The convex powerdomain in a category of posets realized by cpos

. We construct a powerdomain in a category whose objects are posets of data equipped with a cpo of "intensional" representations of the data, and whose morphisms are those monotonic functions between posets that are "realized" by continuous functions between the associated cpos. The category of cpos is contained as a full subcategory that is preserved by lifting, sums, products and function spaces. The construction of the powerdomain uses a cpo of binary trees, these being intensional representations of nondeterministic computation. The powerdomain is characterized as the free semilattice in the category. In contrast to the other type constructors, the powerdomain does not preserve the subcategory of cpos. Indeed we show that the powerdomain has interesting computational properties that differ from those of the usual convex powerdomain on cpos. We end by considering the solution of recursive domain equations. The surprise here is that the limit-colimit coincidence fails. Nevertheless, ..

A representable approach to finite nondeterminism

AbstractWe reformulate denotational semantics for nondeterminism, taking a nondeterministic operation V on programs, and sequential composition, as primitive. This gives rise to binary trees. We analyse semantics for both type and program constructors such as products and exponential types, conditionals and recursion, in this setting. In doing so, we define new category-theoretic structures, in particular premonoidal categories. We also account for equivalences of programs such as those induced by associativity, symmetry and idempotence of V, and we study finite approximation by enrichment over the category of ω-cpos with least element. We also show how to recover the classical powerdomains, especially the convex powerdomain, as three instances of a general, computationally natural, construction

Topological Domain Theory

This thesis presents Topological Domain Theory as a powerful and flexible framework for denotational semantics. Topological Domain Theory models a wide range of type constructions and can interpret many computational features. Furthermore, it has close connections to established frameworks for denotational semantics, as well as to well-studied mathematical theories, such as topology and computable analysis.We begin by describing the categories of Topological Domain Theory, and their categorical structure. In particular, we recover the basic constructions of domain theory, such as products, function spaces, fixed points and recursive types, in the context of Topological Domain Theory.As a central contribution, we give a detailed account of how computational effects can be modelled in Topological Domain Theory. Following recent work of Plotkin and Power, who proposed to construct effect monads via free algebra functors, this is done by showing that free algebras for a large class of parametrised equational theories exist in Topological Domain Theory. These parametrised equational theories are expressive enough to generate most of the standard examples of effect monads. Moreover, the free algebras in Topological Domain Theory are obtained by an explicit inductive construction, using only basic topological and set-theoretical principles.We also give a comparison of Topological and Classical Domain Theory. The category of omega-continuous dcpos embeds into Topological Domain Theory, and we prove that this embedding preserves the basic domain-theoretic constructions in most cases. We show that the classical powerdomain constructions on omega-continuous dcpos, including the probabilistic powerdomain, can be recovered in Topological Domain Theory.Finally, we give a synthetic account of Topological Domain Theory. We show that Topological Domain Theory is a specific model of Synthetic Domain Theory in the realizability topos over Scott's graph model. We give internal characterisations of the categories of Topological Domain Theory in this realizability topos, and prove the corresponding categories to be internally complete and weakly small. This enables us to show that Topological Domain Theory can model the polymorphic lambda-calculus, and to obtain a richer collection of free algebras than those constructed earlier.In summary, this thesis shows that Topological Domain Theory supports a wide range of semantic constructions, including the standard domain-theoretic constructions, computational effects and polymorphism, all within a single setting

Semantic Domains and Denotational Semantics

The theory of domains was established in order to have appropriate spaces on which to define semantic functions for the denotational approach to programming-language semantics. There were two needs: first, there had to be spaces of several different types available to mirror both the type distinctions in the languages and also to allow for different kinds of semantical constructs - especially in dealing with languages with side effects; and second, the theory had to account for computability properties of functions - if the theory was going to be realistic. The first need is complicated by the fact that types can be both compound (or made up from other types) and recursive (or self-referential), and that a high-level language of types and a suitable semantics of types is required to explain what is going on. The second need is complicated by these complications of the semantical definitions and the fact that it has to be checked that the level of abstraction reached still allows a precise definition of computability

Events in computation

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