@MISC{_departmentof, author = {}, title = {Department of Computer Science,}, year = {} }

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Abstract

In the theory of computation on topological algebras there is a considerable gap between so-called abstract and concrete models of computation. With an abstract model of computation on an algebra, the computations do not depend on any representation of the algebra. With a concrete model of computation, the computations depend on the choice of a representation of the algebra. First, we show that to compute functions on topological algebras using an abstract model, it is necessary to use 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, and extend it with a nondeterministic assignment of “countable choice”. This is called the WhileCC ∗ model. Using this, we introduce the notion 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, any function that is WhileCC ∗ approximable over A is computable in the effective representation of A; and conversely, (2) under certain reasonable conditions on A, any function that is computable in the effective representation of A is also WhileCC ∗ approximable. 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.