A complete calculus for the multialgebraic and functional semantics of nondeterminism

Abstract

The current algebraic models for nondeterminism focus on the notion of possibility rather than necessity and consequently equate (nondeterministic) terms that one would intuitively 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 article 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.

Algebraic approaches to nondeterminism—an overview

“…It is easy to abstract such a structure of computations and look only at the result sets produced by a nondeterministic operation [Walicki and Meldal 1995c].…”

“…The domains of indexed sets used in Bekic [1984] instead of power domains remind one of indexed functions. A more elaborate investigation of the functional models is reported in Walicki and Meldal [1995c] where the relationships to the next kind of models are studied as well. We denote functional models by FMod.…”

“…This introduces highly nonstandard features into logics of nondeterminism. The only, to our knowledge, calculus for direct reasoning about nondeterminism that has been shown complete wrt the multialgebraic semantics is given in Walicki and Meldal [1995c] and Walicki [1993]. The calculus NEQ can be used for reasoning about disjunctive clauses where each clause C is a set of atomic formulae (i.e., the ordering and multiplicity of the atomic formulae do not matter), written as e 1 , .…”

Convergent approximate solving of first-order constraints by approximate quantifiers

“…Languages for modeling nondeterminism in various other forms have been introduced within the frame of abstract data types-see for example [43,42,21,26]. There one uses nondeterministic specifications to either model nondeterminism occurring in reality, or to abstract away unnecessary details of the behavior of a real or desired system; these details might be specified later.…”