On completeness and A-completeness of S-sets of determinate functions containing all one-place determinate S-functions

We consider the problem on completeness of sets of S-functions, the determinate functions such that the automaton calculating them realises in each state functions which emanate no value. We assume that each set of S-functions whose completeness is checked in this paper contains all S-functions depending on at most one variable. We describe all A-precomplete classes of such sets. It is shown that there exists an algorithm recognising A-completeness of S-sets of one-place determinate functions containing all one-place determinate S-functions. Let k >= 2, tau >= 1, and P-k(tau) be a functional system of determinate functions defined on words of length T over the alphabet E-k ={0,1,...k-1} equipped with the superposition operations. Note that any determinate function in P-k(tau) can be calculated by some finite automaton during the first tau steps of its work. Let m(k) subset of P-k(tau) and [V] denote the closure of the set all with respect to superposition operations. A set all c PIT, is called closed if it coincides with its closure.