Simultaneous checking of completeness and ground confluence

Algebraic specifications provide a powerful method for the specification of abstract data types in programming languages and software systems. Completeness and ground confluence are fundamental notions for building algebraic specifications in a correct and modular way. In this paper, we present a procedure for simultaneously checking completeness and ground confluence for specifications with free/non-free constructors and parameterized specifications. If the specification is not complete or not ground-confluent, then our procedure outputs the set of patterns on whose ground instances a function is not defined and it can easily identify the rules that break ground confluence. Our procedure is complete and always terminates under the assumption of an oracle for deciding (joinable) inductive properties. In contrast to previous work, our method does not rely on completion techniques and does not require the computation of critical pairs of axioms. The method has been implemented in the prover SPIKE. This system has allowed us to prove the completeness and the ground confluence of many specifications in a completely automatic way, where related techniques diverge or generate very complex proofs.