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Title: Concrete and abstract models of computation over metric algebras
Author: Stewart, K. J.
Awarding Body: University of Wales Swansea
Current Institution: Swansea University
Date of Award: 1999
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In this thesis we study the computability theory of partial functions defined over metric algebras. We integrate aspects of two broad approaches to generalising the theory of computability on the natural numbers to uncountable algebraic structures - abstract models that use generalised models of computation to compute over structures which are considered independently of any representation or implementation, and concrete models for which connections with computability on the naturals and physical realisability are the main focus. The abstract models with which we work are those of the imperative style 'while' programming language and generalisations of Kleene's μ-recursive function schemes from the naturals to abstract algebras. We study computation by such models over effective metric partial algebras. The effectivity of such algebras is based on the choice of a computable dense metric sub-space. The notion of a computable function over these types of spaces is well established. We review some simple connections with other models. We prove the closure of these functions under abstract 'while' language computation and approximation. We prove analogous results for recursive function schemes. We consider some of these ideas in several applications connected with the real numbers, matrix algebra, deterministic parallel models of computation and Hilbert and Banach spaces. We analyse the connection between sets with computable projection functions and computable distance functions in Hilbert space.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available