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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 ..."
Abstract

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...
Logical and Computational Aspects of Programming with Sets/Bags/Lists
 In LNCS 510: Proceedings of 18th International Colloquium on Automata, Languages, and Programming
, 1991
"... . We study issues that arise in programming with primitive recursion over nonfree datatypes such as lists, bags and sets. Programs written in this style can lack a meaning in the sense that their outputs may be sensitive to the choice of input expression. We are, thus, naturally lead to a settheor ..."
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Cited by 79 (2 self)
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. We study issues that arise in programming with primitive recursion over nonfree datatypes such as lists, bags and sets. Programs written in this style can lack a meaning in the sense that their outputs may be sensitive to the choice of input expression. We are, thus, naturally lead to a settheoretic denotational semantics with partial functions. We set up a logic for reasoning about the definedness of terms and a deterministic and terminating evaluator. The logic is shown to be sound in the model, and its recursion free fragment is shown to be complete for proving definedness of recursion free programs. The logic is then shown to be as strong as the evaluator, and this implies that the evaluator is compatible with the provable equivalence between different set (or bag, or list) expressions . Oftentimes, the same nonfree datatype may have different presentations, and it is not clear a priori whether programming and reasoning with the two presentations are equivalent. We formulate t...