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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.
RANGE CATEGORIES I: GENERAL THEORY
"... Abstract. Inthis twopartpaper, weundertakeasystematicstudyofabstractpartial map categories in which every map has both a restriction (domain of definition) and a range (image). In this first part, we explore connections with related structures such as inverse categories and allegories, and establis ..."
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Abstract. Inthis twopartpaper, weundertakeasystematicstudyofabstractpartial map categories in which every map has both a restriction (domain of definition) and a range (image). In this first part, we explore connections with related structures such as inverse categories and allegories, and establish two representational results. The first of these explains how every range category can be fully and faithfully embedded into a category of partial maps equipped with a suitable factorization system. The second is a generalization of a result from semigroup theory by Boris Schein, and says that every small range category satisfying the additional condition that every map is an epimorphism onto its range can be faithfully embedded into the category of sets and partial functions with the usual notion ofrange. Finally, we give an explicit construction
SUBMITTED TO THE FACULTY OF GRADUATE STUDIES
, 2012
"... Restriction categories provide a convenient abstract formulation of partial functions. However, restriction categories can have a variety of structures such as finite partial products (cartesianess), joins, meets, and ranges which are of interest in computability theory, semigroup theory, topology, ..."
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Restriction categories provide a convenient abstract formulation of partial functions. However, restriction categories can have a variety of structures such as finite partial products (cartesianess), joins, meets, and ranges which are of interest in computability theory, semigroup theory, topology, and algebraic geometry. This thesis studies these structures. For finite partial products (cartesianess), a construction to add finite partial products to an arbitrary restriction category freely is provided. For joins, we introduce the notion of join restriction categories, describe a construction for the join completion of a restriction category, and show the completeness of join restriction categories in partial map categories using Madhesive categories and Mgaps. As the join completion for inverse semigroups is wellknown in semigroup theory, we show the relationships between the join completion for restriction categories and the join completion for inverse semigroups by providing adjunctions among restriction categories, join restriction categories, inverse categories, and join inverse categories.
What is a differential partial combinatory algebra?
, 2011
"... In this thesis we combine Turing categories with Cartesian left additive restriction categories and again with differential restriction categories. The result of the first combination is a new structure which models nondeterministic computation. The result of the second combination is a structure wh ..."
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In this thesis we combine Turing categories with Cartesian left additive restriction categories and again with differential restriction categories. The result of the first combination is a new structure which models nondeterministic computation. The result of the second combination is a structure which models the notion of linear resource consumption. We also study the structural background required to understand what new features Turing structure should have in light of addition and differentiation – most crucial to this development is the way in which idempotents split. For the combination of Turing categories with Cartesian left additive restriction categories we will also provide a model.