UDC 519.95

CYBERNETICS
AND CONTROL THEORY

V. S. GRINBERG

SOME NEW ESTIMATES IN THE THEORY OF FINITE AUTOMATA

(Presented by Academician V. M. Glushkov, 5 VI 1965)

Let a finite system of directed graphs with a common set \(S\) of vertices be given,

\[ G=\{(S,\Gamma_1),(S,\Gamma_2),\ldots,(S,\Gamma_k)\}. \]

Select some \(S_0\subseteq S\) and call it the initial set. We extend in the natural way the mappings \(\Gamma_i\) to the set \(\Sigma\) of all subsets of the set \(S\):

\[ \Gamma_i M=\bigcup_{\sigma\in M}\Gamma_i\sigma \qquad (M\subseteq S). \]

By \(\widetilde{\Sigma}\) we shall denote the smallest subset of the set \(\Sigma\), closed with respect to all mappings \(\Gamma_i\), such that \(S_0\in \widetilde{\Sigma}\). Generalizing the definition given in (1), we shall call the transformation of the system \(G\) into the system

\[ \widetilde{G}=\{(\widetilde{\Sigma},\Gamma_1),\ldots,(\widetilde{\Sigma},\Gamma_k)\} \]

an optimal determinization of the system \(G\).

Everywhere below we shall consider systems of graphs possessing the following orthogonality property:

\[ \Gamma_i S\cap \Gamma_j S=\varnothing \qquad (i\ne j). \]

Many problems of automaton synthesis reduce to determinization of an orthogonal system of graphs. In this connection there naturally arises the question of estimating the number \(d(G)\) of elements of the set \(\widetilde{\Sigma}\).

Denoting by \(|A|\) the cardinality of the set \(A\), put \(n=|S|\), \(n_i=|\Gamma_i S|\). By orthogonality,

\[ \sum_{i=1}^{k} n_i \leq n. \tag{1} \]

Since \(\widetilde{\Sigma}\) contains only \(S_0\) and subsets of the sets \(\Gamma_iS\),

\[ d(G)\leq 2^{n_1}+2^{n_2}+\cdots+2^{n_k}+1. \tag{2} \]

Let us now denote by \(d_k(n)\) the maximum of \(d(G)\) over all orthogonal systems \(G\) with fixed \(k\) and \(n\). Inequality (2) gives for \(d_k(n)\) the trivial estimate

\[ d_k(n)\leq 2^n+1. \tag{3} \]

In the work (1), Yu. I. Lyubich found that for \(k=1\) the estimate (3) can be replaced by the substantially more exact estimate

\[ d_1(n)\leq M(n) \qquad (n\geq 6), \tag{4} \]

where

\[ M(n)=\max\left\{m(N_1,N_2,\ldots,N_s)+\sum_{k=2}^{s} m(N_1,N_2,\ldots,N_k)\right\}+n^2-2n+6, \]

the maximum is taken over all systems \((N_1,N_2,\ldots,N_s)\) \((s=2,3,\ldots)\) of natural numbers satisfying the condition

\[ N_1+N_2+\cdots+N_s\leq n; \]

\(m(\ ,\ )\) denotes the least common multiple. There was also given there a found—

the asymptotic formula obtained by I. V. Ostrovskii

\[ \ln M(n)\sim \sqrt{n\ln n}\qquad (n\to\infty) \tag{5} \]

and it was proved that

\[ \ln d_1(n)\sim \sqrt{n\ln n}\qquad (n\to\infty). \]

Relying on these results, in the present note we shall establish that estimate (3) can be substantially lowered also for \(k>1\). We shall then use this to obtain asymptotically exact estimates in some problems of automata synthesis. Everywhere in what follows it is assumed, for definiteness, that

\[ n_1\leq n_2\leq \ldots \leq n_k. \tag{6} \]

Theorem 1. The inequality holds

\[ d(G)\leq (2^{n_1}+2^{n_2}+\ldots+2^{n_{k-1}}+1)(d_1(n_k)+1). \tag{7} \]

Proof. Denote by \(\Sigma_1\) the class of all sets of the form \(\Gamma_k pS_0\) \((p=0,1,2,\ldots)\). Denote by \(\Sigma_2\) the class of all sets of the form \(\Gamma_k pM\) \((p=0,1,2,\ldots;\ M\subseteq \Gamma_iS;\ i=1,2,\ldots,k-1)\). Clearly, \(\Sigma\subseteq \Sigma_1\cup\Sigma_2\). But

\[ |\Sigma_1|\leq d_1(n_k)+1,\qquad |\Sigma_2|\leq (2^{n_1}+2^{n_2}+\ldots+2^{n_{k-1}})(d_1(n_k)+1). \]

The theorem is proved.

It is easy to verify that, under conditions (1) and (6), the inequality

\[ 2^{n_1}+2^{n_2}+\ldots+2^{n_{k-1}}\leq 2^{n/2}\qquad (n\geq 8) \tag{8} \]

is valid.

From (4), (7), and (8) it follows

Corollary 1. The inequality holds

\[ d(G)\leq (2^{n/2}+1)(M(n)+1)\qquad (n\geq 8). \tag{9} \]

Since this estimate uses no information about the orthogonal system \(G\) except the number \(n\), the inequality

\[ d_k(n)\leq (2^{n/2}+1)(M(n)+1)\qquad (n\geq 8) \tag{10} \]

holds.

