On the Satisfiability and Validity Problems in the Propositional Gödel Logic

Guller, Dušan

doi:10.1007/978-3-642-27534-0_14

Cited by 8 publications

(5 citation statements)

References 14 publications

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“…For example, an interpretation I satisfies an assertion a : (A B) ≥ p iff A I (a I ) ≥ p and B I (a I ) ≥ p. Thus, rather than building a model directly, we first create an abstract representation of this model by encoding, for each domain element, only the order between concepts. A similar approach has been previously used for fuzzy extensions of other logics based on Gödel semantics [35,36]. Here we show how the abstraction to ordering affects the construction of Hintikka trees and Hintikka automata recognizing them.…”

Section: Local Consistency In G-nalmentioning

confidence: 79%

Reasoning in Fuzzy Description Logics using Automata

Borgwardt

Peñaloza

2016

Fuzzy Sets and Systems

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Automata-based methods have been successfully employed to prove tight complexity bounds for reasoning in many classical logics, and in particular in Description Logics (DLs). Very recently, the ideas behind these automata-based approaches were adapted for reasoning also in fuzzy extensions of DLs, with semantics based either on finitely many truth degrees or the Gödel t-norm over the interval [0, 1]. Clearly, due to the different semantics in these logics, the construction of the automata for fuzzy DLs is more involved than for the classical case. In this paper we provide an overview of the existing automata-based methods for reasoning in fuzzy DLs, with a special emphasis on explaining the ideas and the requirements behind them. The methods vary from deciding emptiness of automata on infinite trees to inclusions between automata on finite words. Overall, we provide a comprehensive perspective on the automata-based methods currently in use, and the many complexity results obtained through them.

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Section: Local Consistency In G-nalmentioning

confidence: 79%

Reasoning in Fuzzy Description Logics using Automata

Borgwardt

Peñaloza

2016

Fuzzy Sets and Systems

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“…We exploit this property of the Gödel operators in the following constructions, by using order structures and order assertions to represent the semantics of concepts. This idea has also been previously used for other reasoning problems based on the Gödel semantics [24].…”

Section: A First Examplementioning

confidence: 99%

“…The tableau algorithm presented in [48] for ALC under Łukasiewicz semantics can also deal with order assertions, and even linear inequalities over assertions, but does not consider TBoxes. Similar order assertions have been used to decide propositional Gödel logic, albeit without involutive negation [24].…”

Section: Related Workmentioning

confidence: 99%

Algorithms for reasoning in very expressive description logics under infinitely valued Gödel semantics

Borgwardt

Pealoza

2017

International Journal of Approximate Reasoning

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Fuzzy description logics (FDLs) are knowledge representation formalisms capable of dealing with imprecise knowledge by allowing intermediate membership degrees in the interpretation of concepts and roles. One option for dealing with these intermediate degrees is to use the so-called Gödel semantics, under which conjunction is interpreted by the minimum of the degrees of the conjuncts. Despite its apparent simplicity, developing reasoning techniques for expressive FDLs under this semantics is a hard task. In this paper, we introduce two new algorithms for reasoning in very expressive FDLs under Gödel semantics. They combine the ideas of a previous automata-based algorithm for Gödel FDLs with the known crispification and tableau approaches for FDL reasoning. The results are the two first practical algorithms capable of reasoning in infinitely valued FDLs supporting general concept inclusions.

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“…D is a factor iff, for all i < i ≤ n, l i = l i . We now describe some generalisation of the translation in (Guller, 2010;Guller, 2012) to the first-order case. A similar approach exploiting the renaming subformulae technique can be found in (Plaisted and Greenbaum, 1986;de la Tour, 1992;Nonnengart et al, 1998;Sheridan, 2004)…”

Section: Translation To Order Clausal Formmentioning

confidence: 99%

An Order Hyperresolution Calculus for Gödel Logic - General First-order Case

2012

Proceedings of the 4th International Joint Conference on Computational Intelligence

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This paper addresses the deduction problem of a formula from a countable theory in the first-order Gödel logic from a perspective of automated deduction. Our approach is based on the translation of a formula to an equivalent satisfiable CNF one, which contains literals of the augmented form: either a or a → b or (a → b) → b or Qx c → a or a → Qx c where a, c are atoms different from 0 (the false), 1 (the true); b is an atom different from 1; Q ∈ {∀, ∃}; x is a variable occurring in c. A CNF formula is further translated to an equivalent satisfiable finite order clausal theory, which consists of order clauses-finite sets of order literals of the form: either a b or Qx c a or a Qx c or a ≺ b or Qx c ≺ a or a ≺ Qx c where a, b, c are atoms; Q ∈ {∀, ∃}; x is a variable occurring in c. and ≺ are interpreted by the equality and strict linear order on [0, 1], respectively. For an input theory, the proposed translation produces a so-called semantically admissible order clausal theory. An order hyperresolution calculus, operating on semantically admissible order clausal theories, is devised. The calculus is proved to be refutation sound and complete for the countable case.

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On the Satisfiability and Validity Problems in the Propositional Gödel Logic

Cited by 8 publications

References 14 publications

Reasoning in Fuzzy Description Logics using Automata

Reasoning in Fuzzy Description Logics using Automata

Algorithms for reasoning in very expressive description logics under infinitely valued Gödel semantics

An Order Hyperresolution Calculus for Gödel Logic - General First-order Case

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