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127
Notions of Computation and Monads
, 1991
"... The i.calculus is considered a useful mathematical tool in the study of programming languages, since programs can be identified with Iterms. However, if one goes further and uses bnconversion to prove equivalence of programs, then a gross simplification is introduced (programs are identified with ..."
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Cited by 730 (15 self)
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The i.calculus is considered a useful mathematical tool in the study of programming languages, since programs can be identified with Iterms. However, if one goes further and uses bnconversion to prove equivalence of programs, then a gross simplification is introduced (programs are identified with total functions from calues to values) that may jeopardise the applicability of theoretical results, In this paper we introduce calculi. based on a categorical semantics for computations, that provide a correct basis for proving equivalence of programs for a wide range of notions of computation.
Programming with bananas, lenses, envelopes and barbed wire
 In FPCA
, 1991
"... We develop a calculus for lazy functional programming based on recursion operators associated with data type definitions. For these operators we derive various algebraic laws that are useful in deriving and manipulating programs. We shall show that all example Functions in Bird and Wadler's "Introdu ..."
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Cited by 299 (11 self)
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We develop a calculus for lazy functional programming based on recursion operators associated with data type definitions. For these operators we derive various algebraic laws that are useful in deriving and manipulating programs. We shall show that all example Functions in Bird and Wadler's "Introduction to Functional Programming " can be expressed using these operators. 1
Universal coalgebra: a theory of systems
, 2000
"... In the semantics of programming, nite data types such as finite lists, have traditionally been modelled by initial algebras. Later final coalgebras were used in order to deal with in finite data types. Coalgebras, which are the dual of algebras, turned out to be suited, moreover, as models for certa ..."
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Cited by 298 (31 self)
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In the semantics of programming, nite data types such as finite lists, have traditionally been modelled by initial algebras. Later final coalgebras were used in order to deal with in finite data types. Coalgebras, which are the dual of algebras, turned out to be suited, moreover, as models for certain types of automata and more generally, for (transition and dynamical) systems. An important property of initial algebras is that they satisfy the familiar principle of induction. Such a principle was missing for coalgebras until the work of Aczel (NonWellFounded sets, CSLI Leethre Notes, Vol. 14, center for the study of Languages and information, Stanford, 1988) on a theory of nonwellfounded sets, in which he introduced a proof principle nowadays called coinduction. It was formulated in terms of bisimulation, a notion originally stemming from the world of concurrent programming languages. Using the notion of coalgebra homomorphism, the definition of bisimulation on coalgebras can be shown to be formally dual to that of congruence on algebras. Thus, the three basic notions of universal algebra: algebra, homomorphism of algebras, and congruence, turn out to correspond to coalgebra, homomorphism of coalgebras, and bisimulation, respectively. In this paper, the latter are taken
Domain Theory in Logical Form
 Annals of Pure and Applied Logic
, 1991
"... The mathematical framework of Stone duality is used to synthesize a number of hitherto separate developments in Theoretical Computer Science: • Domain Theory, the mathematical theory of computation introduced by Scott as a foundation for denotational semantics. • The theory of concurrency and system ..."
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Cited by 231 (10 self)
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The mathematical framework of Stone duality is used to synthesize a number of hitherto separate developments in Theoretical Computer Science: • Domain Theory, the mathematical theory of computation introduced by Scott as a foundation for denotational semantics. • The theory of concurrency and systems behaviour developed by Milner, Hennessy et al. based on operational semantics. • Logics of programs. Stone duality provides a junction between semantics (spaces of points = denotations of computational processes) and logics (lattices of properties of processes). Moreover, the underlying logic is geometric, which can be computationally interpreted as the logic of observable properties—i.e. properties which can be determined to hold of a process on the basis of a finite amount of information about its execution. These ideas lead to the following programme:
Semantic Domains
, 1990
"... this report started working on denotational semantics in collaboration with Christopher Strachey. In order to fix some mathematical precision, he took over some definitions of recursion theorists such as Kleene, Nerode, Davis, and Platek and gave an approach to a simple type theory of highertype fu ..."
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Cited by 148 (3 self)
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this report started working on denotational semantics in collaboration with Christopher Strachey. In order to fix some mathematical precision, he took over some definitions of recursion theorists such as Kleene, Nerode, Davis, and Platek and gave an approach to a simple type theory of highertype functionals. It was only after giving an abstract characterization of the spaces obtained (through the construction of bases) that he realized that recursive definitions of types could be accommodated as welland that the recursive definitions could incorporate function spaces as well. Though it was not the original intention to find semantics of the socalled untyped calculus, such a semantics emerged along with many ways of interpreting a very large variety of languages. A large number of people have made essential contributions to the subsequent developments, and they have shown in particular that domain theory is not one monolithic theory, but that there are several different kinds of constructions giving classes of domains appropriate for different mixtures of constructs. The story is, in fact, far from finished even today. In this report we will only be able to touch on a few of the possibilities, but we give pointers to the literature. Also, we have attempted to explain the foundations in an elementary wayavoiding heavy prerequisites (such as category theory) but still maintaining some level of abstractionwith the hope that such an introduction will aid the reader in going further into the theory. The chapter is divided into seven sections. In the second section we introduce a simple class of ordered structures and discuss the idea of fixed points of continuous functions as meanings for recursive programs. In the third section we discuss computable functions and...
