On the Largest Cartesian Closed Category of Stable Domains

AbstractLet SABC (resp., SABC˜) be the category of algebraic bounded complete domains with conditionally multiplicative mappings, that is, Scott-continuous mappings preserving meets of pairs of compatible elements (resp., stable mappings). Zhang showed that the category of dI-domains is the largest cartesian closed subcategory of ω-SABC and ω-SABC˜, with the exponential being the stable function space, where ω-SABC and ω-SABC˜ are full subcategories of SABC and SABC˜ respectively which contain countablly based algebraic bounded complete domains as objects. This paper shows that:i)The exponentials of any full subcategory of SABC or SABC˜ are exactly function spaces;ii)SDABC˜ the category of distributive algebraic bounded complete domains, is the largest cartesian closed subcategory of SABC˜; The compact elements of function spaces in the category SABC are also studied

The Largest Cartesian Closed Category of Domains, Considered Constructively

A conjecture of Smyth is discussed which says that if D and [D → D] are effectively algebraic directed-complete partial orders with least element (cpo's), then D is an effectively strongly algebraic cpo, where it was not made precise what is meant by an effectively algebraic and an effectively strongly algebraic cpo. Notions of an effectively strongly algebraic cpo and an effective SFP domain are introduced and shown to be (effectively) equivalent. Moreover, the conjecture is shown to hold if instead of being effectively algebraic, [D → D] is only required to be ω-algebraic and D is forced to have a completeness test, that is a procedure which decides for any two finite sets X and Y of compact cpo elements whether X is a complete set of upper bounds of Y . As a consequence, the category of effective SFP objects and continuous maps turns out to be the largest Cartesian closed full subcategory of the category of ω-algebraic cpo's that have a completeness test. It is then studied whether such a result also holds in a constructive framework, where one considers categories with constructive domains as objects, that is, domains consisting only of the constructive (computable) elements of an indexed ω-algebraic cpo, and computable maps as morphisms. This is indeed the case: the category of constructive SFP domains is the largest constructively Cartesian closed weakly indexed effectively full subcategory of the category of constructive domains that have a completeness test and satisfy a further effectivity requirement

Towards a Convenient Category of Topological Domains

We propose a category of topological spaces that promises to be convenient for the purposes of domain theory as a mathematical theory for modelling computation. Our notion of convenience presupposes the usual properties of domain theory, e.g. modelling the basic type constructors, fixed points, recursive types, etc. In addition, we seek to model parametric polymorphism, and also to provide a flexible toolkit for modelling computational effects as free algebras for algebraic theories. Our convenient category is obtained as an application of recent work on the remarkable closure conditions of the category of quotients of countably-based topological spaces. Its convenience is a consequence of a connection with realizability models

A Convenient Category of Domains

We motivate and define a category of "topological domains", whose objects are certain topological spaces, generalising the usual $omega$-continuous dcppos of domain theory. Our category supports all the standard constructions of domain theory, including the solution of recursive domain equations. It also supports the construction of free algebras for (in)equational theories, provides a model of parametric polymorphism, and can be used as the basis for a theory of computability. This answers a question of Gordon Plotkin, who asked whether it was possible to construct a category of domains combining such properties

On Linear Information Systems

Scott's information systems provide a categorically equivalent, intensional description of Scott domains and continuous functions. Following a well established pattern in denotational semantics, we define a linear version of information systems, providing a model of intuitionistic linear logic (a new-Seely category), with a "set-theoretic" interpretation of exponentials that recovers Scott continuous functions via the co-Kleisli construction. From a domain theoretic point of view, linear information systems are equivalent to prime algebraic Scott domains, which in turn generalize prime algebraic lattices, already known to provide a model of classical linear logic

Exhaustible sets in higher-type computation

We say that a set is exhaustible if it admits algorithmic universal quantification for continuous predicates in finite time, and searchable if there is an algorithm that, given any continuous predicate, either selects an element for which the predicate holds or else tells there is no example. The Cantor space of infinite sequences of binary digits is known to be searchable. Searchable sets are exhaustible, and we show that the converse also holds for sets of hereditarily total elements in the hierarchy of continuous functionals; moreover, a selection functional can be constructed uniformly from a quantification functional. We prove that searchable sets are closed under intersections with decidable sets, and under the formation of computable images and of finite and countably infinite products. This is related to the fact, established here, that exhaustible sets are topologically compact. We obtain a complete description of exhaustible total sets by developing a computational version of a topological Arzela--Ascoli type characterization of compact subsets of function spaces. We also show that, in the non-empty case, they are precisely the computable images of the Cantor space. The emphasis of this paper is on the theory of exhaustible and searchable sets, but we also briefly sketch applications