A simple “mechanical” procedure is described for checking equality of regular expressions. The procedure, based on the work of A. Salomaa, uses derivatives of regular expressions and transition graphs.
Given a regular expression
R
, a corresponding transition graph is constructed. It is used to generate a finite set of left-linear equations which characterize
R
. Two regular events
R
and
S
are equal if and only if each constant term in the set of left-linear equations formed for the pair (
R S
) is (φ φ) or (^ ^).
The procedure does not involve any computations with or transformations of regular expressions and is especially appropriate for the use of a computer.
Introduction. The first part of this paper (Theorems 1-3) gives a short, unified treatment of (Mealy type [12]) automata (sequential machines). By associating with every input two binary relations ("next-state" and "output" relations) we obtain an easy and concise algebraic method for the description and study of complete or partial, finite or infinite automata.In the second part (Theorems 4-7) we develop further the algebraic decomposition theory of automata, continuing previous work by J. In [18] the concept of semi-automaton (see Section I) was introduced and methods for its decomposition by means of overlapping partitions were derived. In the present paper these investigations are extended to (Mealy type) automata and the problems of covering specified automata by direct and cascade products are studied.This approach leads to an interesting new algebraic concept, namely that of a weak (i.e., generalized) homomorphism denned by overlapping partitions. Recently this concept and its applicability to partial algebras has been further investigated [19] and generalizations of well-known results on homomorphisms and subdirect products of partial algebras have
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