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24
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
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...
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,
Lambda Calculus
"... 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 folk ..."
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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. Synopsis Formal theory, xed point theorems, combinatory logic: combinatory completeness, translations between lambda calculus and combinatory logic � reduction: ChurchRosser theorem � Bohm's theorem and applications � basic recursion theory � lambda calculi considered as programming languages � simple type theory and pcf: correspondence between operational and denotational semantics � current developments. Relationship with other courses Basic knowledge of logic and computability in paper B1 is assumed.
Petri Net Semantics of Bunched Implications
"... Engberg and Winskel's Petri net semantics of linear logic is reconsidered, from the point of view of the logic BI of bunched implications. We first show how BI can be used to overcome a number of difficulties pointed out by Engberg and Winskel, and we argue that it provides a more natural logic for ..."
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Engberg and Winskel's Petri net semantics of linear logic is reconsidered, from the point of view of the logic BI of bunched implications. We first show how BI can be used to overcome a number of difficulties pointed out by Engberg and Winskel, and we argue that it provides a more natural logic for the net semantics. We then briefly consider a more expressive logic based on an extension of BI with classical and modal features.
Topological Representation of the &ambda;Calculus
, 1998
"... The calculus can be represented topologically by assigning certain spaces to the types and certain continuous maps to the terms. Using a recent result from category theory, the usual calculus of conversion is shown to be deductively complete with respect to such topological semantics. It is al ..."
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The calculus can be represented topologically by assigning certain spaces to the types and certain continuous maps to the terms. Using a recent result from category theory, the usual calculus of conversion is shown to be deductively complete with respect to such topological semantics. It is also shown to be functionally complete, in the sense that there is always a "minimal" topological model, in which every continuous function is definable. These results subsume earlier ones using cartesian closed categories, as well as those employing socalled Henkin and Kripke models. Introduction The calculus originates with Church [6]; it is intended as a formal calculus of functional application and specification. In this paper, we are mainly interested in the version known as simply typed calculus ; as is now wellknown, the untyped version can be treated as a special case of this ([17]). We present here a topological representation of the calculus: types are represented by cert...
The Bulletin of Symbolic Logic
, 1999
"... In program synthesis, we transform a specification into a system that is guaranteed to satisfy the specification. When the system is open, then at each moment it reads input signals and writes output signals, which depend on the input signals and the history of the computation so far. The specifi ..."
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In program synthesis, we transform a specification into a system that is guaranteed to satisfy the specification. When the system is open, then at each moment it reads input signals and writes output signals, which depend on the input signals and the history of the computation so far. The specification considers all possible input sequences. Thus, if the specification is linear, it should hold in every computation generated by the interaction, and if the specification is branching, it should hold in the tree that embodies all possible input sequences.
Part II Local Realizability Toposes and a Modal Logic for
"... 5.1 Definition and Examples 5.1.1 Definition and Definability Results A tripos is a weak tripos with disjunction which has a (weak) generic object. Explicitly we define: ..."
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5.1 Definition and Examples 5.1.1 Definition and Definability Results A tripos is a weak tripos with disjunction which has a (weak) generic object. Explicitly we define:
Universal Domains For Sequential Computation
, 1995
"... Classical recursion theory asserts that all conventional programming languages are equally expressive because they can define all partial recursive functions over the natural numbers. However, most real programming languages support some form of higherorder data such as potentially infinite streams ..."
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Classical recursion theory asserts that all conventional programming languages are equally expressive because they can define all partial recursive functions over the natural numbers. However, most real programming languages support some form of higherorder data such as potentially infinite streams, lazy trees, and functions. Since these objects do not have finite canonical representations, computations over these objects cannot be accurately modeled as ordinary computations over the natural numbers. In my thesis, I develop a theory of higher order computability based on a new formulation of domain theory. This new formulation interprets elements of any data domain as lazy trees. Like classical domain theory, it provides a universal domain T and a universal language KL. A rich class of domains called observably sequential domains can be specified in T with functions definable in KL. Such an embedding of a data domain enables the operations on the domain to be defined in the universa...