Complete Calculus for the Multialgebraic and Functional Semantics of Nondeterminism

Keywords: Nondeterminism; Multialgebras; Equational logic

Abstract: The current algebraic models for nondeterminism focus on the notion of possibility rather than necessity, and consequently equate (nondeterministic) terms that one intuitively would not consider equal. Furthermore, existing models for nondeterminism depart radically from the standard models for (equational) specifications of deterministic operators. One would prefer that a specification language for nondeterministic operators be based on an extension of the standard model concepts, preferably in such a way that the reasoning system for (possibly nondeterministic) operators becomes the standard equational one whenever restricted to the deterministic operators - the objective should be to minimize the departure from the standard frameworks.

In this paper we define a specification language for nondeterministic operators and multialgebraic semantics. The first complete reasoning system for such specifications is introduced. We also define a transformation of specifications of nondeterministic operators into derived specifications of deterministic ones, obtaining a *computational* semantics of nondeterministic specification by adopting the standard semantics of the derived specification as the semantics of the original one. This semantics turns out to be a refinement of multialgebra semantics. The calculus is shown to be sound and complete also with respect to the new semantics.