CYBERNETICS AND CONTROL THEORY

A. G. Lunts

A METHOD FOR ANALYZING FINITE AUTOMATA

(Presented by Academician A. N. Kolmogorov, 10 VIII 1964)

This article describes a method for the analysis of finite automata, analogous to the third method for analyzing contact circuits from (²).

Let \(X=\{x_1,x_2,\ldots,x_n\}\) be the (input) alphabet; let \(V\) be the set of all (finite) words in this alphabet, including the empty word \(e\); let \(\Delta\) be the algebra of events in this alphabet, i.e., the set of all subsets of the set \(V\), on which three operations are defined: disjunction \(a \cup b\), product \(ab\), and iteration \(\langle a\rangle\) (³). We shall consider matrices over the algebra \(\Delta\). The three operations listed above are naturally transferred to such matrices: disjunction \(A \cup B=C\) (where \(c_{\alpha\beta}=a_{\alpha\beta}\cup b_{\alpha\beta}\)), product \(AB=C\) (where \(c_{\alpha\beta}=\bigcup_k a_{\alpha k}b_{k\beta}\)), and iteration

\[ \langle A\rangle=\bigcup_{k=0}^{\infty} A^k \]

(where \(A^0\) is the identity matrix, i.e., the matrix having the event \(e\) on the main diagonal and the empty events \(\Lambda\) elsewhere, \(A^2=AA\), \(A^3=AAA,\ldots\)) (³). We shall also denote the iteration \(\langle A\rangle\) by \(\chi(A)\), and its elements by \(\chi_{\alpha\beta}(A)\). By analogy with contact circuits (²), we shall interpret a square matrix \(A\) as a multipole with poles \(z_1,z_2,\ldots,z_s\) (\(s\) is the order of the matrix \(A\)); its element \(a_{\alpha\beta}\) (\(\alpha,\beta=1,2,\ldots,s\)) will be called the direct conductance from pole \(z_\alpha\) to pole \(z_\beta\), and the element \(\chi_{\alpha\beta}(A)\) of the iteration the complete conductance from \(z_\alpha\) to \(z_\beta\).

Let us note that if the matrix \(A\) satisfies the conditions:
1) the events \(a_{\alpha\beta}\) contain only one-letter words (not excluding the case \(a_{\alpha\beta}=\Lambda\));
2) \(a_{\alpha\beta}\cap a_{\alpha\gamma}=\Lambda\) for \(\beta\ne\gamma\);
3) \(\bigcup_{\beta=1}^{s} a_{\alpha\beta}=X\),

then the matrix \(A\) is the transition matrix for a finite automaton with states \(z_1,\ldots,z_s\).* In this case the complete conductance \(\chi_{\alpha\beta}(A)\) is the event that is represented in the automaton by the state \(z_\beta\), if the state \(z_\alpha\) is taken as the initial state of the automaton (³).

Let \(A\) be an arbitrary square matrix of order \(s\) over the algebra \(\Delta\). Consider a matrix \(B\) of order \((s-1)\), whose elements are determined by the equalities

\[ b_{\alpha\beta}=a_{\alpha\beta}\cup a_{\alpha s}\langle a_{ss}\rangle a_{s\beta},\quad \alpha,\ \beta=1,2,\ldots,s-1, \tag{1} \]

i.e.

\[ B= \begin{bmatrix} a_{1,1}\cdots a_{1,s-1}\\ \cdots\cdots\cdots\cdots\\ a_{s-1,1}\cdots a_{s-1,s-1} \end{bmatrix} \cup \begin{bmatrix} a_{1,s}\\ \cdots\\ a_{s-1,s} \end{bmatrix} \langle a_{ss}\rangle \begin{bmatrix} a_{s,1}\cdots a_{s,s-1} \end{bmatrix}, \]

* If condition 3) is not satisfied, then the automaton will be partial; if condition 2) is not satisfied, then the automaton will be nondeterministic.

which is analogous to formula (1). We shall say that the \((s-1)\)-terminal network \(B\) is obtained from the \(s\)-terminal network \(A\) by eliminating the terminal \(z_s\). The following theorem, entirely analogous to the theorem from (2), holds.

Theorem. When a terminal is eliminated from a multi-terminal network, the complete conductances corresponding to pairs of the remaining terminals remain unchanged, i.e.

\[ \chi_{\alpha\beta}(B)=\chi_{\alpha\beta}(A),\qquad \alpha,\beta=1,2,\ldots,s-1. \tag{2} \]

Successively eliminating, in some order, the terminals \(z_2, z_3, \ldots, z_s\), we arrive at a first-order matrix \(B=[b_{11}]\). According to the theorem, we shall have \(\chi_{11}(A)=\chi_{11}(B)=\langle b_{11}\rangle\). The quantities \(\chi_{\alpha\alpha}(A)\), \(\alpha=2,3,\ldots,s\), are found analogously. To find the complete conductance \(\chi_{\alpha\beta}\), where \(\alpha\ne\beta\), we successively eliminate all terminals except \(z_\alpha\) and \(z_\beta\). As a result we obtain a second-order matrix

\[ B= \begin{bmatrix} b_{\alpha\alpha} & b_{\alpha\beta}\\ b_{\beta\alpha} & b_{\beta\beta} \end{bmatrix}. \]

After this the desired conductance can be found by any of the following three formulas:

\[ \chi_{\alpha\beta}(A)=\langle b_{\alpha\alpha}\rangle b_{\alpha\beta}\chi_{\beta\beta}(A) =\chi_{\alpha\alpha}(A)b_{\alpha\beta}\langle b_{\beta\beta}\rangle =\chi_{\alpha\alpha}(A)b_{\alpha\beta}\chi_{\beta\beta}(A). \tag{3} \]

Applied to a finite automaton with transition matrix \(A\), the foregoing gives a method for analyzing the automaton.

