We construct a checking test for a read-once alternative in a median-augmented element basis. We prove that the augmentation of the element basis with a median conserves the linearity of the Shannon function for the test length with respect to a read-once alternative.

is shown that if weakly repetitive functions are added to an elementary base, the Shannon function for the length of each test with respect to a read-once alternative remains linear.

The following problem is considered: find a couple of sets (a certificate) with which we can verify if the functions of n variables in a given basis are read-once functions. This work obtains the logarithmic lower bound estimates of the Shannon function of certificate length for all functions of n variables in bases consisting of conjunction, disjunction, negation, and one of Stecenko monotone functions. It is thus shown that the elementary basis is the only one for which read-many certificate length is bound by a constant.

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.

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.