Boolean functions are among the fundamental objects of discrete mathematics, especially in those of its subdisciplines which fall under mathematical logic and mathematical cybernetics. The language of Boolean functions is convenient for describing the operation of many discrete systems such as contact networks, Boolean circuits, branching programs, and some others. An important parameter of discrete systems of this kind is their complexity. This characteristic has been actively investigated starting from Shannon's works. There is a large body of scientific literature presenting many fundamental results. The purpose of this survey is to give an account of the main results over the last sixty years related to the complexity of computation (realization) of Boolean functions by contact networks, Boolean circuits, and Boolean circuits without branching. Bibliography: 165 titles.

Computational complexity of Boolean functions

The problem of constructing simple disjunctive normal forms (DNFs) of Boolean functions with a small number of zeros is considered. The problem is of interest in the complexity analysis of Boolean functions and in its applications to data analysis. The method used is a further development of the reduction approach to the construction of DNFs of Boolean functions. A key idea of the reduction method is that a Boolean function is represented as a disjunction of Boolean functions with fewer zeros. In a number of practically important cases, this technique makes it possible to considerably reduce the complexity of DNF implementations of Boolean functions.

Implementation of Boolean functions with a bounded number of zeros by disjunctive normal forms

The well-known lower bound for the maximum number of prime implicants of a Boolean
function (the length of the reduced DNF) differs by $$\Theta (\sqrt {n})
$$
 times from the upper bound and is asymptotically
attained at a symmetric belt function with belt width $$n/3
$$
 . To study the properties of connected Boolean
functions with many prime implicants, we introduce the notion of a locally extremal function in
a certain neighborhood in terms of the number of prime implicants. Some estimates are obtained
for the change in the number of prime implicants as the values of the belt function range over a

 $$d
$$
 -neighborhood. We prove that the belt function for
which the belt width and the number of the lower layer of unit vertices are asymptotically equal to

 $$n/3
$$
 is locally extremal in some neighborhood for

 $$d \le \Theta (n)
$$
 and not locally extremal if $$d \ge 2^{\Theta (n)}
$$
 . A similar statement is true for the functions that
have prime implicants of different ranks. The local extremality property is preserved after
applying some transformation to the Boolean function that preserves the distance between the
vertices of the unit cube.

Connected Boolean Functions with a Locally Extremal Number of Prime Implicants

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 n2 n 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.

On irredundant complexes of faces in the unit cube

We investigate the Boolean functions that combine various properties: the extremal values of complexity characteristics ofminimization, the inapplicability of local methods for reducing the complexity of the exhaustion, and the impossibility to efficiently use sufficient minimality conditions. Some quasicyclic functions are constructed that possess the properties of cyclic and zone functions, the dominance of vertex sets, and the validity of sufficient minimality conditions based on independent families of sets. For such functions, we obtain the exponential lower bounds for the extent and special sets and also a twice exponential lower bound for the number of shortest and minimal complexes of faces with distinct sets of proper vertices.

On the Complexity of Minimizing Quasicyclic Boolean Functions

The survey focuses on minimization of boolean functions in the class of disjunctive normal forms (d.n.f.s) and covers the publications from 1953 to 1986. The main emphasis is on the mathematical direction of research in boolean function minimization: bounds of parameters of boolean functions and algorithmic difficulties of minimal d.n.f. synthesis). The survey also presents a classification of minimization algorithms and gives some examples of minimization heuristics with their efficiency bounds.

Boolean function minimization in the class of disjunctive normal forms

Abstract The known lower bound for the maximum number of prime implicants (maximal faces) of a Boolean function differs from the upper bound by a factor of $$ O(\sqrt {n}) $$ and is asymptotically attained on a symmetric belt function. To study the properties of extremal functions, we define subsets of functions that have the minimum and maximum vertices of the maximal faces in the belts $$ n/3 \pm r_n $$ and $$ 2n/3 \pm r_n $$ , respectively. Here the fraction of the number of vertices in each layer is at least $$ \varepsilon _n $$ , and for any such vertex the ratio of the number of maximal faces to the maximum possible number is at least $$ \varepsilon _n $$ . Conditions on the parameters $$ \varepsilon _n $$ and $$ r_n $$ are obtained under which the number of prime implicants of a Boolean function in such a subset is equal to the maximum value asymptotically or in the order of growth.

Properties of Boolean Functions with Extremal Number of Prime Implicants

The article is written on the basis of reports presented at the XII International Seminar. Discrete mathematics and its applications named after academician O. B. Lupanov (Moscow, Moscow State University, June 20–25, 2016) and contains a survey of results for the problem of minimizing Boolean functions, in which methods are considered for proving the minimality and nonconstructive approaches to obtaining estimates of parameters that characterize the complexity of minimization with respect to classes of measures difficulties.

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I. P. Chukhrov

Papers

Boolean function minimization in the class of disjunctive normal forms

Properties of Boolean Functions with Extremal Number of Prime Implicants

Boolean functions minimization problem: conditions of minimality and the probabilistic method

Properties of Boolean Functions with Extremal Number of Prime Implicants