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**1 - 7**of**7**### Theory of real computation according to EGC

- In Proceedings of Dagstuhl seminar on Reliable Implementation on Real Numer Algorithms: Theory and Practice
, 2006

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### Computable total functions on metric algebras, universal algebraic specifications and dynamical systems

- THE JOURNAL OF LOGIC AND ALGEBRAIC PROGRAMMING
, 2005

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### Unifying computers and dynamical systems using the theory of synchronous concurrent algorithms

- Applied Mathematics and Computation

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### Author's personal copy Unifying computers and dynamical systems using the theory of synchronous concurrent algorithms

"... a b s t r a c t A synchronous concurrent algorithm (SCA) is a parallel deterministic algorithm based on a network of modules and channels, computing and communicating data in parallel, and synchronised by a global clock with discrete time. Many types of algorithms, computer architectures, and mathe ..."

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a b s t r a c t A synchronous concurrent algorithm (SCA) is a parallel deterministic algorithm based on a network of modules and channels, computing and communicating data in parallel, and synchronised by a global clock with discrete time. Many types of algorithms, computer architectures, and mathematical models of physical and biological systems are examples of SCAs. For example, conventional digital hardware is made from components that are SCAs and many computational models possess the essential features of SCAs, including systolic arrays, neural networks, cellular automata and coupled map lattices. In this paper we formalise the general concept of an SCA equipped with a global clock in order to analyse precisely (i) specifications of their spatio-temporal behaviour; and (ii) the senses in which the algorithms are correct. We start the mathematical study of SCA computation, specification and correctness using methods based on computation on manysorted topological algebras and equational logic. We show that specifications can be given equationally and, hence, that the correctness of SCAs can be reduced to the validity of equations in certain computable algebras. Since the idea of an SCA is general, our methods and results apply to each of the particular classes of algorithms and dynamical systems above.

### IOS Press First and Second Order Recursion on Abstract Data Types

"... Abstract. This paper compares two scheme-based models of computation on abstract many-sorted algebras A: Feferman’s system ACP(A) of “abstract computational procedures ” based on a least fixed point operator, and Tucker and Zucker’s system µPR(A) based on primitive recursion on the naturals together ..."

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Abstract. This paper compares two scheme-based models of computation on abstract many-sorted algebras A: Feferman’s system ACP(A) of “abstract computational procedures ” based on a least fixed point operator, and Tucker and Zucker’s system µPR(A) based on primitive recursion on the naturals together with a least number operator. We prove a conjecture of Feferman that (assuming A contains sorts for natural numbers and arrays of data) the two systems are equivalent. The main step in the proof is showing the equivalence of both systems to a system Rec(A) of computation by an imperative programming language with recursive calls. The result provides a confirmation for a Generalized Church-Turing Thesis for computation on abstract data types.

### Department of Computer Science,

"... 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 computati ..."

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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.

### Models of Computation on Abstract Data Types based on Recursive Schemes By

"... This thesis compares two scheme-based models of computation on abstract manysorted algebras A: Feferman’s system ACP(A) of “abstract computational procedures” based on a least fixed point operator, and Tucker and Zucker’s system µPR(A) based on primitive recursion on the naturals together with a lea ..."

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This thesis compares two scheme-based models of computation on abstract manysorted algebras A: Feferman’s system ACP(A) of “abstract computational procedures” based on a least fixed point operator, and Tucker and Zucker’s system µPR(A) based on primitive recursion on the naturals together with a least number operator. We prove a conjecture of Feferman that (assuming A contains sorts for natural numbers and arrays of data) the two systems are equivalent. The main step in the proof is showing the equivalence of both systems to a system Rec(A) of computation by an imperative programming language with recursive calls. The result provides a confirmation for a Generalized Church-Turing Thesis for computation on abstract data types. iii Acknowledgements I would first like to express my sincere thanks and appreciation to my supervisor, Dr. J. I. Zucker, for his insight, thoughtful guidance and constant encouragement