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...

The semantics of Evaluation Logic proposed in [14] relies on additional properties of monads. This paper proposes an alternative semantics, which drops all additional requirement on monads, at the expense of stronger assumptions on the underlying category. These assumptions are satised by any topos, but not by the category of cpos. However, in the setting of Synthetic Domain Theory (see [7, 23]) it is possible to reconcile the needs of Denotational Semantics with those of Logic. 1 Introduction Evaluation logic (EL T ) is a typed predicate logic originally proposed by [18], which is based on the metalanguage ML T for computational monads (see [12]) and permits statements about the evaluation of programs to values by the use of evaluation modalities: necessity and possibility. In particular, EL T might be used for axiomatizing computation-related properties of a monad or devising computationally adequate theories (see [18]), and it appears useful when addressing the question of logic...

This paper is concerned with models of SDT encompassing traditional categories of domains used in denotational semantics [7,18], showing that the synthetic approach generalises the standard theory of domains and suggests new problems to it. Consider a (locally small) category of domains D with a (small) dense generator G equipped with a Grothendieck topology. Assume further that every cover in G is effective epimorphic in D. Then, by Yoneda, D embeds fully and faithfully in the topos of sheaves on G for the canonical topology, which thus provides a set-theoretic universe for our original category of domains. In this paper we explore such a situation for two traditional categories of domains and, in particular, show that the Grothendieck toposes so arising yield models of SDT. In a subsequent paper we will investigate intrinsic characterizations, within our models, of these categories of domains. First, we present a model of SDT embedding the category !-Cpo of posets with least upper bounds of countable chains (hence called !-complete) and

this paper: (internal version) if C 1 is a quasi-topos, then S

. We place simple axioms on an elementary topos which suffice for it to provide a denotational model of call-by-value PCF with sum and product types. The model is synthetic in the sense that types are interpreted by their set-theoretic counterparts within the topos. The main result characterises when the model is computationally adequate with respect to the operational semantics of the programming language. We prove that computational adequacy holds if and only if the topos is 1-consistent (i.e. its internal logic validates only true \Sigma 0 1 -sentences). 1 Introduction One axiomatic approach to domain theory is based on axiomatizing properties of the category of predomains (in which objects need not have a "least" element). Typically, such a category is assumed to be bicartesian closed (although it is not really necessary to require all exponentials) with natural numbers object, allowing the denotations of simple datatypes to be determined by universal properties. It is well known...

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.

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

The contravariant powerset, and its generalisations ΣX to the lattices of open subsets of a locally compact topological space and of recursively enumerable subsets of numbers, satisfy the Euclidean principle that φ ∧ F (φ) =φ ∧ F (⊤). Conversely, when the adjunction Σ (−) ⊣ Σ (−) is monadic, this equation implies that Σ classifies some class of monos, and the Frobenius law ∃x.(φ(x) ∧ ψ) =(∃x.φ(x)) ∧ ψ) for the existential quantifier. In topology, the lattice duals of these equations also hold, and are related to the Phoa principle in synthetic domain theory. The natural definitions of discrete and Hausdorff spaces correspond to equality and inequality, whilst the quantifiers considered as adjoints characterise open (or, as we call them, overt) and compact spaces. Our treatment of overt discrete spaces and open maps is precisely dual to that of compact Hausdorff spaces and proper maps. The category of overt discrete spaces forms a pretopos and the paper concludes with a converse of Paré’s theorem (that the contravariant powerset functor is monadic) that characterises elementary toposes by means of the monadic and Euclidean properties together with all quantifiers, making no reference to subsets.