Basic relations of the S-classification of functions of the multi-valued logic

Марченков, С. С.

doi:10.1515/dma.1996.6.2.149

Cited by 9 publications

(16 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We act similarly when the relation p 0 is of the form (9), and the relation pj is of the form (15). It remains only to note that for i^(d) = c we set d = 7^(b\ and in the case 7^(0) # c we set b = τ% ι (c) and d = 7^(0).…”

Section: Lemma 6 Let π Be a Set Of Monadic And Binary Relations On Ementioning

confidence: 99%

“…The principal difficulty of the S-classification consists in determination of S-closed classes contained in Ik. To solve this problem, in [15] some standard relations, called basic, were introduced and it was shown that every S-closed class of I k , given by at most binary relations, can be also given by the basic relations. In this paper we complete the reduction of relations started in [15], proving that every S-closed class of I k can be determined by a suitable system of the basic relations.…”

mentioning

confidence: 99%

“…To solve this problem, in [15] some standard relations, called basic, were introduced and it was shown that every S-closed class of I k , given by at most binary relations, can be also given by the basic relations. In this paper we complete the reduction of relations started in [15], proving that every S-closed class of I k can be determined by a suitable system of the basic relations. A small number of the basic relations and their transparent structure permit to find exact formulae for the numbers of S-closed classes of different types contained in I k and also for their total number which is a cubic polynomial in k. With the help of relations we also describe all S-closed classes of QL 4 .…”

mentioning

confidence: 99%

“…The main results concerning the S-classification of functions of I k in terms of the universal algebra are presented in [16]. Brought In terminology and notations we follow the paper [15]. Π* denotes the set of all relations on E k = {0, 1, ...,/: -1}.…”

mentioning

confidence: 99%

“…The following lemma is proved in [15]. (1) where σ/(.τ/) is the projection of the relation p(x\, ...,*") with respect to all variables different from x it ?ij(x it Xj) is the similar projection with respect to all variables different Proof.…”

mentioning

confidence: 99%

See 4 more Smart Citations

The S-classification of functions of many-valued logic

Марченков¹

1997

Discrete Mathematics and Applications

View full text Add to dashboard Cite

The set of functions of many-valued logic is proposed to be classified with respect to two operations: superposition and transition to dual functions (the ^-classification). The contensive description of all 5-closed classes, which was begun by the author in 1979-82, was completed by Nguen Van Hoa. If fc > 5, then the set of functions of /:-valued logic has only two 5-precomplete classes: the class h of idempotent functions and the Slupecki class SLPk. In this paper the key properties determining the 5-closed classes are found and formalized in the form of the so-called basic relations. Using the Galois theory for Post algebras, it is shown that every 5-closed class of functions, which is not contained in SLPk, can be described by the basic relations. In the set of all systems of the basic relations all independent systems are determined which correspond to all 5-closed classes not contained in SLPk. An exact formula for the number of 5-closed classes contained in Ik is obtained which is a cubic polynomial in k.Let P k be the set of all functions of ^-valued logic [1,2]. Any analysis of functions of many-valued logic is necessarily based on structuring (classification) of subsets of P k . In discrete mathematics the approach based on generative capabilities of functions with respect to superposition has got the most prevalence for classification of subsets of P k . For k = 2 this approach gives the well-known countable Post classification [3,4]. For k > 3 the continuity of arising closed classes [5] is an objective obstacle for constructing and effective using of a similar classification. It compels to seek other, wider, closure operators which would include superposition and permit to avoid familiar undesirable effects.In the papers [6-10] the classifications of functions of many-valued logic self-dual with respect to permutations of the symmetric group and of the alternating group were constructed. The ideas underlying these classifications can be transferred from individual functions to classes of functions. More precisely, operations giving dual functions for permutations of a fixed group should be added to superposition. Above all, it is natural to consider the largest permutation group, that is, the symmetric group, which under this approach leads to the largest closed classes and, therefore, to the least number of closed classes. Several names were offered for the corresponding closed classes and the classification; we call them the 5-closed classes and the 5-classification respectively.

show abstract

Section: Lemma 6 Let π Be a Set Of Monadic And Binary Relations On Ementioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations