Decision Procedures for Families of Deterministic Pushdown Automata
Decision Procedures for Families of Deterministic Pushdown Automata
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A deterministic pushdown automaton (dpda) is described as finite-turn if there is a bound on the number of times the direction of the stack movement can change in the set of all derivations from the starting configuration. The purpose of this paper is to show that there exists a procedure for deciding whether two such finite-turn machines recognize the same language. By virtue of a direct correspondence between a restricted class of one-turn dpda and deterministic two-tape acceptors (Valiant (1973)), our proof also provides a solution to the equivalence problem for the latter, alternative to that of Bird (1973). Since some of the ideas we introduce are not related exclusively to the finite-turn property, or even pushdown machines, it is hoped that our methods can be adapted for constructing equivalence tests for other classes of deterministic automata. Our main technique can be regarded as a generalization in several directions of one introduced by Rosenkrantz and Stearns (1970). They consider a class of pushdown automata for which a natural valuation can be placed on each stack segment, and deduce that for any input word, two equivalent machines in that class must have closely related stack movements. They show how, under such circumstances, for any two machines a single pushdown automaton can be constructed to simulate them both, and used to decide their equivalence. What we shall show is that even for a class with no such stack valuation known, and in which two equivalent machines can have totally dissimilar stack movements, pushdown automata, now nondeterministic, can be found to perform the required simulations.