Powerdomains and nondeterministic recursive definitions

Abstract

A nondeterministic Language for recursive definitions, L, is defined. It includes simple functions for manipulating data values, in this case the natural numbers, a choice operator OR and recursive procedures. Various kinds of parameter-passing mechanisms are allowed: the well-known call-by-value, run-time choice which models the Algo160 substitution rule and call-time choice in which no nondeterminism is allowed in the actual parameters once the procedure has been called.

An operational semantics is defined for this language in the form of a next state relation. Using this relation three different operational preorders are defined between programs. The difference in these orders reflect different views of divergent computations.

On the other hand we define three different mathematical models, in the sense of Scott-Theory and show that they are fully-abstract; that is programs are identified in the model if and only if they are identified by the corresponding operational preorder. The three different models are defined using three different powerdomains and these are shown to arise naturally by considering different properties of the semantic counter-part of the operator OR.

Finally we discuss the suitability of L for defining elements of the various domains. An element of a domain is computable if it is the least upper bound of a recursively enumerable set of finite elements. We show that for certain domains all computable elements are definable in L and point to difficulties in extending L so as to define all such elements in the remaining domains.