More on Geometric Morphisms between Realizability Toposes

Geometric morphisms between realizability toposes are studied in terms of morphisms between partial combinatory algebras (pcas). The morphisms inducing geometric morphisms (the {\em computationally dense\/} ones) are seen to be the ones whose `lifts' to a kind of completion have right adjoints. We characterize topos inclusions corresponding to a general form of relative computability. We characterize pcas whose realizability topos admits a geometric morphism to the effective topos.Comment: 20 page

Just below the surface: developing knowledge management systems using the paradigm of the noetic prism

In this paper we examine how the principles embodied in the paradigm of the noetic prism can illuminate the construction of knowledge management systems. We draw on the formalism of the prism to examine three successful tools: frames, spreadsheets and databases, and show how their power and also their shortcomings arise from their domain representation, and how any organisational system based on integration of these tools and conversion between them is inevitably lossy. We suggest how a late-binding, hybrid knowledge based management system (KBMS) could be designed that draws on the lessons learnt from these tools, by maintaining noetica at an atomic level and storing the combinatory processes necessary to create higher level structure as the need arises. We outline the “just-below-the-surface” systems design, and describe its implementation in an enterprise-wide knowledge-based system that has all of the conventional office automation features

Inductive Definition and Domain Theoretic Properties of Fully Abstract

A construction of fully abstract typed models for PCF and PCF^+ (i.e., PCF + "parallel conditional function"), respectively, is presented. It is based on general notions of sequential computational strategies and wittingly consistent non-deterministic strategies introduced by the author in the seventies. Although these notions of strategies are old, the definition of the fully abstract models is new, in that it is given level-by-level in the finite type hierarchy. To prove full abstraction and non-dcpo domain theoretic properties of these models, a theory of computational strategies is developed. This is also an alternative and, in a sense, an analogue to the later game strategy semantics approaches of Abramsky, Jagadeesan, and Malacaria; Hyland and Ong; and Nickau. In both cases of PCF and PCF^+ there are definable universal (surjective) functionals from numerical functions to any given type, respectively, which also makes each of these models unique up to isomorphism. Although such models are non-omega-complete and therefore not continuous in the traditional terminology, they are also proved to be sequentially complete (a weakened form of omega-completeness), "naturally" continuous (with respect to existing directed "pointwise", or "natural" lubs) and also "naturally" omega-algebraic and "naturally" bounded complete -- appropriate generalisation of the ordinary notions of domain theory to the case of non-dcpos.Comment: 50 page

Regular Functors and Relative Realizability Categories

Relative realizability toposes satisfy a universal property that involves regular functors to other categories. We use this universal property to define what relative realizability categories are, when based on other categories than of the topos of sets. This paper explains the property and gives a construction for relative realizability categories that works for arbitrary base Heyting categories. The universal property shows us some new geometric morphisms to relative realizability toposes too

The Unbearable Lightness of Being a Copula in Latin: on a Textual Problem in Cicero's Oration in Defence of Publius Sestius

A textual problem from Cicero's oration In defence of Publius Sestius is discussed in order to determine syntactic and semantic consequences of editorial choices. The discussion involves semantic and syntactic properties of Latin past participles within a minimalist setting. A detailed investigation of the passage indicates that a textual variant not accepted by most editors is to be preferred

On Probabilistic Applicative Bisimulation and Call-by-Value λ\lambda-Calculi (Long Version)

Probabilistic applicative bisimulation is a recently introduced coinductive methodology for program equivalence in a probabilistic, higher-order, setting. In this paper, the technique is applied to a typed, call-by-value, lambda-calculus. Surprisingly, the obtained relation coincides with context equivalence, contrary to what happens when call-by-name evaluation is considered. Even more surprisingly, full-abstraction only holds in a symmetric setting.Comment: 30 page