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George Gargov - One of the best experts on this subject based on the ideXlab platform.
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elements of intuitionistic fuzzy logic part i
Fuzzy Sets and Systems, 1998Co-Authors: Krassimir T Atanassov, George GargovAbstract:The definition of the notion of intuitionistic fuzzy set is the basis for defining intuitionistic fuzzy logics of different kinds. In this paper, we construct two versions of intuitionistic fuzzy Propositional Calculus (IFPC) and a version of intuitionistic fuzzy predicate logic (IFPL).
Krassimir T Atanassov - One of the best experts on this subject based on the ideXlab platform.
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elements of intuitionistic fuzzy Propositional Calculus
2017Co-Authors: Krassimir T AtanassovAbstract:In classical logic (e.g., [1, 2, 3, 4]), to each proposition (sentence) we juxtapose its truth value: truth – denoted by 1, or falsity – denoted by 0. In the case of fuzzy logic [5], this truth value is a real number in the interval [0, 1] and it is called “truth degree” or “degree of validity”.
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elements of intuitionistic fuzzy logic part i
Fuzzy Sets and Systems, 1998Co-Authors: Krassimir T Atanassov, George GargovAbstract:The definition of the notion of intuitionistic fuzzy set is the basis for defining intuitionistic fuzzy logics of different kinds. In this paper, we construct two versions of intuitionistic fuzzy Propositional Calculus (IFPC) and a version of intuitionistic fuzzy predicate logic (IFPL).
Yang Bingru - One of the best experts on this subject based on the ideXlab platform.
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probabilistic Propositional logic is the event semantics for classical formal system of Propositional Calculus
Journal of Chinese Computer Systems, 2011Co-Authors: Yang BingruAbstract:The well formed formulars(wffs) in classical formal system of Propositional Calculus(CPC) are only some formal symbols,whose meanings are given by a interpretation.Probabilistic logic,based on a standard probabilistic space,is the event semantics for CPC,in which set operations are the semantic interpretation for connectives,event functions are the semantic interpretation for wffs,the event(set) inclusion is the semantic interpretation for logical implication,and the event equality = is the semantic interpretation for logical equivalence.We can perform event Calculus instead of probability Calculus in CPC.CPC is applicable to probabilistic Propositional Calculus completely.
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set algebra is semantic interpretation for classical formal system of Propositional Calculus
Computer Science, 2010Co-Authors: Yang BingruAbstract:The well formed formulas(wffs) in classical formal system of Propositional Calculus(CPC) are only some formal symbols,whose meanings are given by a interpretation.Both logic algebra and set algebra are Boolean algebra,and are interpretations for CPC.A set algebra is a set semantics for CPC,in which set operations are the interpretation for connectives,set functions are the interpretation for wffs,the set inclusion ■ is the interpretation for logical implication,and the set equality=is the interpretation for logical equivalence.Standard probabilistic logic is based on a standard probabilistic space,a proposition describes a random event which is a set,the event domain in a probabilistic space is a set algebra,probabilistic logic is just the practical application of the set semantics for CPC.We can perform event Calculus instead of probability Calculus in CPC.CPC is applicable to probabilistic Propositional Calculus completely.
Santocanale Luigi - One of the best experts on this subject based on the ideXlab platform.
