A Procedure for Checking Equality of Regular Expressions

Abstract

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.

Word problems requiring exponential time(Preliminary Report)

“…Our results already imply that previous efforts to find efficient procedures for testing equivalence of regular expressions or minimizing nondeterministic finite automata (cf. [14], [15], [16]) were foredoomed• Recent studies by ourselves and coworkers of decision procedures for logical theories show that our methods are applicable to nearly all of the classical decidability results in logic,and that moreover with the exception of the propositional calculus and some theories resembling the first order theory of equality, all these decidable theories can be proved to require exponential or greater time• Although certain of the word problems considered in this paper are somewhat arbitrarily constructed, we have studied them in the hope that the methods of proof will extend to algebra, topology and other areas where decision procedures arise, and will curtail wasted effort in searching for efficient procedures when none exist.…”

Some remarks on the KH algorithm fors-grammars

“…As every regular set with the addition of an endmarker can be generated by some right linear s-grammar, this algorithm can be used to prove or disprove the equivalence of two regular expressions, using a technique similar to that in [2]. As every regular set with the addition of an endmarker can be generated by some right linear s-grammar, this algorithm can be used to prove or disprove the equivalence of two regular expressions, using a technique similar to that in [2].…”

The Complexity of Regular(-Like) Expressions

“…We discuss the computational complexity of the decision problems membership (MEMBER), inequivalence (INEQ), non-empty complement (NEC), for regularlike expressions over the alphabet Σ and set of operations ϕ. Algorithms that solve the equivalence problem have been given, for example, in [11,23], where the former also handles regular expressions extended by the operations intersection and complementation.…”