On expressibility of functions of many-valued logic in some logical-functional languages

Abstract

For each k> k > 2, three logical-functional languages are introduced for the set of functions of k-valued logic: the positive expressibility language Pos*, the first-order language l L*, and the second-order language 2L*. On the basis of the notion of expressibility in a language, the corresponding closure operators are defined. It is proved that the operators of 1 L*-closure and 2L*-closure coincide. The 1 L^-closed and Pos^-closed classes are described with the help of symmetric groups and symmetric semigroups. The expressibility in the languages 1L* and Pos* is compared with the parametric expressibility and the expressibility by terms.

On the structure of equationally closed classes

“…Let us turn to the definition of the operator of positive closure [6]. As the initial symbols of the language Pos of positive expressibility we consider individual variables, functional constants f .n/ i for notation of the functions in P .n/ k , the equality sign D, the logical connectives ², _, and the existence quantifier 9.…”

“…In Theorem 4 we give a criterion of equational completeness in P k . Many results valid for the operator of equational closure appear to be valid for the operator of positive closure [6]. The interrelations between these operators are given in Corollary 3.…”

“…In a series of papers [1][2][3][4][5][6][7][8][9][12][13][14][15][16][17][18][19][20][21], on the set of functions of many-valued logic closure operators are defined, which are much more strong than the usually considered superposition operator. In particular, for any k 2 these operators generate on the set P k of the functions of k-valued logic finite (occasionally, countable) classifications.…”

Definition of positively closed classes by endomorphism semigroups

“…Later, a more formalised definition of the parametric closure operator from the logic function viewpoint has been suggested in [4] (see also [5]). This definition permits to introduce more new strong closure operators.…”

“…The first of them is the positive closure operator (which is even more strong than the parametric closure operator). The positive closure operator has been investigated in [4][5][6][7][8][9]. In particular, it has been seen (see [4]) that there are precisely six positively closed classes of Boolean functions and that for any k 3 the number of positively closed classes in P k is strictly less than the number of subsemigroups with unity in the symmetric semigroup of degree k.…”