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Axiomatic Domain Theory
 in Categories of Partial Maps. Distinguished Dissertation Series
, 1995
"... The denotational semantics approach to the semantics of programming languages interprets the language constructions by assigning elements of mathematical structures to them. The structures form socalled categories of domains and the study of their closure properties is the subject of domain theory ..."
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The denotational semantics approach to the semantics of programming languages interprets the language constructions by assigning elements of mathematical structures to them. The structures form socalled categories of domains and the study of their closure properties is the subject of domain theory [Sco70, Sco82, Plo83, GS90, AJ94]. Typically, categories of domains consist of suitably complete partially ordered sets together with continuous maps. But, what is a category of domains? The main aim of axiomatic domain theory is to answer this question by axiomatising the structure needed on a mathematical universe so that it can be considered a category of domains. Criteria required from categories of domains can be of the most varied sort. For example, we could ask them to * have a rich collection of type constructors: sums, products, exponentials, powerdomains, dependent types, polymorphic types, etc; * have fixedpoint operators for programs and type constructors; * have only computable maps [Sco76, Smy77, Mul81, McC84, Ros86, Pho90, Lon95]; * have a Stone dual providing a logic of observable properties [Abr87, Vic89, Zha91]. An additional aim of the axiomatic approach is to relate these mathematical criteria with computational criteria. As we indicate below an axiomatic treatment of various of the above aspects is now available but much research remains to be done.
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 Domains and Denotational Semantics: History, Accomplishments and Open Problems, Bulletin of the EATCS
, 1996
"... In this collection we try to give anoverview of some selected topics in Domain Theory and Denotational Semantics. In doing so, we rst survey the mathematical universes which have been used as semantic domains. The emphasis is on those ordered structures which have beenintroduced by Dana Scott in 196 ..."
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In this collection we try to give anoverview of some selected topics in Domain Theory and Denotational Semantics. In doing so, we rst survey the mathematical universes which have been used as semantic domains. The emphasis is on those ordered structures which have beenintroduced by Dana Scott in 1969 and which gure under the name (Scott) domains. After surveying developments in the concrete theory of domains we describe two newer developments, the axiomatic and the synthetic approach. In the second part we look at three computational phenomena in detail, namely, sequential computation, polymorphism, and mutable state, and at the challenges that these pose for a mathematical model. This presentation does by no means exhaust the various approaches to denotational semantics and it certainly does not describe all possible mathematical techniques which havebeen used to describe various aspects of programs. We hope that, nevertheless, it illustrates how a particular challenge (namely the modelling of recursive de nitions) has given rise to an immensely rich theory, both in its general parts and in its applications.
Domains and Denotational Semantics: History, Accomplishments and Open Problems
, 1996
"... categorytheoretic accounts of these issues can be found in [Fio93, HJ95]. In type theory. In [CP92], Crole and Pitts introduced a higherorder typed predicate logic for fixedpoint computations. This was done by exploiting Moggi's treatment of computations using monads [Mog91], and by introducing ..."
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categorytheoretic accounts of these issues can be found in [Fio93, HJ95]. In type theory. In [CP92], Crole and Pitts introduced a higherorder typed predicate logic for fixedpoint computations. This was done by exploiting Moggi's treatment of computations using monads [Mog91], and by introducing the key notion of fixpoint object . Fixpoint objects were partly inspired by MartinLof's nonstandard "iteration type" [ML83], and give a categorical characterisation of general recursion at higher types similar to the characterisation of primitive recursion at higher types in terms of Lawvere's concept of natural number object [LS86]. A typetheoretic approach to domain theory is that of [Plo93]. There, rather than considering directly possible categorical structure, the idea is to work within a type theory pursuing the analogies: intuitionistic exponential = function space, and linear exponential = strict function space. More precisely, the basic setting is that of secondorder intuition...