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Variations on Realizability: Realizing the Propositional Axiom of Choice
 Math. Structures Comput. Sci
, 2000
"... Introduction 1.1 Historical background Early investigators of realizability were interested in metamathematical questions. In keeping with the traditions of the time they concentrated on interpretations of one formal system in another. They considered an ad hoc collection of increasingly ingenious ..."
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Introduction 1.1 Historical background Early investigators of realizability were interested in metamathematical questions. In keeping with the traditions of the time they concentrated on interpretations of one formal system in another. They considered an ad hoc collection of increasingly ingenious interpretations mainly to establish consistency, independence and conservativity results. van Oosten's contribution to the Workshop (see van Oosten [56] and the extended account van Oosten [57]) gave inter alia an account of these concerns from a modern perspective. (One should also draw attention to realizability used to provide interpretations of Brouwer's theory of Choice Sequences. An early approach is in Kleene Vesley [28]; for modern work in the area consult Moschovakis [35], [36], [37].) In the early days of categorical logic one considered realizability as providing models for constructive mathematics; while the metamathematics could be retrieved by `coding' the mod
Parametricity as Isomorphism
 Theoretical Computer Science
, 1993
"... . We investigate a simple form of parametricity, based on adding "abstract" copies of preexisting types. Connections are made with the ReynoldsMa theory of parametricity by logical relations, with the theory of parametricity via dinaturality, and with the categorical notion of equivalence. Introdu ..."
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. We investigate a simple form of parametricity, based on adding "abstract" copies of preexisting types. Connections are made with the ReynoldsMa theory of parametricity by logical relations, with the theory of parametricity via dinaturality, and with the categorical notion of equivalence. Introduction In his fundamental paper on the notion of parametricity in connection with type theories [Rey83], John Reynolds links the notion of parametricity firmly to the notion of data abstraction. This, unlike Strachey's earlier characterization via algorithm reuse, is a needdriven analysis. We need things to be parametric because otherwise our data abstractions will no longer be abstract. In his subsequent paper with Ma [MR91], two further points are made. One is that the problems reside more at the level of parametrized types than at the level of the quantified polymorphic types, and the other is that the notion of parametricity is not absolute, but relative. The MaReynolds work produces ...
Categorical Properties of Logical Frameworks
, 1993
"... In this paper we give a new presentation of ELF which is wellsuited for semantic analysis. We introduce the notions of internal codability, internal definability, internal typed calculi and frame languages. These notions are central to our perspective of logical frameworks. We will argue that a ..."
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In this paper we give a new presentation of ELF which is wellsuited for semantic analysis. We introduce the notions of internal codability, internal definability, internal typed calculi and frame languages. These notions are central to our perspective of logical frameworks. We will argue that a logical framework is a typed calculus which formalizes the relationship between internal typed languages and frame languages. In the second half of the paper, we demonstrate the advantage of our logical framework by showing some categorical properties of it and of encodings in it. By doing so we hope to indicate a sensible model theory of encodings. Copyright c fl1993. All rights reserved. Reproduction of all or part of this work is permitted for educational or research purposes on condition that (1) this copyright notice is included, (2) proper attribution to the author or authors is made and (3) no commercial gain is involved. Technical Reports issued by the Department of Computer Sc...
Two Probabilistic Powerdomains in Topological Domain Theory
"... We present two probabilistic powerdomain constructions in topological domain theory. The first is given by a free ”convex space” construction, fitting into the theory of modelling computational effects via free algebras for equational theories, as proposed by Plotkin and Power. The second is given b ..."
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We present two probabilistic powerdomain constructions in topological domain theory. The first is given by a free ”convex space” construction, fitting into the theory of modelling computational effects via free algebras for equational theories, as proposed by Plotkin and Power. The second is given by an observationally induced approach, following Schröder and Simpson. We show the two constructions coincide when restricted to ωcontinuous dcppos, in which case they yield the space of (continuous) probability valuations equipped with the Scott topology. Thus either construction generalises the classical domaintheoretic probabilistic powerdomain. On more general spaces, the constructions differ, and the second seems preferable. Indeed, for countablybased spaces, we characterise the observationally induced powerdomain as the space of probability valuations with weak topology. However, we show that such a characterisation does not extend to non countablybased spaces.
