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Oosten. Ordered partial combinatory algebras
 Mathematical Proceedings of the Cambridge Philosophical Society
, 1992
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Aspects of predicative algebraic set theory II: Realizability. Accepted for publication in Theoretical Computer Science
 In Logic Colloquim 2006, Lecture Notes in Logic
, 2009
"... This is the third in a series of papers on algebraic set theory, the aim of which is to develop a categorical semantics for constructive set theories, including predicative ones, based on the notion of a “predicative category with small maps”. 1 In the first paper in this series [8] we discussed how ..."
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This is the third in a series of papers on algebraic set theory, the aim of which is to develop a categorical semantics for constructive set theories, including predicative ones, based on the notion of a “predicative category with small maps”. 1 In the first paper in this series [8] we discussed how these predicative categories
Computability structures, simulations and realizability
, 2011
"... We generalize the standard construction of realizability models (specifically, of categories of assemblies) to a very wide class of computability structures, broad enough to embrace models of computation such as labelled transition systems and process algebras. We also discuss a general notion of si ..."
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We generalize the standard construction of realizability models (specifically, of categories of assemblies) to a very wide class of computability structures, broad enough to embrace models of computation such as labelled transition systems and process algebras. We also discuss a general notion of simulation between such computability structures, and show that such simulations correspond precisely to certain functors between the realizability models. Furthermore, we show that our class of computability structures has good closure properties — in particular, it is ‘cartesian closed ’ in a slightly relaxed sense. We also investigate some important subclasses of computability structures and of simulations between them. We suggest that our 2category of computability structures and simulations may offer a framework for a general investigation of questions of computational power, abstraction and simulability for a wide range of computation models from across computer science.
Partial Combinatory Algebras of Functions
, 2009
"... We employ the notions of ‘sequential function ’ and ‘interrogation ’ (dialogue) in order to define new partial combinatory algebra structures on sets of functions. These structures are analyzed using J. Longley’s preorderenriched category of partial combinatory algebras and decidable applicative st ..."
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We employ the notions of ‘sequential function ’ and ‘interrogation ’ (dialogue) in order to define new partial combinatory algebra structures on sets of functions. These structures are analyzed using J. Longley’s preorderenriched category of partial combinatory algebras and decidable applicative structures. We also investigate total combinatory algebras of partial functions. One of the results is, that every realizability topos is a quotient of a realizability topos on a total combinatory algebra. AMS Subject Classification (2000): 03B40,68N18
Regular Functors and Relative Realizability Categories
, 2012
"... The relative realizability toposes that Awodey, Birkedal and Scott introduced in [1] 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 ..."
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The relative realizability toposes that Awodey, Birkedal and Scott introduced in [1] 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. 1
Realizability Toposes and Ordered PCA's
, 2001
"... Introduction Partial combinatory algebras (pca's, for short), are wellknown to form the basic ingredient for the construction of various realizability toposes, of which the Effective Topos is undoubtedly the most famous. There is more than one way to present the realizability topos associated ..."
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Introduction Partial combinatory algebras (pca's, for short), are wellknown to form the basic ingredient for the construction of various realizability toposes, of which the Effective Topos is undoubtedly the most famous. There is more than one way to present the realizability topos associated to a pca; one may take the exact completion of the category of partitioned assemblies (see [7]), or one can use tripos theory. Triposes built from pca's are, together with those from locales, the most important and most extensively studied instances of triposes, but from a structural point of view, there are important differences between the two; whereas locales are organized in a wellbehaved category, which is a reflective subcategory of the category of toposes, it is not immediately clear what an appropriate category for pca's may look like. Moreover, there are various nice properties in the localic case, such as the fact that there is a onetoone correspondence between maps of local