Abstract versus concrete computation on metric partial algebras

Abstract

In the theory of computation on topological algebras there is a considerable gap between so-called abstract and concrete models of computation. In concrete models, unlike abstract models, the computations depend on the representation of the algebra. First, we show that with abstract models, one needs algebras with <i>partial operations</i>, and computable functions that are both <i>continuous</i> and <i>many-valued</i>. This many-valuedness is needed even to compute single-valued functions, and so <i>abstract models must be nondeterministic even to compute deterministic problems</i>. As an abstract model, we choose the "while"-array programming language, extended with a nondeterministic "countable choice" assignment, called the <i><b>WhileCC*</b></i> model. Using this, we introduce the concept of <i>approximable many-valued computation</i> on metric algebras. For our concrete model, we choose metric algebras with <i>effective representations</i>. We prove:(1) for any metric algebra <i>A</i> with an effective representation α, <i><b>WhileCC*</b></i> approximability implies computability in α, and (2) also the converse, under certain reasonable conditions on <i>A</i>. From (1) and (2) we derive an equivalence theorem between abstract and concrete computation on metric partial algebras. We give examples of algebras where this equivalence holds.