Relational Limits in General Polymorphism
, 1993
"... Parametric models of polymorphic lambda calculus have the structure of enriched categories with cotensors and ends in some generalized sense, and thus have many categorical data types induced by them. The !order minimum model is a parametric model. 1 Introduction Higher order quantifier of polymor ..."
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Parametric models of polymorphic lambda calculus have the structure of enriched categories with cotensors and ends in some generalized sense, and thus have many categorical data types induced by them. The !order minimum model is a parametric model. 1 Introduction Higher order quantifier of polymorphic lambda calculus has several meanings. Two inventors of the calculus use different symbols. When Girard wrote V X:F (X) [10] (\PiX:F (X) in [12]), it corresponded to a higher order quantified formula 8X:F (X) via CurryHoward isomorphism. When Reynolds wrote \DeltaX:F (X) [33], it was the type of polymorphism, especially of parametric polymorphism [34]. The third interpretation leaded by categorical semantics is that the quantified type, we write 8X:F (X), is a kind of limits. The notation \PiX:F (X) suggests that it might be regarded as a product of all F (X) where X ranges over all types. That is to say, \PiX:F (X) is the collection of all sections (a section is a function sending a...
On the Failure of FixedPoint Theorems for Chaincomplete Lattices in the Effective Topos
, 2009
"... In the effective topos there exists a chaincomplete distributive lattice with a monotone and progressive endomap which does not have a fixed point. Consequently, the BourbakiWitt theorem and Tarski’s fixedpoint theorem for chaincomplete lattices do not have constructive (toposvalid) proofs. 1 ..."
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In the effective topos there exists a chaincomplete distributive lattice with a monotone and progressive endomap which does not have a fixed point. Consequently, the BourbakiWitt theorem and Tarski’s fixedpoint theorem for chaincomplete lattices do not have constructive (toposvalid) proofs. 1
Partial Combinatory Algebras and Realizability Toposes
, 2004
"... These are the lecture notes for a tutorial at FMCS 2004 in Kananaskis. The aim is to give a first introduction to Partial Combinatory Algebras and the construction of Realizability Toposes. The first part, where Partial Combinatory Algebras are discussed, requires no specific background (except for ..."
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These are the lecture notes for a tutorial at FMCS 2004 in Kananaskis. The aim is to give a first introduction to Partial Combinatory Algebras and the construction of Realizability Toposes. The first part, where Partial Combinatory Algebras are discussed, requires no specific background (except for some of the examples perhaps), although familiarity with combinatory logic and lambda calculus will not hurt. The second part on realizability toposes presupposes some knowledge of category theory; more specifically, we will assume that the reader knows what a topos is. Apart from that the material is selfcontained. 1 Partial Combinatory Algebras We give the basic definitions and properties of Partial Combinatory Algebras in the first subsection. Next, we discuss some of the important examples. Finally, we touch upon the theory of Partial Combinatory Algebras. 1.1 Partial Applicative Structures and Combinatory Completeness We first introduce the basic concept of a Partial Applicative Structure, which may be viewed as a universe for computation. Then look at terms over an applicative structure, we formulate
A Notion of Homotopy for the Effective Topos
, 2010
"... We define a notion of homotopy in the effective topos. AMS Subject Classification (2000): 18B25 (Topos Theory),55U35 (Abstract and axiomatic homotopy theory) ..."
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We define a notion of homotopy in the effective topos. AMS Subject Classification (2000): 18B25 (Topos Theory),55U35 (Abstract and axiomatic homotopy theory)
Abstract
, 2010
"... maps is introduced to provide an instance of an algebraically compact category relative to a restricted class of functors. Algebraic compactness is a synthetic condition on a category which ensures solutions of recursive equations involving endofunctors of the category. We extend that result to incl ..."
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maps is introduced to provide an instance of an algebraically compact category relative to a restricted class of functors. Algebraic compactness is a synthetic condition on a category which ensures solutions of recursive equations involving endofunctors of the category. We extend that result to include all internal functors on C when C is viewed as a full internal category of the effective topos. This is done using two general results: one about internal functors in general, and one about internal functors in the effective topos. The paper “Extensional PERs ” by P.Freyd, P.Mulry, G.Rosolini and D.Scott [2]