Ground reducibility is EXPTIME-complete

We prove that ground reducibility is EXPTIME-complete in the general case. EXPTIME-hardness is proved by encoding the emptiness problem for the intersection of recognisable tree languages. It is more difficult to show that ground reducibility belongs to DEXPTIME. We associate first an automaton with disequality constraints AR,t to a rewrite system R and a term t. This automaton is deterministic and accepts at least one term iff t is not ground reducible by R. The number of states of AR,t is O(2||R||×||t||) and the size of its constraints is polynomial in the size of R, t. Then we prove some new pumping lemmas, using a total ordering on the computations of the automaton. Thanks to these lemmas, we can show that emptiness for an automaton with disequality constraints can be decided in a time which is polynomial in the number of states and exponential in the size of the constraints. Altogether, we get a simply exponential time deterministic algorithm for ground reducibility decision.

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