Suppose we have a finite automaton \(z(t)=\varphi(z(t-1),x(t))\) with initial state \(z(0)=z_1\), and with output either in the form a) \(y(t)=g(z(t))\) (Moore automaton) or in the form b) \(y(t)=f(z(t-1),x(t))\) (Mealy automaton). Then the event \(\chi(z_1,y_j)\), represented in the automaton by the output signal \(y_j\), is found by the formulas: a) \(\chi(z_1,y_j)=\bigcup_k \chi_{1k}(A)\) for a Moore automaton, where \(k\) runs through those values for which \(g(z_k)=y_j\);* b) \(\chi(z_1,y_j)=\bigcup_{(k,i)}\chi_{1k}(A)x_i\) for a Mealy automaton, where \((k,i)\) runs through those pairs of values \(k\) and \(i\) for which \(f(z_k,x_i)=y_j\). However, for analyzing an automaton with output it is convenient to use the following computation scheme.

To the terminals \(z_1,z_2,\ldots,z_s\), corresponding to the states of the automaton, we add (output) terminals \(y_1,y_2,\ldots,y_m\), corresponding to the various output signals, and we add additional connections: we connect the terminal \(z_k\) with the terminal \(y_j\) \((k=1,\ldots,s;\ j=1,\ldots,m)\): a) by an edge with conductance \(e\), if \(g(z_k)=y_j\) (for a Moore automaton); b) by an edge with conductance \(x_i\), if \(f(z_k,x_i)=y_j\) (for a Mealy automaton).

As a result we obtain an \((s+m)\)-terminal network \(C\). The event \(\chi(z_1,y_j)\), represented in the automaton by the output signal \(y_j\), will be equal to the complete conductance of the multi-terminal network \(C\) from the terminal \(z_1\) to the output terminal \(y_j\), i.e. \(\chi(z_1,y_j)=\chi_{1,s+j}(C)\). In the actual computation, the rows of the matrix \(C\) corresponding to the output terminals need not be written down (these rows will remain zero throughout the entire computation), so that instead of the square matrix \(C\) one may operate with a rectangular matrix \(D\) of size \(s\times(s+m)\). After eliminating the terminals \(z_2,z_3,\ldots,z_s\), we arrive at the row matrix \([b_{z_1,z_1}, b_{z_1,y_1},\ldots,b_{z_1,y_m}]\), and this completes the analysis process, since

\[ \chi(z_1,y_j)=\langle b_{z_1,z_1}\rangle b_{z_1,y_j},\qquad j=1,2,\ldots,m. \tag{4} \]

Here, for greater clarity, as indices we have used the designations of the corresponding terminals.

* If \(g(z_1)=y_j\), then \(e\in\chi(z_1,y_j)\).

Example (see (3), p. 93). A Mealy automaton with two states 1, 2, two input signals \(x, y\), and two output signals \(z\) and \(v\) is given by the transition table (Table 1a) and the output table (Table 1b). Find regular expressions for the events \(\chi(1,z)\) and \(\chi(1,v)\), represented in the automaton by the output signals \(z\) and \(v\), with initial state 1.

Table 1

a)

b)

The transition matrix has the form

\[ A= \begin{bmatrix} y & x\\ y & x \end{bmatrix}. \]

Completing it with output columns, we shall have

\[ D= \begin{bmatrix} y & x & x\cup y & \Lambda\\ y & x & \Lambda & x\cup y \end{bmatrix}. \]

After eliminating, by formulas (1), state 2, to which the second row and the second column of the matrix \(D\) correspond, we arrive at the row matrix

\[ [y\cup x\langle x\rangle y,\quad x\cup y\cup x\langle x\rangle\Lambda,\quad \Lambda\cup x\langle x\rangle(x\cup y)], \]

whence, by formulas (4),

\[ \chi(1,z)=\langle y\cup x\langle x\rangle y\rangle(x\cup y),\quad \chi(1,v)=\langle y\cup x\langle x\rangle y\rangle x\langle x\rangle(x\cup y). \]

Leningrad Electrotechnical Institute
named after V. I. Ulyanov (Lenin)

Received
6 VII 1964

CITED LITERATURE

1. A. G. Lunts, DAN, 75, No. 2 (1950).
2. A. G. Lunts, Izv. AN SSSR, ser. matem., 16, No. 5, 409 (1952).
3. V. M. Glushkov, Synthesis of Digital Automata, Moscow, 1962.