By Julius T. Tou (Eds.)

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C , Representation of events in nerve nets and finite automata. "Automata Studies" (Annals of Mathematical Studies, Vol. 34) (C. E. Shannon and J. ), Princeton, New Jersey, pp. 3-42. Princeton Univ. Press, 1956. 3. , Regular expressions and state graphs for automata. IRE Trans. EC-9, 39-47 (1960); also in "Sequential Machines-Selected Papers" (E. F. ), pp. 157-174. Addison-Wesley, Reading, Massachusetts, 1964. 4. , Axiom systems for regular expressions of finite automata. Turun Yliopiston Julkaisuda Ann.

An example of such a graph is given in Fig. 4. Such graphs are graphs in which there is no hierarchy of loops as AN INTRODUCTION TO REGULAR EXPRESSIONS er (e) -Oc 49 000001 001 001 000001 (f) 0 ""^^^ooooörSo poi T (g) 000001 -o (0 U 0*000001(001)* U (001)* F I G . 7e-g. there is in regular expression. A loop in a regular expression is always given by a star; and for every pair of stars, either one is inside the scope of the other or their scopes are completely disjoint. A loop in a graph is any directed path that ends where it begins, and otherwise does not repeat any node.

DEFINITIONS To begin with, regular expressions are expressions standing for regular events (or regular languages), which are certain sets of words. A word is a 35 36 ROBERT MCNAUGHTON string of symbols over some alphabet, denoted by Σ. Although Σ is in general any finite set, very often the alphabet whose symbols are 0 and 1 will be used for examples. Regular expressions are made up of letters of the alphabet, and signs standing for certain operators. The three operators are union, concatenation, and star (or closure).