this paper we adopt the most popular choice, the internal logic of an elementary topos (with nno), also chosen, e.g., in [23, 8, 26]. The principal benefits are that models of the logic (toposes) are ubiquitous, and the methods for constructing and analysing them are very well-established. For the purposes of the axiomatic part of this paper, we believe that it would also be

This paper provides a unifying axiomatic account of the interpretation of recursive types that incorporates both domain-theoretic and realizability models as concrete instances. Our approach is to view such models as full subcategories of categorical models of intuitionistic set theory. It is shown that the existence of solutions to recursive domain equations depends upon the strength of the set theory. We observe that the internal set theory of an elementary topos is not strong enough to guarantee their existence. In contrast, as our first main result, we establish that solutions to recursive domain equations do exist when the category of sets is a model of full intuitionistic Zermelo-Fraenkel set theory. We then apply this result to obtain a denotational interpretation of FPC, a recursively typed lambda-calculus with callby-value operational semantics. By exploiting the intuitionistic logic of the ambient model of intuitionistic set theory, we analyse the relationship between operational and denotational semantics. We first prove an “internal ” computational adequacy theorem: the model always believes that the operational and denotational notions of termination agree. This allows us to identify, as our second main result, a necessary and sufficient condition for genuine “external ” computational adequacy to hold, i.e. for the operational and denotational notions of termination to coincide in the real world. The condition is formulated as a simple property of the internal logic, related to the logical notion of 1-consistency. We provide useful sufficient conditions for establishing that the logical property holds in practice. Finally, we outline how the methods of the paper may be applied to concrete models of FPC. In doing so, we obtain computational adequacy results for an extensive range of realizability and domain-theoretic models.

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We show that an enriched version of Freyd's principle of versality holds in the Kleisli category of a commutative strong monad with fixed-point object. This gives a general categorical setting in which it is possible to model recursive types involving the usual datatype constructors.

This paper studies the notions of approximation and passage to the limit in an axiomatic setting. Our axiomatisation is subject to the following criteria: the axioms should be natural (so that they are available in as many contexts as possible) and non-order-theoretic (so that Research supported by SERC grant RR30735 and EC project Programming Language Semantics and Program Logics grant SC1000 795 they explain the order-theoretic structure). Our aim is 1. to provide a justification of Scott's original consideration of ordered structures, and 2. to deepen our understanding of the notion of passage to the limit

We present a category-theoretic framework for providing intensional semantics of programming languages and establishing connections between semantics given at different levels of intensional detail. We use a comonad to model an abstract notion of computation, and we obtain an intensional category from an extensional category by the co-Kleisli construction; thus, while an extensional morphism can be viewed as a function from values to values, an intensional morphism is akin to a function from computations to values. We state a simple category-theoretic result about cartesian closure. We then explore the particular example obtained by taking the extensional category to be Cont, the category of Scott domains with continuous functions as morphisms, with a computation represented as a non-decreasing sequence of values. We refer to morphisms in the resulting intensional category as algorithms. We show that the category Alg of Scott domains with algorithms as morphisms is cartesian closed. We...

We prove two embedding and extension theorems in the context of the constructive theory of metric spaces. The first states that Cantor space embeds in any inhabited complete separable metric space (CSM) without isolated points, X, in such a way that every sequentially continuous function from Cantor space to Z extends to a sequentially continuous function from X to R. The second asserts an analogous property for Baire space relative to any inhabited locally non-compact CSM. Both results rely on having careful constructive formulations of the concepts involved. As a first application, we derive new relationships between “continuity principles ” asserting that all functions between specified metric spaces are pointwise continuous. In particular, we give conditions that imply the failure of the continuity principle “all functions from X to R are continuous”, when X is an inhabited CSM without isolated points, and when X is an inhabited locally non-compact CSM. One situation in which the latter case applies is in models based on “domain realizability”, in which the failure of the continuity principle for any inhabited locally non-compact CSM, X, generalizes a result previously obtained by Escardó and Streicher in the special case X = C[0, 1]. As a second application, we show that, when the notion of inhabited complete separable metric space without isolated points is interpreted in a recursion-theoretic setting, then, for any such space X, there exists a Banach-Mazur computable function from X to the computable real numbers that is not Markov computable. This generalizes a result obtained by Hertling in the special case that X is the space of computable real numbers.

this paper, is the analysis of notions of approximation aiming at explaining and justifying (order-theoretic) properties of categories of domains. For example, in [Fio94c, Fio94a], while studying the interaction between partiality and order-enrichment we considered contextual approximation which, in the framework we were working in, coincided with the specialisation preorder . But in the applications carried out in [FP94, Fio94a] we had to work with an axiomatised notion of approximation, instead of the aforementioned one, for the following two reasons: first, the specialisation preorder is not appropriate in categories of domains and stable functions (see [Fio94c]) and, second, we do not know of non-order-theoretic axioms making the specialisation preorder !-complete. To overcome these drawbacks another notion of approximation was to be considered. And, it was the second problem that motivated the intensional notion of approximation provided by the path relation. In fact, it is shown in [Fio94b] that under suitable axioms the path relation can be equipped with a canonical passage-to-the-limit operator appropriate for fixed-point computations; stronger axioms make this operator be given by lubs of !-chains

Introduction In (Plotkin 1985) a revitalised approach to domain theory was initiated. Roughly, the idea was to eliminate the bottom from the domains and to keep the functions partially defined. Thus replacing Cppo (the category of small cppos ---posets with a least element and closed under lubs of !-chains--- and continuous functions) with pCpo (the category of small cpos ---posets closed under lubs of !-chains--- and partial continuous functions --- see Subsection 3.1). One important point in the reformulation is the recognition of pCpo Research partially supported by Fundaci'on Antorchas and The British Council grant ARG 2281/14/6, and SERC grant RR30735. as a category of partial maps as, for example, such presentation fits better with standard formulations of recursion theory and it allows a categorical description of data types (via partial cartesian closed categories (Longo and Moggi 1984) with finite coproducts) in the presence of fixed-point operators. Following the main moti

We introduce the notion of an equational lifting monad: a commutative strong monad satisfying one additional equation (valid for monads arising from partial map classifiers). We prove that any equational lifting monad has a representation by a partial map classifier such that the Kleisli category of the former fully embeds in the partial category of the latter. Thus equational lifting monads precisely capture the (partial) equational properties of partial map classifiers. The representation theorem also provides a tool for transferring non-equational properties of partial map classifiers to equational lifting monads. It is proved using a direct axiomatization of the Kleisli categories of equational lifting monads as abstract Kleisli categories with extra structure. This axiomatization is of interest in its own right. 1