Positively closed classes of partial Boolean functions

Марченков, С. С.; Popova, A. A.

doi:10.3103/s0278641908030059

Cited by 10 publications

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“…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.…”

Section: Introductionmentioning

confidence: 99%

On the structure of equationally closed classes

Марченков¹

2006

Discrete Mathematics and Applications

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We study the structure of equationally closed classes. We prove a theorem on representation of the graph of a function in an equationally closed class in the form of a union of the sets of values of special vector functions. For any k 2 we establish the equational generability of any equationally closed class in P k by the set of all its k-place functions. We find all equationally precomplete classes in P k and prove a criterion of equational completeness. Some results are extended from equationally closed classes to positively closed classes.

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Section: Introductionmentioning

confidence: 99%

On the structure of equationally closed classes

Марченков¹

2006

Discrete Mathematics and Applications

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“…All twenty five parametrically closed classes of Boolean functions are found in [3] (see also [5]); the finiteness of the number of parametrically closed classes in P 3 is proved in [2], and for P k with k 4 it is established in [11]. It is worth noticing that for k 3 there is no complete description of parametrically closed classes in P k yet.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.…”

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“…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.…”

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“…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.…”

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Definition of positively closed classes by endomorphism semigroups

Марченков¹

2012

Discrete Mathematics and Applications

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The positively closed classes of multivalued logic are characterised with the use of endomorphism semigroups. All subsemigroups of the symmetric semigroup of degree k are described which are semigroups of endomorphisms of positively closed classes in P k . On the base of the obtained results, we find all positively precomplete classes in P k .Prominent among a number of ways to classify the functions of multivalued logic are those based on the closure operators. On the set P k of the functions of k-valued logic, some operator O is defined; then the sets of functions in P k closed with respect to the operator O (usually referred to as the O-closed classes) form the O-classification of the set P k .The most known classification of such a kind is the classification based on the superposition operator. This classification has long been in use. But there exists an objective difficulty in its use: for any k 3 the number of closed classes in P k is continual (see [10]). In this connection, a great body of 'strong' closure operators have been introduced which for any k induce either finite or countable classifications on the set P k .One of the first strong closure operators is the parametric closure operator introduced by Kuznetsov in [3]. All twenty five parametrically closed classes of Boolean functions are found in [3] (see also [5]); the finiteness of the number of parametrically closed classes in P 3 is proved in [2], and for P k with k 4 it is established in [11]. It is worth noticing that for k 3 there is no complete description of parametrically closed classes in P k yet.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

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Extensions of the Minimum Cost Homomorphism Problem

Takhanov

2010

Lecture Notes in Computer Science

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Assume D is a finite set and R is a finite set of functions from D to the natural numbers. An instance of the minimum R-cost homomorphism problem (M inHomR) is a set of variables V subject to specified constraints together with a positive weight cvr for each combination of v ∈ V and r ∈ R. The aim is to find a function f : V → D such that f satisfies all constraints and v∈V r∈R cvrr(f (v)) is minimized. This problem unifies well-known optimization problems such as the minimum cost homomorphism problem and the maximum solution problem, and this makes it a computationally interesting fragment of the valued CSP framework for optimization problems. We parameterize M inHomR (Γ ) by constraint languages, i.e. sets Γ of relations that are allowed in constraints. A constraint language is called conservative if every unary relation is a member of it; such constraint languages play an important role in understanding the structure of constraint problems. The dichotomy conjecture for M inHomR is the following statement: if Γ is a constraint language, then M inHomR (Γ ) is either polynomial-time solvable or NP-complete. For M inHom the dichotomy result has been recently obtained [Takhanov, STACS, 2010] and the goal of this paper is to expand this result to the case of M inHomR with conservative constraint language. For arbitrary R this problem is still open, but assuming certain restrictions on R we prove a dichotomy. As a consequence of this result we obtain a dichotomy for the conservative maximum solution problem.

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Positively closed classes of partial Boolean functions

Cited by 10 publications

References 1 publication

On the structure of equationally closed classes

On the structure of equationally closed classes

Definition of positively closed classes by endomorphism semigroups

Extensions of the Minimum Cost Homomorphism Problem

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