Programming with Equations, Subsets, and Relations

We discuss the declarative and computational issues in combining equa-tional, subset, and relational assertions in a logic programming language. The novel feature in this work is the subset assertion, whose interactions with equational and relational assertions are discussed in this paper. The semantics of subset assertions incorporate a collect all capability, which is expressed formally by the completion of the program. When used in conjunction with equational assertions, subset assertions serve to deene set-valued functions, and the resulting paradigm is called subset-equational programming. We also present the class of stratiied subset-equational programs for formalizing the class of closure functions, which are useful in deening various transitive-closure sets. The declarative and operational semantics of simple and stratiied subset-equational programs are the main focus of this paper. The operational semantics of closures is based on memo-tables (or extension tables). When subset assertions are used in conjunction with rela-tional assertions, the paradigm is called subset-relational programming. We present both simple and stratiied subset-relational programming, and show how they provide a declarative way to deene Prolog's setof construct.