Ground Reducibility is EXPTIME-Complete

We prove that ground reducibility is EXPTIME-complete in the general case. EXPTIME-hardness is proved by encoding the computations of an alternating Turing machine whose space is polynomially bounded. 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 a term u if and only if t is not ground reducible by R. The number of states of AR;t is O(2kRk ktk) and the size of the constraints are 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 give an upper bound to the number of distinct subtrees of a minimal successful computation of an automaton with disequality constraints. It follows that emptiness of such an automaton can be decided in time polynomial in the number of its states and exponential in the size of its constraints. Altogether, we get a simply exponential deterministic algorithm for ground reducibility.