A relation between additive and multiplicative complexity of Boolean functions

In the present note we prove an asymptotically tight relation between additive and multiplicative complexity of Boolean functions with respect to implementation by circuits over the basis {⊕,∧, 1}. To start, consider a problem of computation of polynomials over a semiring (K,+,×) by circuits over the arithmetic basis {+,×} ∪K. It’s a common knowledge that a polynomial of n variables with nonscalar multiplicative complexity M (i.e. the minimal number of multiplications to implement the polynomial, not counting multiplications by constants) has total complexity O(M(M + n)). Generally speaking, the bound could not be improved for infinite semirings. For instance, it follows from results by E. G. Belaga [1] and V. Ya. Pan [8] (there exist 1-variable complex and real polynomials of degree n with additive complexity n; at the same time, each such polynomial has nonscalar multiplicative complexity O( √ n) [9]). An analogous standard bound for finite semirings is O(M(M+n)/ logM). Generally speaking, this bound is also tight in order. A result of such sort was proven in [11]. We prove a similar but asymptotically tight result. Theorem 1. If a Boolean function of n variables can be implemented by a circuit over the basis {⊕,∧, 1} of multiplicative complexity M = Ω(n), then it can be implemented by a circuit of total complexity (1/2 + o(1))M(M + 2n)/ log2M over the same basis. The bound is asymptotically optimal. ∗Research supported in part by RFBR, grants 11–01–00508, 11–01–00792, and OMN RAS “Algebraic and combinatorial methods of mathematical cybernetics and information systems of new generation” program (project “Problems of optimal synthesis of control systems”). †e-mail: [email protected] [11] deals with monotone Boolean circuits.