Проблема полноты в функциональной системе линейных полиномов с целыми коэффициентами © 2010 г. А. И. Мамонтов, Д. Г. Мещанинов В функциональной системе полиномов первой степени с целыми коэффициентами устанавливается критерий полноты в терминах предполных классов, определяется мощность базисов.

We introduce the notion of the second p-differences for a function of p a -valued logic (p is a prime number and a > 3) and also define the property that a function preserves the second p-differences. We establish a necessary and sufficient conditions for a representation of the functions by polynomials modulo p 3 and a sufficient condition for a representation of the functions by polynomials modulo p a , a > 3. We give a method of constructing the polynomial and also estimate the complexity of an algorithm to recognize a possibility of such a polynomial representation. Using the second p-differences, we describe a class which is closed in p 3 -valued logic, and find its place in the lattice of all the closed classes.

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Functions of &-valued logic are considered. Necessary and sufficient conditions for representability of functions by polynomials modulo k are established for any composite integer k. The canonical form of a polynomial and the way of its construction are given. The complexity of the algorithm for recognizing polynomiality of a function and constructing its polynomial is estimated.