Relational Algebra and Equational Proofs

We show that two concepts involving equational provability can be elegantly formalized in terms of a relational algebra, equipped with two special-purpose mappings. We derive some calculus in order to prove that these concepts are equivalent, and that they are sound and complete. We illustrate the use of the relational framework by a few examples. We show how decidability of provability of an equation from a finite set of variable-free equations, where all equations are variable-free. Then we discuss a method by Reeves ([3]) to deal with equations in semantic tableaux, that we can now prove to be complete in a very simple way. Finally we discuss an equational proof format that is naturally induced by the relational formulation, and serves as a guideline in finding proofs. The relation between Reeves' rules and the construction of such proofs is made explicit.