On irredundant complexes of faces in the unit cube

Abstract

The study of properties of irredundant complexes of faces is connected with the problem of minimisation of Boolean functions in the class of disjunctive normal forms (d.n.f.). In researches by S. V. Yablonskii, Yu. I. Zhuravlev, V. V. Glagolev, Yu. L. Vasilyev, A. A. Sapozhenko, on the base of construction and investigation of properties of particular Boolean functions, estimates of the maximum length and number of irredundant d.n.f. have been obtained.

The author suggests a different approach to investigations of these objects based on constructing and estimating the cardinality of sets of irredundant complexes of faces. In this paper, with the use of the probabilistic approach, new methods of construction and estimation of characteristics of irredundant complexes of faces are suggested, which give a possibility to improve the known estimates. On the base of a method of construction of irredundant complexes of faces in a belt of the unit cube Bⁿ of width k, we obtain estimates of the maximum number of faces and the number of irredundant complexes for the faces of dimension k < (1/4 – ε)n, where ε is as small as wished positive constant. By the optimal choice of the parameters we obtain for the logarithm of the number of irredundant complexes of faces the lower bound of order n2ⁿ with constant 1.355 · 2^–5 for the dimension of the faces k ≈ 0.0526n.

Because of equivalence of the problem of minimisation of Boolean functions in the class of d.n.f. and the problem of construction of complexes of faces covering subsets of vertices of the unit cube, the obtained results can be used for estimation of the maximum values of the length and the number of irredundant d.n.f.