Multiple Congruence Relations, First-Order Theories on Terms, and the Frames of the Applied Pi-Calculus

International audienceWe investigate the problem of deciding first-order theories of finite trees with several distinguished congruence relations, each of them given by some equational axioms. We give an automata-based solution for the case where the different equational axiom systems are linear and variable-disjoint (this includes the case where all axioms are ground), and where the logic does not permit to express tree relations x=f(y,z). We show that the problem is undecidable when these restrictions are relaxed. As motivation and application, we show how to translate the model-checking problem of Apil, a spatial equational logic for the applied pi-calculus, to the validity of first-order formulas in term algebras with multiple congruence relations

Ground Reducibility is EXPTIME-complete

International audienceWe prove that ground reducibility is EXPTIME-complete in the general case. EXPTIME-hardness is proved by encoding the emptiness problem for the intersection of recognizable tree languages. It is more difficult to show that ground reducibility belongs to DEXPTIME. We associate first an automaton with disequality constraints A(R,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 A(R,t) is O(2^|R|x|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

Encompassment Properties and Automata with Constraints

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