Dmitriy Zhuk

The predicate method to construct the Post lattice

Abstract We suggest a new way to construct the structure of all closed classes of two-valued logic. In contrast to classical approaches, in this research the functions of two-valued logic are auxiliary objects and the construction starts from the set of predicates.

The Lattice of the Clones of Self-Dual Functions in Three-Valued Logic

The lattice of all clones of self-dual functions in three-valued logic is described in the paper. Even though this lattice contains a continuum of clones, a simple description was found. Using this description different properties of the lattice and of the clones were derived. Pair wise inclusion of the clones into each other was described, and all finitely generated clones were found. Also, for each clone the relation degree, the cardinality of the set of all clones containing this clone, and the cardinality of the set of all clones that are contained in this clone were determined.

On Key Relations Preserved by a Weak Near-Unanimity Function

In the paper we introduce a notion of a key relation, which almost coincides with the notion of a critical relation introduced by Keith A. Kearnes and Ágnes Szendrei. We show that we need only key relations to describe all clones on finite sets. We describe the set of all key relations on 2 elements, this set consists of all relations that can be defined as a disjunction of linear equations. Then we show that in general key relations do not have such a nice characterization, but we can get a characterization of key relations preserved by a weak near unanimity function.

On the Clones Containing a Near-Unanimity Function

In previous papers we introduced the notion of an essential predicate, that is a predicate that cannot be presented as a conjunction of predicates with smaller arities. We showed that all clones of functions can be defined by a set of essential predicates. We introduced several operations on the set of essential predicates and defined a closure operator under these operations. There exists a one-to-one correspondence (a Galois connection) between clones and closed sets of essential predicates. Thus, these closed sets provide a valuable tool to studying clones. If the closure of a set of essential predicates is finite then it can be calculated using a computer. As it follows from Baker-Pixley theorem, a closed set corresponding to a clone is finite if and only if the clone contains a near-unanimity function. In the paper we present an approach to constructing a lattice of clones containing a given near-unanimity function. Particularly, all clones on three elements containing a majority function were calculated by a computer program. There turned out to be 1 918 040 clones. To analyze the possibility of constructing the lattices of clones in other cases, we give estimates on the number of clones containing a near-unanimity function of arity n. Also we prove estimates on the maximal size of a chain of clones containing a near-unanimity function. We conclude that it is impossible in practice to obtain the description for clones on four elements, as well as for a near-unanimity function of arity four.

An Algorithm for Constraint Satisfaction Problem

Many natural combinatorial problems can be expressed as constraint satisfaction problems. This class of problems is known to be NP-complete in general, but certain restrictions on theform of the constraints can ensure tractability. The standard way to parameterize interesting subclasses of the constraint satisfaction problem is via finite constraint languages. The mainproblem is to classify those subclasses that are solvable in polynomial time and those that are NP-complete. It was conjectured that if a core of a constraint language has a weak near unanimity polymorphism then the corresponding constraint satisfaction problem is tractable, otherwise it is NP-complete. In the paper we present an algorithm that solves Constraint Satisfaction Problem in polynomial time for constraint languages having a weak near unanimity polymorphism, which proves the remaining part of the conjecture.

On the classification of Post automaton bases by the decidability of the A-completeness property for definite automata

Abstract We consider systems of the form M = F ∪ ν, where F is some Post class and ν is a finite system of definite automata. We divide the Post classes into those for which the problem of A-completeness of such systems of definite automata is algorithmically decidable and those for which the problem of A-completeness is algorithmically undecidable.