An improvement on Valiant's decision procedure for equivalence of deterministic finite-turn pushdown automata

In this paper Valiant's decision procedure for equivalence of deterministic finite-turn pushdown machines is improved upon. The improved equivalence test is: Given two mahcines, one constructs a pushdown machine that simulates them simultaneously and accepts a string iff it is accepted by exactly one of them. The given machines are equivalent iff the simulating pda accepts the empty language. The simulating machine uses its pushdown store to hold the contents of the stores of the two simulated machines. In order ot prevent the tops of the two stores to get too far apart the simulating machine sometimes replaces the contents of one of the stores. Valiant 1,2 has proved the existence of a suitable function which determines the replacements. The crux of this paper is an algorithm for constructing such a function. The existence of such an algorithm enables us to calculate an upper bound on the time complexity of the improved algorithm. We obtain the results that equivalence of deterministic finite-turn pushdown automata can be tested in super-exponential time and equivalence of deterministic two-tape automata can be tested in exponential time.