Upper bounds on the multiplicative complexity of symmetric Boolean functions

Abstract

A special metric of interest about Boolean functions is multiplicative complexity (MC): the minimum number of AND gates sufficient to implement a function with a Boolean circuit over the basis {XOR, AND, NOT}. In this paper we study the MC of symmetric Boolean functions, whose output is invariant upon reordering of the input variables. Based on the Hamming weight method from Muller and Preparata (J. ACM 22(2), 195–201, 1975), we introduce new techniques that yield circuits with fewer AND gates than upper bounded by Boyar et al. (Theor. Comput. Sci. 235(1), 43–57, 2000) and by Boyar and Peralta (Theor. Comput. Sci. 396(1–3), 223–246, 2008). We generate circuits for all such functions with up to 25 variables. As a special focus, we report concrete upper bounds for the MC of elementary symmetric functions ${{\Sigma }^{n}_{k}}$ and counting functions ${E^{n}_{k}}$ with up to n = 25 input variables. In particular, this allows us to answer two questions posed in 2008: both the elementary symmetric ${{\Sigma }^{8}_{4}}$ and the counting ${E^{8}_{4}}$ functions have MC 6. Furthermore, we show upper bounds for the maximum MC in the class of n-variable symmetric Boolean functions, for each n up to 132.

Appendices

Appendix A: MC upper-bounds for special classes of symmetric Boolean functions

Tables 5 and 6 show MC upper bounds, respectively for all elementary-symmetric Boolean functions ${\Sigma }^n_k$ and all exactly-counting Boolean functions $E^n_k$, with any number n of variables up to 25, and with k up to n.

Table 5 Upper bounds on the MC of elementary symmetric functions ${{\Sigma }_{k}^{n}}$

Table 6 Upper bounds on the MC of (exactly-k) counting functions ${E}_{k}^{n}$

Appendix B: Description of $\text {MC}_{\max \nolimits }$ upper bounds

Table 7 shows, for each n ≤ 132, the $\text {MC}_{\max \nolimits }$ upper bound we found for the set ${\mathcal {S}}_{n}$ of n-variable symmetric Boolean functions. For each n, the table identifies an encoding H of the Hamming weight, and a method G for finding an MC upper bound of the corresponding g. We checked five different practical combinations of H and G:

Table 7 Upper bounds (UB) obtained for $\text {MC}_{\max \nolimits }({\mathcal {S}}_{n})$

C₁.:

H = H_BR and G = gen, where “gen” uses for g the MC upper bound from Table 1. This was used for n ∈{1, 3, 7, 15, 29–31, 48–63, 99–127}.

C₂.:

H = H₀ (i.e., using only full adders) and G = exp, where “exp” is an exhaustive computation “experimentally” determining the MC of each g corresponding to each $f \in {\mathcal {S}}_{n}$. This was used for n ∈{14, 18, 20, 21}.

C₃.:

H = H_j (possibly using some (j ≥ 0) half adders, but not computing the full H_BR) and G = gen. This was used for n ∈{2, 4 −− 6, 8 −− 13, 16 −− 17, 19, 22 −− 28, 32 −− 47}.

C₄.:

H = H_BR and G = G_i, where G_i applies the concatenation method to function g, to obtain g₂ with only i variables (if i ≥ 1), or to use g₂ = g₁ + g(0,…, 0, 1) (if i = 0). This was used for n ∈{64 −− 79, 81 −− 95, 128 −− 132}.

C₅.:

H = H_BR and G = G_i,j, where G_i,j applies G_i to g and then applies G_j to the corresponding g₂. This was used for n ∈{80, 96 −− 98}.

The best combination varies with n, but sometimes several combinations yield the same best upper bound. Table 7 shows H and G only for the first best combination in the order C₁ < C₂ < C₃ < C₄ < C₅.

Column “H” shows the number of used half adders as a subscript j in H_j. When said encoding is H_BR, an asterisk is added as suffix ($H^{\ast }_{j}$). Column “D” shows the difference to the upper bound that would be obtained with the reference method C₁. Column “UB” shows the upper bound in bold when it is equal to the degree bound (n − 1).

Example 3

The case n = 72 (using combination C₄) indicates an encoding $H=H_{5}^{\ast } = H_{\text {BR}}$ with 5 half adders, and a method G₄ for g. The encoding H_BR produces an output of seven variables (z₁,…,z₇), upon which the function g can be written as g₁(z₁,…,z₆) ⊕ z₇ ∧ (g₁(z₁,…,z₆) ⊕ g₂(z₁,…,z₄)). Since the MC for H_BR(x₁,...,x₇₂) is 70, the overall upper bound is equal to 79 = 70 + 6 + 1 + 3, where 6 and 3 are the generic MC upper bounds for g₁ and g₂ (functions of 6 and 4 variables, respectively), and the extra 1 is the AND used to multiply z₇ with (g₁ ⊕ g₂).

Example 4

The case n = 80 (using combination C₅) indicates the use of $H=H_{5}^{\ast } = H_{\text {BR}}$ and G_5,0. The H_BR encoding outputs 7 variables. Then, G₅ decomposes g into g₁(z₁,…,z₆) ⊕ y₆ ∧ (g₁(z₁,…,z₆) ⊕ g₂(z₁,...,z₅)). Since for n = 80 there are 81 possible weights, the function g₂ is a 5-variable function with 17 (= 81 − 64) defined entries and 15 free entries. For the second decomposition, the number of defined entries of the second component will be 1(= 17 − 16). Thus, G₀ can be applied (recall the exceptional case described in Section 4.3) to decompose g₂ into $g^{\prime }_{2}(z_{1},\ldots ,z_{4}) \oplus (z_{5} \wedge b)$, where b is the constant g(0,..., 0, 1). Thus, the upper bound for the $\text {MC}_{\max \nolimits }$ for n = 80 is equal to 88 = 78 + 6 + 1 + (3 + 0), where 78 is the MC of H_BR on 80 variables, and where 6, 3 and 0 are the MC majorants for the 6-variable function g₁, the 4-variable function $g^{\prime }_{2}$, and the 1-variable function b ∧ z₅, respectively.