7,8,39] or the resolution-style chaining calculi used in [12]. However, the mentioned calculi cannot be modified in a simple way to include quantifiers, modalities or to formalize related logics.…”

Section: Resultsmentioning

confidence: 99%

Hypersequent Calculi for Godel Logics -- a Survey

Baaz¹,

Ciabattoni²,

Fermüller³

2003

Journal of Logic and Computation

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Hypersequent calculi arise by generalizing standard sequent calculi to refer to whole contexts of sequents instead of single sequents. We present a number of results using hypersequents to obtain a Gentzen-style characterization for the family of Gödel logics. We first describe analytic calculi for propositional finite and infinite-valued Gödel logics. We then show that the framework of hypersequents allows one to move straightforwardly from the propositional level to first-order as well as propositional quantification. A certain type of modalities, enhancing the expressive power of Gödel logic, is also considered.

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Section: Resultsmentioning

confidence: 99%

Hypersequent Calculi for Godel Logics -- a Survey

Baaz¹,

Ciabattoni²,

Fermüller³

2003

Journal of Logic and Computation

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“…Meanwhile, let us give the name DWS(G ) to the proof system of disjunctive winning strategies. Then DWS(G ) is quite close to the sequents-ofrelations calculus RG ∞ , and its extension RG ∞ capturing the -projection operator, as developed in [3,5,6]. The approach there is algebraic rather then game-theoretic.…”

Section: Proof-theoretical Content Of Disjunctive Strategiesmentioning

confidence: 93%

From Semantic Games to Provability: The Case of Gödel Logic

2021

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We present a semantic game for Gödel logic and its extensions, where the players’ interaction stepwise reduces arbitrary claims about the relative order of truth degrees of complex formulas to atomic ones. The paper builds on a previously developed game for Gödel logic with projection operator in Fermüller et al. (in: M.-J. Lesot, S. Vieira, M.Z. Reformat, J.P. Carvalho, A. Wilbik, B. Bouchon-Meunier, and R.R. Yager, (eds.), Information processing and management of uncertainty in knowledge-based systems, Springer, Cham, 2020, pp. 257–270). This game is extended to cover Gödel logic with involutive negations and constants, and then lifted to a provability game using the concept of disjunctive strategies. Winning strategies in the provability game, with and without constants and involutive negations, turn out to correspond to analytic proofs in a version of $$\text{ SeqGZL } $$ SeqGZL (A. Ciabattoni, and T. Vetterlein, Fuzzy Sets and Systems 161(14):1941–1958, 2010) and in a sequent-of-relations calculus (M. Baaz, and Ch.G. Fermüller, in: N.V. Murray, (ed.), Automated reasoning with analytic tableaux and related methods, Springer, Berlin, 1999, pp. 36–51) respectively.

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“…Therefore cut-free derivations even for propositional formulas might lead to long cut-free proofs (recall that the validity problem in intuitionistic logic is P-space complete while the same problem in LC is in co-NP). This is not the case, when eliminating cuts from derivations in the sequent-of-relations calculus for LC defined in [4] (see also [6]). In this latter calculus, all the rules are invertible.…”

Section: Final Remarksmentioning

confidence: 99%

A Schütte-Tait Style Cut-Elimination Proof for First-Order Gödel Logic

Baaz

Ciabattoni

2002

Lecture Notes in Computer Science

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Abstract. We present a Schütte-Tait style cut-elimination proof for the hypersequent calculus HIF for first-order Gödel logic. This proof allows to bound the depth of the resulting cut-free derivation by 4 |d| ρ(d) , where |d| is the depth of the original derivation and ρ(d) the maximal complexity of cut-formulas in it. We compare this Schütte-Tait style cut-elimination proof to a Gentzen style proof.

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Sequent of Relations Calculi: A Framework for Analytic Deduction in Many-valued Logics

Cited by 7 publications

References 21 publications

Hypersequent Calculi for Godel Logics -- a Survey

Hypersequent Calculi for Godel Logics -- a Survey

From Semantic Games to Provability: The Case of Gödel Logic

A Schütte-Tait Style Cut-Elimination Proof for First-Order Gödel Logic

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