Complete restrictions of the intersection type discipline
 Theoretical Computer Science
, 1992
"... In this paper the intersection type discipline as defined in [Barendregt et al. ’83] is studied. We will present two different and independent complete restrictions of the intersection type discipline. The first restricted system, the strict type assignment system, is presented in section two. Its m ..."
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Cited by 104 (41 self)
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In this paper the intersection type discipline as defined in [Barendregt et al. ’83] is studied. We will present two different and independent complete restrictions of the intersection type discipline. The first restricted system, the strict type assignment system, is presented in section two. Its major feature is the absence of the derivation rule (≤) and it is based on a set of strict types. We will show that these together give rise to a strict filter lambda model that is essentially different from the one presented in [Barendregt et al. ’83]. We will show that the strict type assignment system is the nucleus of the full system, i.e. for every derivation in the intersection type discipline there is a derivation in which (≤) is used only at the very end. Finally we will prove that strict type assignment is complete for inference semantics. The second restricted system is presented in section three. Its major feature is the absence of the type ω. We will show that this system gives rise to a filter λImodel and that type assignment without ω is complete for the λIcalculus. Finally we will prove that a lambda term is typeable in this system if and only if it is strongly normalizable.
Relational Properties of Domains
 Information and Computation
, 1996
"... New tools are presented for reasoning about properties of recursively defined domains. We work within a general, categorytheoretic framework for various notions of `relation' on domains and for actions of domain constructors on relations. Freyd's analysis of recursive types in terms of a property o ..."
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Cited by 99 (5 self)
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New tools are presented for reasoning about properties of recursively defined domains. We work within a general, categorytheoretic framework for various notions of `relation' on domains and for actions of domain constructors on relations. Freyd's analysis of recursive types in terms of a property of mixed initiality/finality is transferred to a corresponding property of invariant relations. The existence of invariant relations is proved under completeness assumptions about the notion of relation. We show how this leads to simpler proofs of the computational adequacy of denotational semantics for functional programming languages with userdeclared datatypes. We show how the initiality/finality property of invariant relations can be specialized to yield an induction principle for admissible subsets of recursively defined domains, generalizing the principle of structural induction for inductively defined sets. We also show how the initiality /finality property gives rise to the coinduct...
A Categorical Programming Language
, 1987
"... A theory of data types and a programming language based on category theory are presented. Data types play a crucial role in programming. They enable us to write programs easily and elegantly. Various programming languages have been developed, each of which may use different kinds of data types. Ther ..."
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Cited by 66 (0 self)
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A theory of data types and a programming language based on category theory are presented. Data types play a crucial role in programming. They enable us to write programs easily and elegantly. Various programming languages have been developed, each of which may use different kinds of data types. Therefore, it becomes important to organize data types systematically so that we can understand the relationship between one data type and another and investigate future directions which lead us to discover exciting new data types. There have been several approaches to systematically organize data types: algebraic specification methods using algebras, domain theory using complete partially ordered sets and type theory using the connection between logics and data types. Here, we use category theory. Category theory has proved to be remarkably good at revealing the nature of mathematical objects, and we use it to understand the true nature of data types in programming.
A Semantics for Shape
 Science of Computer Programming
, 1995
"... Shapely types separate data, represented by lists, from shape, or structure. This separation supports shape polymorphism, where operations are defined for arbitrary shapes, and shapely operations, for which the shape of the result is determined by that of the input, permitting static shape checking. ..."
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Cited by 60 (18 self)
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Shapely types separate data, represented by lists, from shape, or structure. This separation supports shape polymorphism, where operations are defined for arbitrary shapes, and shapely operations, for which the shape of the result is determined by that of the input, permitting static shape checking. The shapely types are closed under the formation of fixpoints, and hence include the usual algebraic types of lists, trees, etc. They also include other standard data structures such as arrays, graphs and records. 1 Introduction The values of a shapely type are uniquely determined by their shape and their data. The shape can be thought of as a structure with holes or positions, into which data elements (stored in a list) can be inserted. The use of shape in computing is widespread, but till now it has not, apparently, been the subject of independent study. The body of the paper presents a semantics for shape, based on elementary ideas from category theory. First, let us consider some examp...
Initial Algebra and Final Coalgebra Semantics for Concurrency
, 1994
"... The aim of this paper is to relate initial algebra semantics and final coalgebra semantics. It is shown how these two approaches to the semantics of programming languages are each others dual, and some conditions are given under which they coincide. More precisely, it is shown how to derive initial ..."
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Cited by 55 (9 self)
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The aim of this paper is to relate initial algebra semantics and final coalgebra semantics. It is shown how these two approaches to the semantics of programming languages are each others dual, and some conditions are given under which they coincide. More precisely, it is shown how to derive initial semantics from final semantics, using the initiality and finality to ensure their equality. Moreover, many facts about congruences (on algebras) and (generalized) bisimulations (on coalgebras) are shown to be dual as well.