It was shown earlier that product xy is a universal function for the class of linear functions of two variables for k=6l± 1 . In this work, it is shown that there are no universal polynomials for classes of linear functions of two variables for even k .

It was shown earlier that product xy for k=6l± 1 is a universalfunction in the class of linear functions of two variables, and there exist no universal polynomials for even k in classes of linear functions of two variables. This work proves that polynomial xy+xz+yz is universal for classes of linear functions of three variables for arbitrary odd k and polynomial xy+zw is universal for classes of linear functions of four variables for arbitrary k .

Abstract Earlier, the author introduced the concept of a universal function and proved the existence of universal functions for classes of linear k-valued functions of two variables for k ≥ 5. In this paper, we show that the product modulo k is a universal function for the class of linear k-valued functions of two variables if and only if k = 6l ± 1.

The author considers the classical problem of testing contact circuits implementing a parity counter of n variables. The lower bound of the test depth is raised from 2^n-1 to 2^n-1+n for conditional full diagnostic tests.

A problem is considered in which it is required that a Boolean function $$g$$ of $$n$$ variables be specified such that for any function that essentially depends on exactly $$n-1$$ variables it is possible to present a certain number of tuples of a given function so that this function $$g$$ is the only one that coincides with the given function on these tuples. It is shown that the sum modulo two of $$n$$ variables is the required function.

We consider universal function's construction for classes of sums of two arguments modulo 2. We constructed functions with optimal domain cardinality O(log n).

Absolute lower bound $$2^{n}$$ is established for the length of a full conditional diagnostic test for a Cardot circuit that implements the sum of $$n$$ variables.

We consider the notion of universal function such that a subset of the function’s values defines any function from some set. For the set of linear functions, we consider all the combinations of the number of variables and the number of values, except four-valued functions of two variables.

New upper bound 3n is presented for the cardinality of the definition domain of a universal function for a class of linear Boolean functions in which n is the number of variables.

The following problem is considered: specify a Boolean function of n variables such that every bilinear (polylinear) function is reduced, on a certain number of tuples of the specified function, to a unique bilinear (polylinear) function that is identical with the specified function on these tuples. We show that this is feasible for bilinear functions and for polylinear functions with a fixed number of parentheses k , starting with some n , and we can restrict the analysis to a sequence of functions with definition domain of cardinality O ( n ).

The paper puts forward a nontrivial lower estimate 21/6n for the cardinality of the domain of a universal function for the class of linear Boolean functions, where n is the number of variables.

The article describes the construction of discrete functions which, by some of their values, specify (generate) arbitrary linear functions. The cases of prime and sufficiently large composite k have been considered previously. In the present study we finally solve the problem of existence of such functions for almost all k and n variables. The proof of the probabilistic upper bound and the general approach are due to A. A. Voronenko. The proof for small k has been developed by N. K. Voronova. The proof for k from 21 to 48 is the result of indispensable cooperation of V. P. Il'yutko and A. A. Voronenko.

The problem of constructing discrete functions such that parts of their value sets determine (generate) arbitrary linear functions is considered. A case in which k is a prime number was considered earlier by the author. It is proved that the existence of such partial functions wshen the number of independent variables is no less then two implies they exists for any arbitrary greater number of independent variables. Upper estimates linear with respect to the number of independent variables are proved for the size of the domain of universal functions. The existence of two-variable universal functions is proved for sufficiently large k .

In this article we describe the construction of discrete functions such that some of their values specify (generate) arbitrary pairs of literals under two mutually exclusive assumptions: the function is equal to one of the variables or to its negation. We prove the existence of such functions with at least seven arguments and show that for sufficiently large n this function can be defined on O(n log2 n) tuples. We also consider the problem of simultaneous generation of k literals. We show that with k < n − log2 n+log2(log3 4−1), functions generating arbitrary k literals exist, and if (n−log2 n−k) → ∞ as n → ∞ , then almost all functions generate arbitrary k literals.

A certificate of non-membership for a Boolean function f with respect to a class C, f ∉ C, is a set S of input strings such that the values of f on strings from S are inconsistent with any function h ∈ C. We study certificates of non-membership with respect to several classes of read-once functions, generated by their bases. For the basis {&, ∨, $\neg$}, we determine the optimal certificate size for every function outside the class and deduce that 6 strings always suffice. For the same basis augmented with a function x1... xs ∨ ${\bar x}_1$... ${\bar x}_s$, we show that there exist n-variable functions requiring Ωns-1 strings in a certificate as n → ∞. For s = 2, we show that this bound is tight by constructing certificates of size On for all functions outside the class.

The article examines the construction of discrete functions such that a part of their value set specifies (generates) arbitrary linear functions. For prime k greater than 5, we prove the existence of such partial functions with no fewer than two variables and derive linear upper and lower bounds in the number of variables for the size of their definition domain.

A short proof of V.A. Stetsenko’s theorem on weakly read-many Boolean functions is given, which is based on the technique of representing read-once functions in the form of trees.

The article proves the read-many property of Boolean functions in the basis of all functions of l variables. The length of the minimal read-many certificate in this basis is known to be upper bounded by a polynomial of degree l – 1 in the number of function variables. In this article, we prove that the upper bound on the length of the minimal read-many certificate for functions in this basis is a polynomial of degree not exceeding l.