A mathematical approach to nondeterminism in data types

Abstract

The theory of abstract data types is generalized to the case of nondeterministic operations (set-valued functions). Since the nondeterminism of operations may be coupled, signatures are extended so that operations can have results in Cartesian products. Input/output behavior is used to characterize implementation of one model by another. It is described by means of accumulated arrows, which form a generalization of the term algebra. Morphisms of nondeterministic models are introduced. Both innovations prove to be powerful tools in the analysis of input/output behavior. Extraction equivalence and observable equivalence of values are investigated. Quotient models for such equivalence relations are constructed. The equivalence relations are compared with each other, with separation of values by means of experiments, and with the separation property that characterizes a terminal model. Examples are given to show that the four concepts are different. In deterministic models the concepts coincide.

Nondeterminism in Algebraic Specifications and Algebraic Programs

“…Within the last years there have been attempts to integrate nondeterminism as a basic concept into algebraic specifications ( [Nipkow86], [Hesselink88]). …”

“…Both papers [Nipkow86] and [Hesselink88] treat nondeterministic algebras and basic relations between algebras, but they exclude the question of a well-suited specification language. A nondeterministic specification language is presented in [Kaplan 88], but this approach is based on the classical notion of a model and the classical calculus,…”

“…Homomorphisms for multi-algebras have been defined already in [Pickert 50] and later in [Pickett67], [Hansoul83], [Nipkow86] and [Hesselink88]. These papers always consider only elementvalued homomorphisms instead of the set-valued definition from above.…”

“…Below, only the multi-algebra approach will be followed, which forms the basis for the work of [Nipkow86] and [Hesselink88], too.…”

Generated models and the ω-rule: The nondeterministic case

“…In recent years several authors have tried to provide algebraic specifications of nondeterminism with the semantics based on the notion of rnultialgebras [22,15,17,24,21 ]. It is also the approach we are taking here.…”