Complexity of systems of functions of Boolean algebra and systems of functions of three-valued logic in classes of polarized polynomial forms

Apolarized polynomial form (PPF) (modulo k) is amodulo k sum of products of variables x(1),..., x(n) or their Post negations, where the number of negations of each variable is determined by the polarization vector of the PPF. The length of a PPF is the number of its pairwise distinct summands. The length of a function f(x(1),..., x(n)) of k-valued logic in the class of PPFs is the minimum length among all PPFs realizing the function. The paper presents a sequence of symmetric functions f(n)(x(1),..., x(n)) of three-valued logic such that the length of each function f(n) in the class of PPFs is not less than left perpendicular3(n+1)/4right perpendicular, where left perpendicular a right perpendicular denotes the greatest integer less or equal to the number a. The complexity of a system of PPFs sharing the same polarization vector is the number of pairwise distinct summands entering into all of these PPFs. The complexity L-k(PPF) (F) of a system F ={f(1),..., f(m)} of functions of k-valued logic depending on variables x(1),..., x(n) in the class of PPFs is the minimum complexity among all systems of PPFs {p(1),..., p(m)} such that all PPFs p(1),..., p(m) share the same polarization vector and the PPF p(j) realizes the function f(j), j = 1,..., m. Let L-k(PPF) (m, n)= max(F)L(k)(PPF) (F), where F runs through all systems consisting of m functions of k-valued logic depending on variables x(1),..., x(n). For prime values of k it is easy to derive the estimate L-k(PPF) (m, n) <= k(n). In this paper it is shown that L-2(PPF) (m, n)= 2(n) and L-3(PPF) (m, n)= 3(n) for all m >= 2, n = 1, 2,... Moreover, it is demonstrated that the estimates remain valid when consideration is restricted to systems of symmetric functions only.