Functional programming with sets

Set abstraction, originally introduced in fuctional languages by Turner, is an appealing construct because it leads to concise definitions of many interesting operations. However, existing approaches treat sets as lists for the sake of efficiency, and thereby sacrifice a simple declarative semantics. In this paper, we present a novel language based on sets and equations, where sets are treated as sets, consistent with their semantics. The language is called SEL, for Set-Equation Language. Equations are assumed to define a confluent rewriting system when oriented left to right. Sets are defined in terms of their subsets; these rules define a nonconfluent rewriting system when oriented left to right. We show examples of programs in this language, and provide an operational semantics for such programs. Programs are executed by innermost reduction, which may be nondeterministic or deterministic. Nondeterministic reduction is used when one of the elements of a set is desired. Deterministic reduction is used to simplify a term via an equation or to obtain all the elements of a set. The correctness of the operational semantics is also established.