Completion of rewrite systems with membership constraints

Abstract

We consider a constrained equational logic where the constraints are membership conditions t ∈ s where s is interpreted as a regular tree language. Our logic includes a fragment of second order equational logic (without λ-expressions) where second order variables range over regular sets of contexts. The problem with constrained equational logics is the failure of the critical pair lemma. This is the reason why we propose new deduction rules for which the critical pair lemma is restored. Computing critical pairs requires however to solve some constraints in a second-order logic with membership constraints. This is the most difficult result of the paper: we give a terminating set of transformation rules for these formulas, which decides the existence of a solution.

Since an order-sorted signature is nothing but a bottom-up tree automaton, order-sorted equational logic falls into the scope of our study; our results show how to perform ordersorted completion without regularity and without sort decreasingness. It also shows how to perform unification in the order-sorted case, with some higher-order variables (without any regularity assumption).