MATHEMATICS

B. A. SUBBOTOVSKAYA

ON THE REALIZATION OF LINEAR FUNCTIONS BY FORMULAS IN THE BASIS \(\vee, \&, \overline{\phantom{x}}\)

(Presented by Academician A. I. Berg on 20 VIII 1960)

We shall consider formulas in the basis \(\vee, \&, \overline{\phantom{x}}\). The number of occurrences of variable symbols in a formula \(F\) will be called its complexity and denoted by \(L(F)\), and the number \(\min L(F)\), where the minimum is taken over all formulas \(F\) realizing \(f\) in the basis \(\vee, \&, \overline{\phantom{x}}\), by \(L(f)\). Obviously, in the language of contact circuits \(L(f)\) is the minimum number of contacts sufficient for realizing the function \(f\) by series-parallel circuits.

We shall denote the function \(\sigma + x_1 + \cdots + x_n \pmod 2\) by the symbol \(f_\sigma^n\). As was shown by S. V. Yablonskii \((^1)\),

\[ L(f_\sigma^n) \leq {9 \over 8} n^2 . \]

In this note it will be shown that

\[ L(f_\sigma^n) > c n^{3/2}, \]

where \(c\) is a certain constant.

For the proof, let us consider a certain special class of formulas in the basis \(\&, \vee, \overline{\phantom{x}}, 0, 1\). Let \(\{x_1,\ldots,x_i,\ldots\}\) be the set of variable symbols. We shall call the expressions \(0,1,\overline{0},\overline{1},x_i,\overline{x_i}\) \((i=1,2,\ldots)\) \(\pi\)-formulas and \(\sigma\)-formulas simultaneously. Let \(F_1,\ldots,F_s\) be \(\pi\)-formulas (respectively, \(\sigma\)-formulas); then \((F_1 \vee \cdots \vee F_s)\) will be called a \(\sigma\)-formula (respectively, \((F_1 \& \cdots \& F_s)\) will be called a \(\pi\)-formula). We denote by \(W\) the class of all \(\pi\)-formulas and \(\sigma\)-formulas. We shall call the formula \((F_1 \vee \cdots \vee F_s)\) (respectively, \((F_1 \& \cdots \& F_s)\)) an extension of each of its subformulas \(F_i\), which we shall in turn call a component of this formula.

Obviously, every component is itself a formula from \(W\), and, consequently, every component different from formulas of the form \(0,\overline{0},1,\overline{1},x_i,\overline{x_i}\) is itself an extension of its components.

Two formulas \(F_1\) and \(F_2\) realizing one and the same function will be called equivalent (notation \(F_1 \sim F_2\)). We shall not distinguish two equivalent formulas each of which is obtained from the other by a permutation of the components of some of its subformulas.

Obviously, for every formula \(F\) in the basis \(\vee, \&, \overline{\phantom{x}}\) one can indicate an equivalent formula \(F_1\) from \(W\) such that \(L(F)=L(F_1)\). For example, the formula \((((x_1 \vee x_2)\& \overline{x_3})\& x_4)\) corresponds to the formula \(((x_1 \vee x_2)\& \overline{x_3}\& x_4)\) from \(W\).

For an arbitrary formula \(F\) from \(W\), the symbol \(L(F)\) will denote, as before, the number of occurrences of variables in \(F\).

Let \(\widehat W\) be the set of formulas from \(W\) containing no subformulas of the form \(0,1,\overline{0},\overline{1}\).

Lemma 1. For every formula \(F\), \(F \in W\), not equivalent to a constant, one can indicate an equivalent formula \(\widehat F\) from \(\widehat W\) such that \(L(\widehat F) \leq L(F)\).

The assertion of the lemma follows in a trivial way from the equalities
$0\vee x=x$, $0\&x=0$, $\bar 0=1$, $1\vee x=1$, $1\&x=x$, $\bar 1=0$.

Let $\psi$ be a subformula of the form $x^\sigma$ of a formula $F$ from $W$. Consider its expansion $(\psi\circ\varphi_1\circ\ldots\circ\varphi_s)$ in $F$, where $\circ$ denotes $\vee$ or $\&$. Obviously, for a certain value $\tau$ of the occurrence of the variable $x$ in $\psi$ ($\tau=\sigma$, if $\circ$ is $\vee$, and $\tau=\bar\sigma$, if $\circ$ is $\&$), the value of the formula $(\tau^\sigma\circ\varphi_1\circ\ldots\circ\varphi_s)$ does not depend on the expression $\varphi_1\circ\ldots\circ\varphi_s$. We shall call $\tau$ the determining value of this occurrence of the variable $x$. A formula in which the expansion of every subformula of the form $x_i^\sigma$ $(i=1,2,\ldots)$ has no other occurrences of the variable $x_i$ will be called normal*.

Example. The formula $((x_1\vee x_2)\&(x_1\vee 0))$ is normal, while the formula $(((x_1\&x_1)\vee x_2)\&x_3)$ is not normal.

Remark. If $x^{\sigma_1}$ and $x^{\sigma_2}$ are two distinct subformulas of a normal formula $F$, then their expansions do not intersect.

Let $f$ be a function of the algebra of logic; denote by the symbol ${}^\tau f^x$ the function that is obtained from $f$ if in it the variable $x$ is replaced by the constant $\tau$. Obviously, if $f$ does not depend essentially on $x$, then ${}^\tau f^x=f$. Analogously, if $F$ is a formula, then by the symbol ${}^\tau F^x$ we shall denote the formula obtained from $F$ by substituting the constant $\tau$ for all occurrences of the variable $x$. If $F$ has no occurrences of the variable $x$, then ${}^\tau F^x$ coincides with $F$.