We note that knowledge of the structure of the system \(G\) makes it possible in some cases, using the results obtained in \((^1)\), to improve estimate (9). For example, the following holds:

Corollary 2. If the graph \((\Gamma_kS,\Gamma_k)\) is strongly connected, then

\[ d(G)\leq (2^{n/2}+1)(n^2-2n+4)\qquad (n\geq 8). \]

Let us apply the results obtained to derive estimates in two problems of automata synthesis. Everywhere in what follows Moore automata are meant.

Let \(E\) be a regular event (see \((^3,^4)\)). Denote by \(f(E)\) the number of states of the minimal automaton representing the event \(E\). Next put \(\varphi(n)=\max f(E)\), where the maximum is taken over all events \(E\) enumerable** by automata with \(n\) states. We shall be interested in the question of the rate of growth of \(\varphi(n)\). This question was posed by V. A. Uspenskii (see, for example, \((^6)\)) and for Mealy automata was completely solved independently by Yu. L. Ershov \((^7)\) and O. B. Lupanov \((^5)\).

* For the definition of a strongly connected graph, see, for example, \((^2)\).

** For the definition of an event enumerable by an automaton, see, for example, \((^5)\).

Theorem 2. The following asymptotic equality holds:
\[ \log_2 \varphi(n) \sim n/2 \qquad (n \to \infty). \tag{11} \]

Proof. There is a well-known algorithm (see, for example, (7)) for constructing, from a given automaton \(\mathfrak A\), an automaton \(\widehat{\mathfrak A}\) representing the event that is enumerated by the automaton \(\mathfrak A\). In our terms this algorithm is the determinization of the system of graphs \(G\) obtained if we denote by \(k\) the number of letters of the output alphabet of the automaton \(\mathfrak A\), by \(S\) the set of states of the automaton \(\mathfrak A\), and by \(\Gamma_i\) the mapping specified by those transitions in the automaton \(\mathfrak A\) for which the \(i\)-th alphabet symbol is supplied at the output. In the case of a Moore automaton this system is, obviously, orthogonal. Therefore
\[ \varphi(n) \leq \max_k d_k(n). \]
Taking (5) and (10) into account, we obtain
\[ \log_2 \varphi(n) \leq \frac n2[1+o(1)]. \]

The validity of (11) now follows from the following result of G. M. Kornelevich (6):
\[ \varphi(n) \geq 2^{[n/2]} \qquad (n \geq 4). \]

Theorem 1 also makes it possible to obtain the asymptotics of the maximum amount of memory specified by a regular expression of length \(n\) \((n \to \infty)\) over a \(k\)-letter alphabet for arbitrary \(k>1\). For \(k=1\) the asymptotics was obtained in (8). By the length of a regular expression \(R\) is meant the number of occurrences in it of letters of the alphabet. The weight \(V(R)\) of a regular expression is the number of states of the minimal automaton representing the corresponding event. Put
\[ \psi_k(n)=\max V(R), \]
where the maximum is taken over all regular expressions of length \(n\) over a \(k\)-letter alphabet.

V. M. Glushkov established (8) that
\[ \psi_k(n) \leq 2^n+1. \]

Theorem 3. For \(k \geq 2\) the following asymptotic equality holds:
\[ \log_2 \psi_k(n) \sim n/2 \qquad (n \to \infty). \]

Proof. The known algorithm (4) for constructing an automaton representing an event consists in the following: from a given regular expression of length \(n\), a certain orthogonal system of graphs \(G\) is constructed on a set \(S\) of cardinality \(n+1\), and then it is determinized. Consequently,
\[ \log_2 \psi_k(n) \leq \frac n2[1+o(1)]. \]

To complete the proof it is enough to construct a regular expression over an alphabet of two symbols \(x\) and \(y\), whose length is an arbitrary even \(n=2p\) and whose weight is no less than \(2^{p-1}\). Such an expression is
\[ R_p=(\{x\}(\bar x\{y\})^{-p-1}x). \]

We note that the regular expression \(R_p\) is closed (9). In (9) it was found that for \(k=1\) closed regular expressions of length \(n\) have weight
\[ \leq n^2-2n+4 \qquad (n \geq 5), \]
i.e., substantially smaller than \(\psi_1(n)\), since
\[ \ln \psi_1(n) \sim \sqrt{n\ln n}\qquad (n \to \infty). \]
At the same time, we have seen that for \(k>1\) the maximum of the weight is attained asymptotically on closed regular expressions.

The author expresses his gratitude to Yu. I. Lyubich for supervising the work.

Kharkov State University
named after A. M. Gorky

Received
18 V 1965

REFERENCES

1. Yu. I. Lyubich, Sibirsk. matem. zhurn., 5, No. 2 (1964).
2. C. Berge, The Theory of Graphs and Its Applications, Moscow, 1962.
3. S. C. Kleene, in: Automata, Moscow, 1956.
4. V. M. Glushkov, Sintez tsifrovykh avtomatov, Moscow, 1962.
5. O. B. Lupanov, Problemy kibernetiki, vol. 9, Moscow, 1963.
6. G. M. Kornelevich, DAN, 149, No. 5 (1963).
7. Yu. L. Ershov, Algebra i logika, Seminar of the Institute of Mathematics, Siberian Branch, Academy of Sciences of the USSR, 1, No. 4, 45 (1962).
8. V. M. Glushkov, Ukr. matem. zhurn., 12, No. 2 (1960).
9. Yu. I. Lyubich and E. M. Livshits, Sibirsk. matem. zhurn., 6, No. 1 (1965).