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Semantic Domains
, 1990
"... this report started working on denotational semantics in collaboration with Christopher Strachey. In order to fix some mathematical precision, he took over some definitions of recursion theorists such as Kleene, Nerode, Davis, and Platek and gave an approach to a simple type theory of highertype fu ..."
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Cited by 167 (8 self)
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this report started working on denotational semantics in collaboration with Christopher Strachey. In order to fix some mathematical precision, he took over some definitions of recursion theorists such as Kleene, Nerode, Davis, and Platek and gave an approach to a simple type theory of highertype functionals. It was only after giving an abstract characterization of the spaces obtained (through the construction of bases) that he realized that recursive definitions of types could be accommodated as welland that the recursive definitions could incorporate function spaces as well. Though it was not the original intention to find semantics of the socalled untyped calculus, such a semantics emerged along with many ways of interpreting a very large variety of languages. A large number of people have made essential contributions to the subsequent developments, and they have shown in particular that domain theory is not one monolithic theory, but that there are several different kinds of constructions giving classes of domains appropriate for different mixtures of constructs. The story is, in fact, far from finished even today. In this report we will only be able to touch on a few of the possibilities, but we give pointers to the literature. Also, we have attempted to explain the foundations in an elementary wayavoiding heavy prerequisites (such as category theory) but still maintaining some level of abstractionwith the hope that such an introduction will aid the reader in going further into the theory. The chapter is divided into seven sections. In the second section we introduce a simple class of ordered structures and discuss the idea of fixed points of continuous functions as meanings for recursive programs. In the third section we discuss computable functions and...
Combinatory Models and Symbolic Computation
 Lecture Notes in Computer Science , Springer Verlag 721
, 1992
"... Weintroduce an algebraic model of computation which is especially useful for the description of computations in analysis. On one level the model allows the representation of algebraic computation and on an other level approximate computation is represented. ..."
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Cited by 1 (0 self)
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Weintroduce an algebraic model of computation which is especially useful for the description of computations in analysis. On one level the model allows the representation of algebraic computation and on an other level approximate computation is represented.
Combinatory Differential Fields: An Algebraic Approach to Approximate Computation and Constructive Analysis
, 1991
"... The algebraic structure of combinatory differential fields is constructed to provide a semantics for computations in analysis. In this setting programs, approximations, limits and operations of analysis are represented as algebraic terms. Analytic algorithms can be derived by algebraic methods. The ..."
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The algebraic structure of combinatory differential fields is constructed to provide a semantics for computations in analysis. In this setting programs, approximations, limits and operations of analysis are represented as algebraic terms. Analytic algorithms can be derived by algebraic methods. The main tool in this construction are combinatory models which are inner algebras of Engeler graph models. As an universal domain of denotational semantics the lattice structure of the graph models allows to give a striking simple semantics for computations with approximations. As models of combinatory algebra they provide all essential computational constructs, including recursion. Combinatory models are constructed as extensions of first order theories. The classical first order theory to describe analysis is the theory of differential fields. It turns out that two types of computational constructs, namely composition and piecewise definition of functions, are preferably introduced as extensi...