Equational formulas and pattern operations in initial order-sorted algebras

Abstract

A pattern t, i.e., a term possibly with variables, denotes the set (language) ${\llbracket t \rrbracket}$ of all its ground instances. In an untyped setting, symbolic operations on finite sets of patterns can represent Boolean operations on languages. But for the more expressive patterns needed in declarative languages supporting rich type disciplines such as subtype polymorphism, untyped pattern operations and algorithms break down. We show how they can be properly defined by means of a signature transformation ${\Sigma \mapsto \Sigma^{\#}}$ that enriches the types of ${\Sigma}$. We also show that this transformation allows a systematic reduction of the first-order logic properties of an initial order-sorted algebra supporting subtype-polymorphic functions to equivalent properties of an initial many-sorted (i.e., simply typed) algebra. This yields a new, simple proof of the known decidability of the first-order theory of an initial order-sorted algebra.