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16
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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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
Firstorder Gödel logics
, 2006
"... Firstorder Gödel logics are a family of infinitevalued logics where the sets of truth values V are closed subsets of [0,1] containing both 0 and 1. Different such sets V in general determine different Gödel logics GV (sets of those formulas which evaluate to 1 in every interpretation into V). It i ..."
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Cited by 14 (5 self)
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Firstorder Gödel logics are a family of infinitevalued logics where the sets of truth values V are closed subsets of [0,1] containing both 0 and 1. Different such sets V in general determine different Gödel logics GV (sets of those formulas which evaluate to 1 in every interpretation into V). It is shown that GV is axiomatizable iff V is finite, V is uncountable with 0 isolated in V, or every neighborhood of 0 in V is uncountable. Complete axiomatizations for each of these cases are given. The r.e. prenex, negationfree, and existential fragments of all firstorder Gödel logics are also characterized.
Automated theorem proving by resolution in nonclassical logics
 Annals of Mathematics and Artificial Intelligence
, 2007
"... This paper is an overview of a variety of results, all centered around a common theme, namely embedding of nonclassical logics into first order logic and resolution theorem proving. We present several classes of nonclassical logics, many of which are of great practical relevance in knowledge repre ..."
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This paper is an overview of a variety of results, all centered around a common theme, namely embedding of nonclassical logics into first order logic and resolution theorem proving. We present several classes of nonclassical logics, many of which are of great practical relevance in knowledge representation, which can be translated into tractable and relatively simple fragments of classical logic. In this context, we show that refinements of resolution can often be used successfully for automated theorem proving, and in many interesting cases yield optimal decision procedures. 1
A SchütteTait style cutelimination proof for firstorder Gödel logic
 In Automated Reasoning with Tableaux and Related Methods (Tableaux’02), volume 2381 of LNAI
, 2002
"... Abstract. We present a SchütteTait style cutelimination proof for the hypersequent calculus HIF for firstorder Gödel logic. This proof allows to bound the depth of the resulting cutfree derivation by 4 d ρ(d) , where d is the depth of the original derivation and ρ(d) the maximal complexity o ..."
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Abstract. We present a SchütteTait style cutelimination proof for the hypersequent calculus HIF for firstorder Gödel logic. This proof allows to bound the depth of the resulting cutfree derivation by 4 d ρ(d) , where d is the depth of the original derivation and ρ(d) the maximal complexity of cutformulas in it. We compare this SchütteTait style cutelimination proof to a Gentzen style proof. 1
Characterization of the axiomatizable prenex fragments of firstorder Gödel logics
 IN 33RD INTERNATIONAL SYMPOSIUM ON MULTIPLEVALUED LOGIC. MAY 2003
, 2003
"... The prenex fragments of firstorder infinitevalued Gödel logics are classified. It is shown that the prenex Gödel logics characterized by finite and by uncountable subsets of [0,1] are axiomatizable, and that the prenex fragments of all countably infinite Gödel logics are not axiomatizable. ..."
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The prenex fragments of firstorder infinitevalued Gödel logics are classified. It is shown that the prenex Gödel logics characterized by finite and by uncountable subsets of [0,1] are axiomatizable, and that the prenex fragments of all countably infinite Gödel logics are not axiomatizable.
Herbrand’s theorem, skolemization, and proof systems for firstorder ̷Lukasiewicz logic
 Journal of Logic and Computation
"... Abstract. An approximate Herbrand theorem is established for firstorder infinitevalued Łukasiewicz Logic and used to obtain a prooftheoretic proof of Skolemization. These results are then used to define proof systems in the framework of hypersequents. In particular, a calculus lacking cutelimin ..."
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Abstract. An approximate Herbrand theorem is established for firstorder infinitevalued Łukasiewicz Logic and used to obtain a prooftheoretic proof of Skolemization. These results are then used to define proof systems in the framework of hypersequents. In particular, a calculus lacking cutelimination is defined for the firstorder logic characterized by linearly ordered MValgebras, a cutfree calculus with an infinitary rule for the full firstorder Łukasiewicz Logic, and a cutfree calculus with finitary rules for its onevariable fragment. 1
SAT in Monadic Gödel Logics: a borderline between decidability and undecidability
"... Abstract. We investigate satisfiability in the monadic fragment of firstorder Gödel logics. These are a family of finite and infinitevalued logics where the sets of truth values V are closed subsets of [0, 1] containing 0 and 1. We identify conditions on the topological type of V that determine th ..."
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Cited by 5 (3 self)
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Abstract. We investigate satisfiability in the monadic fragment of firstorder Gödel logics. These are a family of finite and infinitevalued logics where the sets of truth values V are closed subsets of [0, 1] containing 0 and 1. We identify conditions on the topological type of V that determine the decidability or undecidability of their satisfiability problem. 1
Cut elimination for first order Gödel logic by hyperclause resolution
"... Efficient, automated elimination of cuts is a prerequisite for proof analysis. The method CERES, based on Skolemization and resolution has been successfully developed for classical logic for this purpose. We generalize this method to Gödel logic, an important intermediate logic, which is also one ..."
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Efficient, automated elimination of cuts is a prerequisite for proof analysis. The method CERES, based on Skolemization and resolution has been successfully developed for classical logic for this purpose. We generalize this method to Gödel logic, an important intermediate logic, which is also one of the main formalizations of fuzzy logic.
Monadic Fragments of Gödel Logics: Decidability and Undecidability Results
"... Abstract. The monadic fragments of firstorder Gödel logics are investigated. It is shown that all finitevalued monadic Gödel logics are decidable; whereas, with the possible exception of one (G↑), all infinitevalued monadic Gödel logics are undecidable. For the missing case G↑ the decidability of ..."
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Abstract. The monadic fragments of firstorder Gödel logics are investigated. It is shown that all finitevalued monadic Gödel logics are decidable; whereas, with the possible exception of one (G↑), all infinitevalued monadic Gödel logics are undecidable. For the missing case G↑ the decidability of an important subcase, that is well motivated also from an application oriented point of view, is proven. A tight bound for the cardinality of finite models that have to be checked to guarantee validity is extracted from the proof. Moreover, monadic G↑, like all other infinitevalued logics, is shown to be undecidable if the projection operator △ is added, while all finitevalued monadic Gödel logics remain decidable with △. 1
Herbrand Theorems and Skolemization for Prenex Fuzzy Logics
"... Abstract. Approximate Herbrand theorems are established for firstorder fuzzy logics based on continuous tnorms, and used to provide prooftheoretic proofs of Skolemization for their Prenex fragments. Decidability and complexity results for particular fragments are obtained as consequences. ..."
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Abstract. Approximate Herbrand theorems are established for firstorder fuzzy logics based on continuous tnorms, and used to provide prooftheoretic proofs of Skolemization for their Prenex fragments. Decidability and complexity results for particular fragments are obtained as consequences.