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Relating firstorder set theories, toposes and categories of classes
 In preparation
, 2006
"... This paper introduces Basic Intuitionistic Set Theory BIST, and investigates it as a firstorder settheory extending the internal logic of elementary toposes. Given an elementary topos, together with the extra structure of a directed structural system of inclusions (dssi) on the topos, a forcingst ..."
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This paper introduces Basic Intuitionistic Set Theory BIST, and investigates it as a firstorder settheory extending the internal logic of elementary toposes. Given an elementary topos, together with the extra structure of a directed structural system of inclusions (dssi) on the topos, a forcingstyle interpretation of the language of firstorder set theory in the topos is given, which conservatively extends the internal logic of the topos. Since every topos is equivalent to one carrying a dssi, the language of firstorder has a forcing interpretation in every elementary topos. We prove that the set theory BIST+ Coll (where Coll is the strong Collection axiom) is sound and complete relative to forcing interpretations in toposes with natural numbers object (nno). Furthermore, in the case that the structural system of inclusions is superdirected, the full Separation schema is modelled. We show that every cocomplete topos and every realizability topos can be endowed (up to equivalence) with such a superdirected structural system of inclusions. This provides a uniform explanation for why such “realworld ” toposes model Separation. A large part of the paper is devoted to an alternative notion of categorytheoretic model for BIST, which, following the general approach of Joyal and Moerdijk’s Algebraic Set Theory, axiomatizes the structure possessed by categories of classes compatible with ∗Corresponding author. 1Previously, lecturer at HeriotWatt University (2000–2001), and the IT University of
A Categorytheoretic characterization of functional completeness
, 1990
"... . Functional languages are based on the notion of application: programs may be applied to data or programs. By application one may define algebraic functions; and a programming language is functionally complete when any algebraic function f(x 1 ,...,x n ) is representable (i.e. there is a constant a ..."
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. Functional languages are based on the notion of application: programs may be applied to data or programs. By application one may define algebraic functions; and a programming language is functionally complete when any algebraic function f(x 1 ,...,x n ) is representable (i.e. there is a constant a such that f(x 1 ,...,x n ) = (a . x 1 . ... . x n ). Combinatory Logic is the simplest typefree language which is functionally complete. In a sound categorytheoretic framework the constant a above may be considered as an "abstract gödelnumber" for f, when gödelnumberings are generalized to "principal morphisms", in suitable categories. By this, models of Combinatory Logic are categorically characterized and their relation is given to lambdacalculus models within Cartesian Closed Categories. Finally, the partial recursive functionals in any finite higher type are shown to yield models of Combinatory Logic. ________________ (+) Theoretical Computer Science, 70 (2), 1990, pp.193211. A p...
Partial categorical multicombinators and church rosser theorems
, 1992
"... Abstract: Categorical MultiCombinators form a rewriting system developed with the aim of providing efficient implementations of lazy functional languages. The core of the system of Categorical MultiCombinators consists of only four rewriting laws with a very low patternmatching complexity. This s ..."
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Abstract: Categorical MultiCombinators form a rewriting system developed with the aim of providing efficient implementations of lazy functional languages. The core of the system of Categorical MultiCombinators consists of only four rewriting laws with a very low patternmatching complexity. This system allows the equivalent of several βreductions to be performed at once, as functions form frames with all their arguments. Although this feature is convenient for most cases of function application it does not allow partially parameterised functions to fetch arguments. This paper presents Partial Categorical MultiCombinators, a new rewriting system, which removes this drawback. Key Words: functional programming, categorical combinators, explicit substitutions. Category: SD.F.4.1, SD D.3.2
Lambda Calculus
, 1997
"... Recursive functions are representable as lambda terms, and de nability in the calculus may be regarded as a de nition of computability. This forms part of the standard foundations of computer science. Lambda calculus is the commonly accepted basis of functional programming languages; and it is folkl ..."
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Recursive functions are representable as lambda terms, and de nability in the calculus may be regarded as a de nition of computability. This forms part of the standard foundations of computer science. Lambda calculus is the commonly accepted basis of functional programming languages; and it is folklore that the calculus is the prototypical functional language in puri ed form. The course investigates the syntax and semantics of lambda calculus both as a theory of functions from a foundational point of view, and as a minimal programming language.
The Categorical MultiCombinator Machine: CMCM
, 1992
"... this paper we introduce another abstract machine, Categorical MultiCombinator Machine, (CMCM). In this paper we give a thoroughgoing introduction to the machine, in particular as far as the discussion of sharing of computational information is concerned. The approaches of both TIM and the CMCM depen ..."
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this paper we introduce another abstract machine, Categorical MultiCombinator Machine, (CMCM). In this paper we give a thoroughgoing introduction to the machine, in particular as far as the discussion of sharing of computational information is concerned. The approaches of both TIM and the CMCM depend upon the source code being llifted before the translation takes place. This transformation, discovered by Johnsson [Joh], independently of related work by Hughes into supercombinators, has the effect of making flat the environments in which function bodies are interpreted. The transformation only came to light with the work on combinators, mentioned above, so perhaps reflecting the epigraph. In another paper, [LiTh], we discuss in detail the close relationship between the TIM and the CMCM. CMCM 10/7/92 2 CATEGORICAL MULTICOMBINATORS The second author in his thesis [Lins1] introduces the system of categorical multicombinators, which are based on Curien's categorical combinators [Cur], which in turn have their foundation in the theory of Cartesian Closed categories [Sco]. The major innovation of the multicombinators is that a number of breductions can be performed in a single step of rewriting, rather than in a sequence of such steps. This offers the possibility of increasing the efficiency of a rewriting implementation considerably; a similar idea has been discussed by the implementors of the Gmachine [Joh2]. The syntax of categorical multicombinators consists of variables, n,... the constants P,
REFLEXIVITY, EIGENFORM AND FOUNDATIONS OF PHYSICS
"... Abstract: This essay is a discussion of the concept of reflexivity and its relationships with selfreference, reentry, eigenform and the foundations of physics. Reflexive is a term that refers to the presence of a relationship between an entity and itself. One can be aware of one's own thought ..."
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Abstract: This essay is a discussion of the concept of reflexivity and its relationships with selfreference, reentry, eigenform and the foundations of physics. Reflexive is a term that refers to the presence of a relationship between an entity and itself. One can be aware of one's own thoughts. An organism produces itself through its own action and its own productions. A market or a system of finance is composed of actions and individuals, and the actions of those individuals influence the market