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13
Sequent and Hypersequent Calculi for Abelian and Łukasiewicz Logics
 ACM Transactions on Computational Logic
, 2005
"... We present two embeddings of infinitevalued ̷Lukasiewicz logic ̷L into Meyer and Slaney’s abelian logic A, the logic of latticeordered abelian groups. We give new analytic proof systems for A and use the embeddings to derive corresponding systems for ̷L. These include: hypersequent calculi for A a ..."
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Cited by 19 (6 self)
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We present two embeddings of infinitevalued ̷Lukasiewicz logic ̷L into Meyer and Slaney’s abelian logic A, the logic of latticeordered abelian groups. We give new analytic proof systems for A and use the embeddings to derive corresponding systems for ̷L. These include: hypersequent calculi for A and ̷L and terminating versions of these calculi; labelled single sequent calculi for A and ̷L of complexity coNP; unlabelled single sequent calculi for A and ̷L. 1
Herbrand’s theorem for prenex Gödel logic and its consequences for theorem proving
 IN LOGIC FOR PROGRAMMING AND AUTOMATED REASONING LPAR’2001, 201–216. LNAI 2250
, 2001
"... Herbrand’s Theorem £¥ ¤ ¦ for, i.e., Gödel logic enriched by the projection § operator is proved. As a consequence we obtain a “chain normal form” and a translation of £ ¤ ¦ prenex into (order) clause logic, referring to the classical theory of dense total orders with endpoints. A chaining calc ..."
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Cited by 15 (12 self)
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Herbrand’s Theorem £¥ ¤ ¦ for, i.e., Gödel logic enriched by the projection § operator is proved. As a consequence we obtain a “chain normal form” and a translation of £ ¤ ¦ prenex into (order) clause logic, referring to the classical theory of dense total orders with endpoints. A chaining calculus provides a basis for efficient theorem proving.
Hypersequent calculi for Gödel logics: a survey
 Journal of Logic and Computation
, 2003
"... 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 Gentzenstyle characterization for the family of Gödel logics. We first describe analytic calculi for ..."
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Cited by 13 (4 self)
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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 Gentzenstyle characterization for the family of Gödel logics. We first describe analytic calculi for propositional finite and infinitevalued Gödel logics. We then show that the framework of hypersequents allows one to move straightforwardly from the propositional level to firstorder as well as propositional quantification. A certain type of modalities, enhancing the expressive power of Gödel logic, is also considered. 1
A Tableaux System for GödelDummett Logic Based on a Hypersequential Calculus
 In Automated Reasoning with Tableaux and Related Methods (Tableaux’2000), volume 1847 of Lectures Notes in Artificial Intelligence
, 2000
"... We present a terminating contractionfree calculus GLC for the propositional fragment of GodelDummett Logic LC. GLC uses hypersequents, and unlike other Gentzentype calculi for LC, all its rules have at most two premises. These rules are all invertible. Hence it can be used as a basis for a ..."
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Cited by 11 (0 self)
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We present a terminating contractionfree calculus GLC for the propositional fragment of GodelDummett Logic LC. GLC uses hypersequents, and unlike other Gentzentype calculi for LC, all its rules have at most two premises. These rules are all invertible. Hence it can be used as a basis for a deterministic tableaux system for LC. This tableaux system is presented in the last section. I A Review of LC and GLC In [Go33] Godel introduced a sequence fG n g of nvalued logics, as well as an infinitevalued matrix G ! in which all the G n s can be embedded. He used these matrices to show some important properties of intuitionistic logic. The logic of G ! was later axiomatized by Dummett in [Du59] and is known since then as Dummett's LC. It probably is the most important intermediate logic, one that turns up in several places, like the provability logic of Heyting's Arithmetics ([Vi82]) and relevance logic ([DM71]) and recently fuzzy logic([Ha98]). semantically LC corresponds to lin...
CutElimination in a SequentsofRelations Calculus for Gödel Logic
 In International Symposium on Multiple Valued Logic (ISMVL’2001
, 2001
"... In [5] the analytic calculus RG1 for G odel logic has been introduced. RG1 operates on "sequents of relations ". We show constructively how to eliminate cuts from RG1 derivations. The version of the cut rule we consider allows to derive other forms of cut as well as a rule corresponding to the "com ..."
