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A Treatise on ManyValued Logics
 Studies in Logic and Computation
, 2001
"... The paper considers the fundamental notions of many valued logic together with some of the main trends of the recent development of infinite valued systems, often called mathematical fuzzy logics. Besides this logical approach also a more algebraic approach is discussed. And the paper ends with som ..."
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Cited by 84 (5 self)
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The paper considers the fundamental notions of many valued logic together with some of the main trends of the recent development of infinite valued systems, often called mathematical fuzzy logics. Besides this logical approach also a more algebraic approach is discussed. And the paper ends with some hints toward applications which are based upon actual theoretical considerations about infinite valued logics. Key words: mathematical fuzzy logic, algebraic semantics, continuous tnorms, leftcontinuous tnorms, Pavelkastyle fuzzy logic, fuzzy set theory, nonmonotonic fuzzy reasoning 1 Basic ideas 1.1 From classical to manyvalued logic Logical systems in general are based on some formalized language which includes a notion of well formed formula, and then are determined either semantically or syntactically. That a logical system is semantically determined means that one has a notion of interpretation or model 1 in the sense that w.r.t. each such interpretation every well formed formula has some (truth) value or represents a function into
A New Logical Characterisation of Stable Models and Answer Sets
 In Proc. of NMELP 96, LNCS 1216
, 1997
"... This paper relates inference in extended logic programming with nonclassical, nonmonotonic logics. We define a nonmonotonic logic, called equilibrium logic, based on the least constructive extension, N2, of the intermediate logic of "hereandthere". We show that on logic programs equilib ..."
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Cited by 59 (15 self)
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This paper relates inference in extended logic programming with nonclassical, nonmonotonic logics. We define a nonmonotonic logic, called equilibrium logic, based on the least constructive extension, N2, of the intermediate logic of "hereandthere". We show that on logic programs equilibrium logic coincides with the inference operation associated with the stable model and answer set semantics of Gelfond and Lifschitz. We thereby obtain a very simple characterisation of answer set semantics as a form of minimal model reasoning in N2, while equilibrium logic itself provides a natural generalisation of this semantics to arbitrary theories. We discuss briefly some consequences and applications of this result. 1 Introduction By contrast with the minimal model style of reasoning characteristic of several approaches to the semantics of logic programs, the stable model semantics of Gelfond and Lifschitz [8] was, from the outset, much closer in spirit to the styles of reasoning found in othe...
Hypersequents and the proof theory of intuitionistic fuzzy logic
 Computer Science Logic CSL’2000. Proceedings, LNCS 1862
, 2000
"... Abstract. Takeuti and Titani have introduced and investigated a logic they called intuitionistic fuzzy logic. This logic is characterized as the firstorder Gödel logic based on the truth value set [0,1]. The logic is known to be axiomatizable, but no deduction system amenable to prooftheoretic, and ..."
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Cited by 25 (9 self)
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Abstract. Takeuti and Titani have introduced and investigated a logic they called intuitionistic fuzzy logic. This logic is characterized as the firstorder Gödel logic based on the truth value set [0,1]. The logic is known to be axiomatizable, but no deduction system amenable to prooftheoretic, and hence, computational treatment, has been known. Such a system is presented here, based on previous work on hypersequent calculi for propositional Gödel logics by Avron. It is shown that the system is sound and complete, and allows cutelimination. A question by Takano regarding the eliminability of the TakeutiTitani density rule is answered affirmatively. 1
Characterization of Strongly Equivalent Logic Programs in Intermediate Logics
 in Intermediate Logics. Theory and Practice of Logic Programming
, 2001
"... The nonclassical, nonmonotonic inference relation associated with the stable model semantics for logic programs gives rise to a relationship of strong equivalence between logical programs that can be veri ed in the 3valued Godel logic, G3, the strongest nonclassical intermediate proposition ..."
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Cited by 23 (0 self)
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The nonclassical, nonmonotonic inference relation associated with the stable model semantics for logic programs gives rise to a relationship of strong equivalence between logical programs that can be veri ed in the 3valued Godel logic, G3, the strongest nonclassical intermediate propositional logic (see [10]). In this paper we will show that KC (the logic of :p _ ::p), is the weakest intermediate logic for which strongly equivalent logic programs in a language allowing negations are logically equivalent.
