By Julius T. Tou (Eds.)

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**Example text**

The transition diagram of a three-state automaton is illustrated in Fig. 1. ■ FIG. & w 1. The transition matrix of an n-state automaton is an n X n matrix whose ijth entry is the label of the branch ba from the ith state Xi to the j t h state Xj if the branch bij exists, and is equal to zero if the branch does not exist. The transition matrix for the three-state automaton shown in Fig. 1 is ~n/c2 r2/ci 0 0 r2/c2 n/ci n/c2 0 \_r2/ci A finite automaton can also be described by the state matrix PoO/e) = [>ii(r*)], i, j = 1, 2, · · ·, n; k = 1, 2, · · ·, a, (6) where Pij(rk) is conditional probability for the automaton to go from state Xi to state Xj when the input is rk.

The fact that we know there is a nice formula for every regular set is itself an incentive to ask about a set of words we may be interested in, whether it is regular or not, or whether it can be described by referring in some way to regular sets, etc. In spite of all this speculation about possible applications, I would like to suggest that our proper attitude toward the study of regular expressions (and toward the theory of automata, in general) should be that it is part of pure mathematics.

In any operation, replacing an element by an equivalent element results in equivalent elements. This extends at once as follows. If R is a congruence relation and (ai, 6i), (a 2 , 62), · · · , (α Λλ , ί>ηλ) 6 Ä, then [ Λ ( « ι , «2, · · · , αηχ)9 Λ(6ι,62,···,6»λ)]€Α. For example, in the case of switching functions, if Nf and ΛΓ^ are the disjunctive normal forms of f and g, then f = Nf and g = Ng. Here the congruence relation is just the "equality" of switching algebra. Also, / v g = NfvNg and/<7 = NfNg.