Lemma 2. For an arbitrary formula $F$ from $W$, not equivalent to a constant, one can find a normal formula $\tilde F$ from $W$ equivalent to it such that
\[ L(\tilde F)\leq L(F). \]

Proof. Let in the formula $F$ from $W$ there be a subformula $\Phi$ of the form $(\psi\circ\varphi_1\circ\ldots\circ\varphi_s)$, where $\psi$ has the form $x^\sigma$ and $\tau$ is the determining value of the occurrence of the variable $x$ in $\psi$. Suppose further that the expression $\varphi_1\circ\ldots\circ\varphi_s$ contains its occurrences of the variable $x$. Consider the formula $F_1$, obtained from the formula $F$ by replacing in it the subformula $\Phi$ by the formula $\Phi_1$ of the form
\[ (\psi\circ{}^{\bar\tau}\varphi_1^x\circ\ldots\circ{}^{\bar\tau}\varphi_s^x). \]
Obviously, for $F$ and $F_1$ the inequality $L(F_1)<L(F)$ holds. We shall show that $F_1$ and $F$ are equivalent. For this it suffices to show that $\Phi\sim\Phi_1$. But indeed, for $x=\tau$ the values of the formulas $\Phi$ and $\Phi_1$ are determined only by the value of the component ${}^\tau\psi^x$ and are equal to $\tau^\sigma$, while for $x=\bar\tau$ these values coincide by the construction of the formula $\Phi_1$.

Obviously, by applying the described process, one can satisfy the requirement of the lemma. For example, the formula $(((x_1\&x_1)\vee x_2)\&x_3)$ is equivalent to the normal formula $(((x_1\&1)\vee x_2)\&x_3)$ from $W$.

Lemma 3. If a normal formula $F$ from $\hat W$, realizing the function $f$, contains $m$ occurrences of the variable $x$, then there exists a $\tau$ and a formula $F_1$, realizing the function ${}^\tau f^x$, such that the inequality
\[ L(F)\geq \frac{3}{2}\,m+L(F_1). \tag{1} \]
holds.

Proof. Let $F_1,\ldots,F_m$ be the expansions, respectively, of the subformulas $\psi_1,\ldots,\psi_m$ of the formula $F$ of the form $x^{\sigma_i}$, $1\leq i\leq m$, and let $\tau_1,\ldots,\tau_m$ be, respectively, the determining values of the occurrences of the variable $x$ in these subformulas. Obviously, no two of these expansions have common occurrences of variables. Let $\tau$ be a constant occurring in the set $\tau_1,\ldots,\tau_m$ at least $m/2$ times. Consider the formula ${}^\tau F^x$. Obviously, in it no fewer than $m/2$ subformulas ${}^\tau F_i^x$ are equivalent respectively

* $x^\sigma$ is $x$ for $\sigma=1$ and $\bar x$ for $\sigma=0$.

constants \(\tau^{\sigma_i}\). Replace in the formula \(\tau F^x\) each subformula \(\tau F_i^x\) equivalent to the constant \(\tau^{\sigma_i}\) by this constant, and denote the resulting formula by \(F_1\). Taking into account that in each of the subformulas \(F_i\), \(i=1,\ldots,m\), there is at least one occurrence of a variable different from \(x\), we conclude that \(F_1\) satisfies inequality (1).

Corollary 1. Let the variable \(x\) have, in a normal formula \(F\) from \(\hat W\) realizing a function \(f\) of \(n\) variables, a maximal number of occurrences equal to \(m\); then there are a \(\tau\) and a formula \(F_1\) realizing the function \(\tau f^x\) for which the inequality

\[ L(F)\geqslant \frac{L(F_1)}{1-3/2m}. \]

holds.

Obviously, the number \(m\) considered in the hypothesis of the corollary satisfies the inequality \(m\geqslant L(F)/n\). Substituting it in (1), we obtain the required inequality.

Now consider a formula realizing the function \(f_\sigma^n\). Obviously, the formula \(\tau F_n^x\) realizes some function \(f_{\sigma_1}^{\,n-1}\).

Corollary 2. If a formula \(\hat F_n\) from \(\hat W\) realizes the function \(f_\sigma^n\), then

\[ L(\hat F_n)\geqslant \prod_{i=2}^{n}\frac{1}{1-3/2i}\,L(f_\sigma^1). \tag{2} \]

The inequality is proved by induction using Corollary 1 of Lemma 3 and Lemmas 1 and 2.

Theorem. If \(F_n\) is an arbitrary formula in the basis \(\vee,\ \&,\ ^{-}\), realizing the function \(f_\sigma^n\), then

\[ L(F_n)\geqslant Cn^{3/2}. \]

Proof. Obviously, \(L(F_n)\) satisfies inequality (2). We have

\[ \prod_{i=2}^{n}\frac{1}{1-3/2i} = \exp\left[-\sum_{i=2}^{n}\ln\left(1-\frac{3}{2i}\right)\right] = \exp\left[\sum_{k=2}^{n}\frac{3}{2k}+O(1)\right] \geqslant C_1 n^{3/2}, \]

whence

\[ L(F_n)\geqslant Cn^{3/2}. \]

Moscow State University
named after M. V. Lomonosov

Received
5 VIII 1960

REFERENCES

1. S. V. Yablonskii, DAN, 94, No. 5, 805 (1954).