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Cited by 7 (4 self)
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In [5] the analytic calculus RG1 for G odel logic has been introduced. RG1 operates on "sequents of relations ". We show constructively how to eliminate cuts from RG1 derivations. The version of the cut rule we consider allows to derive other forms of cut as well as a rule corresponding to the "communication rule" of Avron's hypersequent calculus for G1 . Moreover, we give an explicit description of all the axioms of RG1 and prove their completeness. 1.
Sequent of Relations Calculi: A Framework for Analytic Deduction in ManyValued Logics
 Beyond Two: Theory and applications of MultipleValued Logics
, 2003
"... We present a general framework that allows to construct systematically analytic calculi for a large family of (propositional) manyvalued logics  called projective logics  characterized by a special format of their semantics. All finitevalued logics as well as infinitevalued Godel logic ..."
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Cited by 5 (3 self)
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We present a general framework that allows to construct systematically analytic calculi for a large family of (propositional) manyvalued logics  called projective logics  characterized by a special format of their semantics. All finitevalued logics as well as infinitevalued Godel logic are projective. As a casestudy, sequent of relations calculi for Godel logics are derived. A comparison with some other analytic calculi is provided.
Classical Gentzentype Methods in Propositional ManyValued Logics
 In Fitting, M., & Orlowska, E. (Eds.), Theory and Applications in MultipleValued Logics
, 2002
"... A classical Gentzentype system is one which employs twosided sequents, together with structural and logical rules of a certain characteristic form. A decent Gentzentype system should allow for direct proofs, which means that it should admit some useful forms of cut elimination and the subformula p ..."
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Cited by 5 (2 self)
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A classical Gentzentype system is one which employs twosided sequents, together with structural and logical rules of a certain characteristic form. A decent Gentzentype system should allow for direct proofs, which means that it should admit some useful forms of cut elimination and the subformula property. In this tutorial we explain the main difficulty in developing classical Gentzentype systems with these properties for manyvalued logics. We then illustrate with numerous examples the various possible ways of overcoming this difficulty. Our examples include practically all 3valued logics, the most important class of 4valued logics, as well as central infinitevalued logics (like GodelDummett logic, S5 and some substructural logics). 1
Graphbased decision for GödelDummett logics
"... Abstract. We present a graphbased decision procedure for GödelDummett logics and an algorithm to compute countermodels. A formula is transformed into a conditional bicolored graph in which we detect some specific cycles and alternating chains using matrix computations. From an instance graph con ..."
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Cited by 4 (0 self)
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Abstract. We present a graphbased decision procedure for GödelDummett logics and an algorithm to compute countermodels. A formula is transformed into a conditional bicolored graph in which we detect some specific cycles and alternating chains using matrix computations. From an instance graph containing no such cycle (resp. no (n + 1)alternating chain) we extract a countermodel in LC (resp. LCn). Keywords: GödelDummett logic, sequent calculus, decision procedures, graphs, countermodels. 1.
A dialogue game for intuitionistic fuzzy logic based on comparison of degrees of truth
 In Proceedings of InTech’03
, 2003
"... Abstract: A dialogue game for fuzzy logic, based on the comparison of truth degrees, is presented. It is shown that the game is adequate for G △ ∞, i.e., intuitionistic fuzzy logic enriched by the projection operator △. Any given countermodel to a formula can be used to construct a winning strategie ..."
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Cited by 4 (2 self)
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Abstract: A dialogue game for fuzzy logic, based on the comparison of truth degrees, is presented. It is shown that the game is adequate for G △ ∞, i.e., intuitionistic fuzzy logic enriched by the projection operator △. Any given countermodel to a formula can be used to construct a winning strategies for one of the players, called Opponent. Conversely, countermodels can be extracted from each winning strategy for Opponent. Winning strategies for the other player, Proponent, correspond to proofs of validity. The systematic construction of socalled complete dialogue trees can be viewed as tableau style proof search procedure.
Fast decision procedure for propositional dummett logic based on a multiple premise tableau calculus
, 2008
"... We present a procedure to decide propositional Dummett logic. This procedure relies on a tableau calculus with a multiple premise rule and optimizations. The resulting implementation outperforms the state of the art graphbased procedure. 1 ..."
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Cited by 3 (2 self)
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We present a procedure to decide propositional Dummett logic. This procedure relies on a tableau calculus with a multiple premise rule and optimizations. The resulting implementation outperforms the state of the art graphbased procedure. 1