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 19 (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
Quantified propositional Gödel logics
 In Proceedings of LPAR’2000, LNAI 1955
, 2000
"... Entailment in propositional Gödel logics can be defined in a natural way. While all infinite sets of truth values yield the same sets of tautologies, the entailment relations differ. It is shown that there is a rich structure of infinitevalued Gödel logics, only one of which is compact. It is also ..."
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Cited by 17 (6 self)
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Entailment in propositional Gödel logics can be defined in a natural way. While all infinite sets of truth values yield the same sets of tautologies, the entailment relations differ. It is shown that there is a rich structure of infinitevalued Gödel logics, only one of which is compact. It is also shown that the compact infinitevalued Gödel logic is the only one which interpolates, and the only one with an r.e. entailment relation. 1
A Survey on Different Triangular NormBased Fuzzy Logics
, 1999
"... Among various approaches to fuzzy logics, we have chosen two of them, which are built up in a similar way. Although starting from different basic logical connectives, they both use interpretations based on Frank tnorms. Different interpretations of the implication lead to different axiomatizati ..."
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Cited by 16 (1 self)
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Among various approaches to fuzzy logics, we have chosen two of them, which are built up in a similar way. Although starting from different basic logical connectives, they both use interpretations based on Frank tnorms. Different interpretations of the implication lead to different axiomatizations, but most logics studied here are complete. We compare the properties, advantages and disadvantages of the two approaches. Key words: Fuzzy logic, manyvalued logic, Frank tnorm 1 Introduction A manyvalued propositional logic with a continuum of truth values modelled by the unit interval [0; 1] is quite often called a fuzzy logic. In such a logic, the conjunction is usually interpreted by a triangular norm. In this context, a (propositional) fuzzy logic is considered as an ordered pair P = (L; Q) of a language (syntax ) L and a structure (semantics) Q described as follows: (i) The language of P is a pair L = (A; C), where A is an at most countable set of atomic symbols and C is ...
Mathematical fuzzy logic as a tool for the treatment of vague information
 Information Sciences
, 2005
"... The paper considers some of the main trends of the recent development of mathematical fuzzy logic as an important tool in the toolbox of approximate reasoning techniques. Particularly the focus is on fuzzy logics as systems of formal logic constituted by a formalized language, by a semantics, and by ..."
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Cited by 16 (1 self)
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The paper considers some of the main trends of the recent development of mathematical fuzzy logic as an important tool in the toolbox of approximate reasoning techniques. Particularly the focus is on fuzzy logics as systems of formal logic constituted by a formalized language, by a semantics, and by a calculus for the derivation of formulas. Besides this logical approach also a more algebraic approach is discussed. And the paper ends with some hints toward applications which are based upon these theoretical considerations. Key words: mathematical fuzzy logic, algebraic semantics, continuous tnorms, leftcontinuous tnorms, Pavelkastyle fuzzy logic, fuzzy set theory, nonmonotonic fuzzy reasoning 1
A Polynomial Translation of Logic Programs with Nested Expressions into Disjunctive Logic Programs: Preliminary Report
 IN PROC. NMR02
, 2002
"... Nested logic programs have recently been introduced in order to allow for arbitrarily nested formulas in the heads and the bodies of logic program rules under the answer sets semantics. Previous results show that nested logic programs can be transformed into standard (unnested) disjunctive logic pro ..."
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Cited by 14 (3 self)
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Nested logic programs have recently been introduced in order to allow for arbitrarily nested formulas in the heads and the bodies of logic program rules under the answer sets semantics. Previous results show that nested logic programs can be transformed into standard (unnested) disjunctive logic programs in an elementary way, applying the negationasfailure operator to body literals only. This is of great practical relevance since it allows us to evaluate nested logic programs by means of offtheshelf disjunctive logic programming systems, like DLV. However, it turns out that this straightforward transformation results in an exponential blowup in the worstcase, despite the fact that complexity results indicate that there is a polynomial translation among both formalisms. In this paper, we take up this challenge and provide a polynomial translation of logic programs with nested expressions into disjunctive logic programs. Moreover, we show that this translation is modular and (strongly) faithful. We have implemented both the straightforward as well as our advanced transformation; the resulting compiler serves as a frontend to DLV and is publicly available on the Web.