Abstract versus concrete computation on metric partial algebras

• Published in: ACM transactions on computational logic 2004-10, Vol.5 (4), p.611-668
• Main Authors: Tucker, J V, Zucker, J I
• Format: Article
• Language: English
• Subjects: Mathematical models; Studies
• ISSN: 1529-3785; 1557-945X
• DOI: 10.1145/1024922.1024924

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 partial operations, and computable functions that are both continuous and many-valued. This many-valuedness is needed even to compute single-valued functions, and so abstract models must be nondeterministic even to compute deterministic problems. As an abstract model, we choose the "while"-array programming language, extended with a nondeterministic "countable choice" assignment, called the WhileCC* model. Using this, we introduce the concept of approximable many-valued computation on metric algebras. For our concrete model, we choose metric algebras with effective representations. We prove:(1) for any metric algebra A with an effective representation α, WhileCC* approximability implies computability in α, and (2) also the converse, under certain reasonable conditions on A. 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.