Abstract not found

this paper we provide a uniform approach to modelling them in categories of modest sets. To do this, we identify appropriate structure for doing "domain theory" in such "realizability models". In Sections 2 and 3 we introduce PCAs and define the associated "realizability" categories of assemblies and modest sets. Next, in Section 4, we prepare for our development of domain theory with an analysis of nontermination. Previous approaches have used (relatively complicated) categorical formulations of partial maps for this purpose. Instead, motivated by the idea that A provides a primitive programming language, we consider a simple notion of "diverging" computation within A itself. This leads to a theory of divergences from which a notion of (computable) partial function is derived together with a lift monad classifying partial functions. The next task is to isolate a subcategory of modest sets with sufficient structure for supporting analogues of the usual domain-theoretic constructions. First, we expect to be able to interpret the standard constructions of total type theory in this category, so it should inherit cartesian-closure, coproducts and the natural numbers from modest sets. Second, it should interact well with the notion of partiality, so it should be closed under application of the lift functor. Third, it should allow the recursive definition of partial functions. This is achieved by obtaining a fixpoint object in the category, as defined in (Crole and Pitts 1992). Finally, although there is in principle no definitive list of requirements on such a category, one would like it to support more complicated constructions such as those required to interpret polymorphic and recursive types. The central part of the paper (Sections 5, 6, 7 and 9) is devoted to establish...

. We relate certain models of Axiomatic Domain Theory (ADT) and Synthetic Domain Theory (SDT). On the one hand, we introduce a class of non-elementary models of SDT and show that the domains in them yield models of ADT. On the other hand, for each model of ADT in a wide class we construct a model of SDT such that the domains in it provide a model of ADT which conservatively extends the original model. Introduction The aim of Axiomatic Domain Theory (ADT) is to axiomatise the structure needed on a category so that its objects can be considered to be domains (see [11, x Axiomatic Domain Theory]). Models of axiomatic domain theory are given with respect to an enrichment base provided by a model of intuitionistic linear type theory [2, 3]. These enrichment structures consist of a monoidal adjunction C \Gamma! ? /\Gamma D between a cartesian closed category C and a symmetric monoidal closed category with finite products D, as well as with an !-inductive fixed-point object (Definition 1...

Introduction In [2] Martin Hyland has proposed the notion of "S-replete object" relative to a given object S as the appropriate generalisation of predomain for the purposes of Synthetic Domain Theory (SDT). The aim of SDT is to provide a constructive logical framework for reasoning about domains and thus about meanings of functional programs where all functions between predomains are continuous and all endofunctions on domains have least fixpoints. Furthermore, SDT allows one to consider predomains as a full reflective subcategory of the ambient category of constructive sets. For this reason the ambient category is assumed to be a model of some sufficiently strong impredicative constructive type theory which will be used as the internal language for the ambient category of constructive sets in order to replace complicated external arguments by simpler proofs in the

Introduction 1.1 Historical background Early investigators of realizability were interested in metamathematical questions. In keeping with the traditions of the time they concentrated on interpretations of one formal system in another. They considered an ad hoc collection of increasingly ingenious interpretations mainly to establish consistency, independence and conservativity results. van Oosten's contribution to the Workshop (see van Oosten [56] and the extended account van Oosten [57]) gave inter alia an account of these concerns from a modern perspective. (One should also draw attention to realizability used to provide interpretations of Brouwer's theory of Choice Sequences. An early approach is in Kleene Vesley [28]; for modern work in the area consult Moschovakis [35], [36], [37].) In the early days of categorical logic one considered realizability as providing models for constructive mathematics; while the metamathematics could be retrieved by `coding' the mod

We give various internal descriptions of the category !-Cpo of !-complete posets and !-continuous functions in the model H of Synthetic Domain Theory introduced in [8]. It follows that the !-cpos lie between the two extreme synthetic notions of domain given by repleteness and well-completeness. Introduction Synthetic Domain Theory aims at giving a few simple axioms to be added to an intuitionistic set theory in order to obtain domain-like sets. The idea at the core of this study was proposed by Dana Scott in the late 70's: domains should be certain "sets" in a mathematical universe where domain theory would be available. In particular, domains would come with intrinsic notions of approximation and passage to the limit with respect to which all functions will be continuous. Various suggestions for the notion of domain (typically within a set-theoretic universe given by an elementary topos with natural numbers object [17]) appeared in the literature, e.g. in [11, 26, 10, 23, 20, 16]. A...

The object of study of the present paper may be considered as a model, in an elementary topos with a natural numbers object, of a non-classical variation of the Peano arithmetic. The new feature consists in admitting, in addition to the constant (zero) s0 2 N and the unary operation (the successor map) s1 : N ! N, arbitrary operations su : N u ! N of arities u `between 0 and 1'. That is, u is allowed to range over subsets of a singleton set.

We consider the notion of replete object in the category of directed complete partial orders and Scott-continuous functions, and show that, contrary to previous expectations, there are non-replete objects. The same happens in the case of ω-complete posets. Synthetic Domain Theory developed from an idea of Dana Scott: it is consistent with intuitionistic set theory that all functions between domains are continuous. He never wrote about this point of view explicitly, though he presented his ideas in many lectures also suggesting that the model offered by Kleene’s realizability was appropriate, and influenced various thesis works, e.g. [10,13,11,8,12], see also [14]. SDT can now be recognized as defining the “good properties ” required on a category C (usually, a topos with a dominance t: 1 ✲ Σ) in order to develop domain theory within a theory of sets. One of the problems addressed early in the theory was the identification of the sets to be considered as the Scott domains. As one would expect in a synthetic approach, the collection of these should be determined by the “good properties ” of the universe, in an intrinsic way. The best suggestion so far for such a collection comes from [6,15,5] and is that of repleteness. It is an orthogonality condition, see [2], and determines the replete objects of C as those which are completely recoverable from their properties detected by Σ. Say that A is replete (wrt. Σ) if it is orthogonal to all f: X ✲ Y in 1

We provide an internal characterization of the category!-Cpo of!-complete posets and!-continuous functions within the model H of SDT recently introduced by the authors. It follows that!-cpos lie between the two extreme synthetic notions of domain given by repleteness and well-completeness.