Reports of the Academy of Sciences of the USSR
1961. Volume 139, No. 6

MATHEMATICS

E. I. NECHIPORUK

ON THE COMPLEXITY OF CIRCUITS IN CERTAIN BASES CONTAINING NONTRIVIAL ELEMENTS WITH ZERO WEIGHTS

(Presented by Academician P. S. Novikov on 16 IV 1961)

Let Boolean functions be realized by circuits \((^{1})\) of a certain type over a basis consisting of a set \(\mathfrak{Z}\) of elements with zero weights, realizing a set of functions \(Z\), and a set \(\mathfrak{E}\) of elements with positive weights, realizing a set of functions \(E\). We consider the cases when arbitrary circuits are allowed and when circuits are superpositions of basis elements \((^{2})\).

As usual, we define the weight of a circuit and of a function in Shannon’s sense \(L(f), L(n)\).

1. Consider the basis \(\mathfrak{A}\) with \(E=\{\neg x\}\), \(Z=\{x_1\&x_2,\ x_1\vee x_2,\ 0,\ 1\}\); the weight of negation is equal to 1.

We shall call a chain a set \(\{\tilde{\sigma}^{j}\}\), \(j=0,1,\ldots,n\), of \(n\)-dimensional Boolean vectors such that \(\tilde{\sigma}^{j'}<\tilde{\sigma}^{j'+1}\), \(0\leq j'\leq n-1\). Renumber all chains for fixed \(n\); denote the \(k\)-th chain by \(\omega_k\), \(\omega_k=\{\tilde{\sigma}^0_k,\ldots,\tilde{\sigma}^n_k\}\). Let \(\xi(\tilde{x})=\xi(x^1,\ldots,x^n)\) be an arbitrary Boolean function. The vector \(\tilde{\sigma}\) will be called a negative (positive) inversion node (n.i.n. or p.i.n.) of the pair \((\xi,\omega_k)\), if \(\tilde{\sigma}=\tilde{\sigma}^{j'}_k\) and \(\xi(\tilde{\sigma}^{j'}_k)=1\), \(\xi(\tilde{\sigma}^{j'+1}_k)=0\) \((\xi(\tilde{\sigma}^{j'}_k)=0,\ \xi(\tilde{\sigma}^{j'+1}_k)=1)\). Denote by \(M^-_k[\xi]\) \((M^+_k[\xi])\) the number of n.i.n. (p.i.n.) of the pair \((\xi,\omega_k)\); introduce the functions \(M^-[\xi]=\max_k M^-_k[\xi]\),

\[ M^+[\xi]=\max_k M^+_k[\xi]. \]

We shall denote by \(\chi(Q)\) the characteristic function of a set of vectors \(Q\).

Lemma 1. 1) Every n.i.n. (p.i.n.) of one of the pairs \((\xi\&\eta,\omega_k)\), \((\xi\vee\eta,\omega_k)\) is an n.i.n. (p.i.n.) of one of the pairs \((\xi,\omega_k)\), \((\eta,\omega_k)\).

2) Every n.i.n. (p.i.n.) of the pair \((\neg\xi,\omega_k)\) is a p.i.n. (n.i.n.) of the pair \((\xi,\omega_k)\).

Lemma 2. \(M^+[\xi]-M^-[\xi]=\operatorname{sign}[\xi(\tilde{1})-\xi(\tilde{0})]\).

Theorem 1. For superpositions in the basis \(\mathfrak{A}\),

\[ L(f)=M^-[f]. \]

Proof. By Lemmas 1 and 2, \(M^-[f]\leq L(f)\). Denote by \(R_k(\tilde{\sigma})\) the number of n.i.n. of the pair \((f,\omega_k)\) not less than the vector \(\tilde{\sigma}\); the number \(\max_k R_k(\tilde{\sigma})\) will be called the depth of the vector \(\tilde{\sigma}\). Denote by \(Q_s\) the set of all vectors of depth \(s\). We shall call a function \(\eta\) monotonically decreasing if \(\eta(\tilde{\sigma}_1)\geq \eta(\tilde{\sigma}_2)\) for \(\tilde{\sigma}_1\leq \tilde{\sigma}_2\). Denote by \(\xi^+\) \((\xi^-)\) the least monotone (monotonically decreasing) function such that \(\xi\leq\xi^+\). Put

\[ g_s=(f\&\chi(\bigcup_{0\leq s''\leq s'} Q_{s''}))^+,\qquad h_s=\neg(f\&\chi(Q_s))^-, \]

\(1 \leq s \leq M^{-}[f],\quad 0 \leq s' \leq M^{-}[f]\). We have

\[ f=g_0 \vee \bigvee_{s=1}^{M^{-}[f]} g_s \,\&\,(\neg h_s), \]

where \(g_s, h_s\) are monotone functions. Consequently, \(L(f) \leq M^{-}[f]\). The theorem is proved.

Corollary. \(L(n)=\left\lfloor \dfrac{n}{2}\right\rfloor\).

If \(f \ne \mathrm{const}\), the theorem remains valid also for superpositions in bases obtained from \(\mathfrak A\) by excluding one or both constants.

1. Let \(\mathfrak G\) be a basis with \(E=\{\neg x\}\), \(Z=\{x_1 \vee x_2\}\); the weight of inversion is equal to 1.

Theorem 2. For superpositions in the basis \(\mathfrak G\)

\[ L(n)\sim \frac{2^n}{n}. \]

The lower estimate is obtained as in \((^3)\). The upper estimate is based on representing a function in the form of a disjunction of conjunctions of two functions, one of which is symmetric and the other monotonically decreasing, with a subsequent proper representation \((^4)\) of the monotonically decreasing function.

Theorem 3. For circuits in the basis \(\mathfrak G\)

\[ L(n)\sim \sqrt{2}\,2^{n/2}. \]

The proof is based on the idea of using graphs, as in \((^5)\).

Theorem 4. For circuits in the basis \(\Lambda_i,\ i=1—5\ ({}^3)\),

\[ L(n)\sim 2^{n/2}. \]

I express my gratitude to O. B. Lupanov for posing the problem.

Received
25 III 1961

REFERENCES

\({}^1\) O. B. Lupanov, Radiophysics, 1, 120 (1958).
\({}^2\) R. E. Krichevskii, Dokl. Akad. Nauk SSSR, 126, No. 6 (1959).
\({}^3\) É. I. Nechiporuk, Dokl. Akad. Nauk SSSR, 136, No. 3 (1961).
\({}^4\) O. B. Lupanov, Dokl. Akad. Nauk SSSR, 119, No. 1 (1958).
\({}^5\) É. I. Nechiporuk, Dokl. Akad. Nauk SSSR, 137, No. 5 (1961).