In his pioneering papers, Igor Zaslavsky started an algorithmic (constructivist) analysis of fuzzy logic. In this paper, we extend this analysis to fuzzy mathematics and fuzzy data processing. Specifically, we show that the two mathematically equivalent representations of a fuzzy number – by a membership function and by α-cuts – are not algorithmically equivalent, and only the α-cut representation enables us to efficiently process fuzzy data. A short version of this paper appeared in [5]. Mathematics Subject Classification: 03F60, 03B52, 03E72

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Membership Functions or alpha-Cuts? Algorithmic (Constructivist) Analysis Justifies an Interval Approach

We introduce a logical system in which the principles of fuzzy logic are interpreted from the point of view of the constructive approach The language of predicate formulas without functional symbols and symbols of constants is considered. The notion of identically trae predicate formula in the framework of the introduced logic is defined; two variants of this definition are given. Theorems concerning identically true predicate formulas are proved. Some connections between the introduced logic and the constructive (intuitionistic) predicate calculus are established. Bibliography: 40 titles.

Fuzzy constructive logic

A logical system is introduced that is similar to the “fuzzy constructive logic” earlier developed by the author, but gives wider possibilities of establishing the truth of predicate formulas and logical deductions in the framework of this logic. The notions of strong and weak FCL∗ -validity of predicate formulas are defined. It is proved that every formula deducible in the constructive (intuitionistic) predicate calculus is strongly FCL∗ -valid. On the other hand, it is proved that some formulas not deducible in this calculus are not weakly FCL∗ -valid. A definition is given for the semantics of the traditional constructive logic on the base of the developed logical apparatus. Theorems are proved showing differences between the extended fuzzy constructive logic and the traditional constructive logic. Bibliography: 38 titles.

Extended Fuzzy Constructive Logic

Algebraic representations of multidimensional recursively enumerable sets which are expressible in formal arithmetical systems based on the 1, are introduced and x , S), where S(x) , S), (0,,), (0,, S,signatures(0, investigated. The equivalence is established between the algebraic and logical representations of multidimensional recursively enumerable sets expressible in the mentioned systems.

On an Algebraic Classification of Multidimensional Recursively Enumerable Sets Expressible in Formal Arithmetical Systems 1

Classical and Non-Classical Logic, Logical Methods for Electronic Circuits Testing

Algebras of operations defined on recursively enumerable sets of different kinds are considered. Every such algebra is specified by a list of operations involved and a list of basic elements. An element of an algebra is said to be representable in this algebra if it can be obtained from given basic elements by operations of the algebra. Two kinds of recursively enumerable sets are considered: recursively enumerable sets in the usual sense and fuzzy recursively enumerable sets. On binary, i.e., two-dimensional recursively enumerable sets of these kinds, algebras of operations are introduced. An algebra θ is constructed in which all binary recursively enumerable sets are representable. A subalgebra θ0 of θ is constructed in which all binary recursively enumerable sets are representable if and only if they are described by formulas of Presburger’s arithmetic system. An algebra Ω is constructed in which all binary recursively enumerable fuzzy sets are representable. A subalgebra Ω0 of the algebra Ω is constructed such that fuzzy recursively enumerable sets representable in Ω0 can be treated as fuzzy counterparts of sets representable by formulas of Presburger’s system. Bibliography: 16 titles.

S. N. Manukian

Papers

Algebras of Recursively Enumerable Sets and Their Applications to Fuzzy Logic