Speciication and Proof in Membership Equational Logic

This paper is part of a long term eeort to increase expres-siveness of algebraic speciication languages while at the same time having a simple semantic basis on which eecient execution by rewriting and powerful theorem-proving tools can be based. In particular, our rewriting techniques provide semantic foundations for Maude's functional sublanguage, where they have been eeciently implemented. Membership equational logic is quite simple, and yet quite powerful. Its atomic formulae are equations and membership assertions to sorts, and its sentences are Horn clauses. It extends in a conservative way both order-sorted equational logic and partial algebra approaches, while Horn logic can be very easily encoded. After introducing the basic concepts of the logic, we give conditions and proof rules with which eecient equational deduction by rewriting can be achieved. We also give completion techniques to transform a speciication in one meeting these conditions. We address the important issue of proving suucient completeness of a speciication. Using tree-automata techniques, we develop a test set based approach for proving inductive theorems about a speciication. Narrowing and proof techniques for parameterized specii-cations are investigated as well. Finally, we discuss the generality of our approach and how it extends several previous approaches.