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Fixed-point elimination in the Intuitionistic Propositional Calculus (extended version)
'Association for Computing Machinery (ACM)', 2019Co-Authors: Ghilardi Silvio, Gouveia, Maria Joao, Santocanale LuigiAbstract:International audienceIt is a consequence of existing literature that least and greatest fixed-points of monotone polynomials on Heyting algebras—that is, the alge- braic models of the Intuitionistic Propositional Calculus—always exist, even when these algebras are not complete as lattices. The reason is that these extremal fixed-points are definable by formulas of the IPC. Consequently, the μ-Calculus based on intuitionistic logic is trivial, every μ-formula being equiv- alent to a fixed-point free formula. We give in this paper an axiomatization of least and greatest fixed-points of formulas, and an algorithm to compute a fixed-point free formula equivalent to a given μ-formula. The axiomatization of the greatest fixed-point is simple. The axiomatization of the least fixed- point is more complex, in particular every monotone formula converges to its least fixed-point by Kleene’s iteration in a finite number of steps, but there is no uniform upper bound on the number of iterations. We extract, out of the algorithm, upper bounds for such n, depending on the size of the formula. For some formulas, we show that these upper bounds are polynomial and optimal
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Fixed-point elimination in the Intuitionistic Propositional Calculus (extended version)
2018Co-Authors: Ghilardi Silvio, Gouveia, Maria Joao, Santocanale LuigiAbstract:It is a consequence of existing literature that least and greatest fixed-points of monotone polynomials on Heyting algebras-that is, the alge- braic models of the Intuitionistic Propositional Calculus-always exist, even when these algebras are not complete as lattices. The reason is that these extremal fixed-points are definable by formulas of the IPC. Consequently, the $\mu$-Calculus based on intuitionistic logic is trivial, every $\mu$-formula being equiv- alent to a fixed-point free formula. We give in this paper an axiomatization of least and greatest fixed-points of formulas, and an algorithm to compute a fixed-point free formula equivalent to a given $\mu$-formula. The axiomatization of the greatest fixed-point is simple. The axiomatization of the least fixed- point is more complex, in particular every monotone formula converges to its least fixed-point by Kleene's iteration in a finite number of steps, but there is no uniform upper bound on the number of iterations. We extract, out of the algorithm, upper bounds for such n, depending on the size of the formula. For some formulas, we show that these upper bounds are polynomial and optimal.Comment: extended version of arXiv:1601.0040
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Fixed-point elimination in the Intuitionistic Propositional Calculus
HAL CCSD, 2016Co-Authors: Ghilardi Silvio, Gouveia, Maria Joao, Santocanale LuigiAbstract:International audienceIt is a consequence of existing literature that least and greatest fixed-points of monotone polynomials on Heyting algebras—that is, the algebraic models of the Intuitionistic Propositional Calculus—always exist, even when these algebras are not complete as lattices. The reason is that these extremal fixed-points are definable by formulas of the IPC. Consequently, the µ-Calculus based on intuitionistic logic is trivial, every µ-formula being equivalent to a fixed-point free formula. We give in this paper an axiomatization of least and greatest fixed-points of formulas, and an algorithm to compute a fixed-point free formula equivalent to a given µ-formula. The axiomatization of the greatest fixed-point is simple. The axiomatization of the least fixed-point is more complex, in particular every monotone formula converges to its least fixed-point by Kleene's iteration in a finite number of steps, but there is no uniform upper bound on the number of iterations. We extract, out of the algorithm, upper bounds for such n, depending on the size of the formula. For some formulas, we show that these upper bounds are polynomial and optimal
Sanmin Wang - One of the best experts on this subject based on the ideXlab platform.
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a fuzzy logic for an ordinal sum t norm
Fuzzy Sets and Systems, 2005Co-Authors: Sanmin Wang, Baoshu Wang, Daowu PeiAbstract:Among the class of residuated fuzzy logics, a few of them have been shown to have standard completeness both for Propositional and predicate Calculus, like Godel, NM and monoidal t-norm-based logic systems. In this paper, a new residuated logic NMG, which aims at capturing the tautologies of a class of ordinal sum t-norms and their residua, is introduced and its standard completeness both for Propositional Calculus and for predicate Calculus are proved.
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a triangular norm based Propositional fuzzy logic
Fuzzy Sets and Systems, 2003Co-Authors: Sanmin Wang, Baoshu Wang, Guojun WangAbstract:In 1997, Wang introduced the formal system L * for a type of fuzzy Propositional Calculus given by a particular t-norm different from the three famous ones (Lukasiewicz, Godel, product). Recently, Pei proved the completeness theorem for L * with respect to W -semantic by developing a theory of algebraic systems. In this paper, we aim to prove it by essentially metamathematical method. Furthermore, we discussed originally the completeness for the schematic extension L n * of L * with respect to W n -semantic.
