Proving Semantical Equivalence of Data Specifications

More than two decades ago, Peter Freyd introduced essentially algebraic specifications, a well-behaved generalization of algebraic specifications, allowing for equational partiality. These essentially algebraic specifications turn out to have a number of very interesting applications in computer science. In this paper, we present a deduction system for essentially algebraic specifications that is very suitable as the underlying deduction system of an automated theorem prover. Using the well-known fact that theories of sketches can be constructed as initial algebras of essentially algebraic specifications, we describe a semi-automatic procedure for proving the equivalence of the theories of two sketches. Next, we demonstrate that sketches are a very suitable formalism for making semantic data specifications, as used in database design and software engineering. Two such data specifications are semantically equivalent iff their model categories in FinSet are equivalent. Equivalence of theories is a sufficient condition for semantical equivalence, and hence the procedure to prove the equivalence of the theories of sketches can be used as a powerful tool to prove semantical equivalence of data specifications. Proving semantical equivalence of data specifications is an important component of the view integration process, i.e. the process of integrating a number of partly overlapping data specifications into one large data specification. The structure of this paper is as follows. In section 1, we consider essentially algebraic specifications. We develop a restricted equational deduction system for these specifications, and prove its soundness and completeness. We show how finitely complete and finitely cocomplete categories can be specified in an essentially algebraic way. In section 2, we define finitary sketches, and we show how their theories can be constructed as initial algebras of an essentially algebraic specification. The essentially algebraic specifications of finitely complete and finitely cocomplete categories, and the construction of the theory of a sketch as an initial algebra of an essentially algebraic specification are of course well-known in the category theory community. Nevertheless, we thought it worthwhile to give a detailed account of these constructions in this paper, because many computer scientists are not familiar with them. In section 3, we develop a semi-automatic procedure for proving the equivalence of the theories of two sketches, using the deduction system for essentially algebraic specifications from section 1. In section 4, we show that sketches are a very suitable formalism for making semantic data specifications. Semantic data specifications are specifications of a mathematical model of that part of the real world that is reflected in a database or application program. One of the most challenging problems that has been studied in the literature on data specifications is the view integration problem, i.e. the ∗Postdoctoral Fellow of the Belgian National Fund for Scientific Research (N.F.W.O.)