A Total Approach to Partial Algebraic Specification

Partiality is a fact of life, but at present explicitly partial algebraic specifications lack tools and have limited proof methods. We propose a sound and complete way to support execution and formal reasoning of explicitly partial algebraic specifications within the total framework of membership equational logic (MEL) which has a highperformance interpreter (Maude) and proving tools. This is accomplished by a sound and complete mapping PMEL ? MEL of partial membership equational (PMEL) theories into total ones. Furthermore, we characterize and give proof methods for a practical class of theories for which this mapping has "almost-zero representational distance," in that the partial theory and its total translation are identical up to minor syntactic sugar conventions. This then supports very direct execution of, and formal reasoning about, partial theories at the total level. In conjunction with tools like Maude and its proving tools, our methods can be used to execute and reason about partial specifications such as those in CASL.