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51
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
A Fold for All Seasons
 IN PROC. CONFERENCE ON FUNCTIONAL PROGRAMMING LANGUAGES AND COMPUTER ARCHITECTURE
, 1993
"... Generic control operators, such as fold, can be generated from algebraic type definitions. The class of types to which these techniques are applicable is generalized to all algebraic types definable in languages such as Miranda and ML, i.e. mutually recursive sumsofproducts with tuples and functio ..."
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Cited by 113 (15 self)
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Generic control operators, such as fold, can be generated from algebraic type definitions. The class of types to which these techniques are applicable is generalized to all algebraic types definable in languages such as Miranda and ML, i.e. mutually recursive sumsofproducts with tuples and function types. Several other useful generic operators, also applicable to every type in this class, also are described. A normalization algorithm which automatically calculates improvements to programs expressed in a language based upon folds is described. It reduces programs, expressed using fold as the exclusive control operator, to a canonical form. Based upon a generic promotion theorem, the algorithm is facilitated by the explicit structure of fold programs rather than using an analysis phase to search for implicit structure. Canonical programs are minimal in the sense that they contain the fewest number of fold operations. Because of this property, the normalization algorithm has important ...
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...
Inductive and Coinductive types with Iteration and Recursion
 Proceedings of the 1992 Workshop on Types for Proofs and Programs, Bastad
, 1992
"... We study (extensions of) simply and polymorphically typed lambda calculus from a point of view of how iterative and recursive functions on inductive types are represented. The inductive types can usually be understood as initial algebras in a certain category and then recursion can be defined in ter ..."
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Cited by 51 (0 self)
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We study (extensions of) simply and polymorphically typed lambda calculus from a point of view of how iterative and recursive functions on inductive types are represented. The inductive types can usually be understood as initial algebras in a certain category and then recursion can be defined in terms of iteration. However, in the syntax we often have only weak initiality, which makes the definition of recursion in terms of iteration inefficient or just impossible. We propose a categorical notion of (primitive) recursion which can easily be added as computation rule to a typed lambda calculus and gives us a clear view on what the dual of recursion, corecursion, on coinductive types is. (The same notion has, independently, been proposed by [Mendler 1991].) We look at how these syntactic notions work out in the simply typed lambda calculus and the polymorphic lambda calculus. It will turn out that in the syntax, recursion can be defined in terms of corecursion and vice versa using polymo...
On the Foundations of Final Semantics: NonStandard Sets, Metric Spaces, Partial Orders
 PROCEEDINGS OF THE REX WORKSHOP ON SEMANTICS: FOUNDATIONS AND APPLICATIONS, VOLUME 666 OF LECTURE NOTES IN COMPUTER SCIENCE
, 1998
"... Canonical solutions of domain equations are shown to be final coalgebras, not only in a category of nonstandard sets (as already known), but also in categories of metric spaces and partial orders. Coalgebras are simple categorical structures generalizing the notion of postfixed point. They are ..."
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Cited by 48 (10 self)
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Canonical solutions of domain equations are shown to be final coalgebras, not only in a category of nonstandard sets (as already known), but also in categories of metric spaces and partial orders. Coalgebras are simple categorical structures generalizing the notion of postfixed point. They are also used here for giving a new comprehensive presentation of the (still) nonstandard theory of nonwellfounded sets (as nonstandard sets are usually called). This paper is meant to provide a basis to a more general project aiming at a full exploitation of the finality of the domains in the semantics of programming languages  concurrent ones among them. Such a final semantics enjoys uniformity and generality. For instance, semantic observational equivalences like bisimulation can be derived as instances of a single `coalgebraic' definition (introduced elsewhere), which is parametric of the functor appearing in the domain equation. Some properties of this general form of equivalence are also studied in this paper.
Constructions, Inductive Types and Strong Normalization
, 1993
"... This thesis contains an investigation of Coquand's Calculus of Constructions, a basic impredicative Type Theory. We review syntactic properties of the calculus, in particular decidability of equality and typechecking, based on the equalityasjudgement presentation. We present a settheoretic notio ..."
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Cited by 31 (2 self)
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This thesis contains an investigation of Coquand's Calculus of Constructions, a basic impredicative Type Theory. We review syntactic properties of the calculus, in particular decidability of equality and typechecking, based on the equalityasjudgement presentation. We present a settheoretic notion of model, CCstructures, and use this to give a new strong normalization proof based on a modification of the realizability interpretation. An extension of the core calculus by inductive types is investigated and we show, using the example of infinite trees, how the realizability semantics and the strong normalization argument can be extended to nonalgebraic inductive types. We emphasize that our interpretation is sound for large eliminations, e.g. allows the definition of sets by recursion. Finally we apply the extended calculus to a nontrivial problem: the formalization of the strong normalization argument for Girard's System F. This formal proof has been developed and checked using the...
General recursion via coinductive types
 Logical Methods in Computer Science
"... Vol. 1 (2:1) 2005, pp. 1–28 ..."
Improving Programs which Recurse over Multiple Inductive Structures
 In ACM SIGPLAN Workshop on Partial Evaluation and SemanticsBased Program Manipulation (PEPM'94
, 1994
"... This paper considers generic recursion schemes for programs which recurse over multiple inductive structures simultaneously, such as equality, zip and the nth element of a list function. Such schemes have been notably absent from previous work. This paper defines a uniform mechanism for defining suc ..."
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Cited by 25 (6 self)
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This paper considers generic recursion schemes for programs which recurse over multiple inductive structures simultaneously, such as equality, zip and the nth element of a list function. Such schemes have been notably absent from previous work. This paper defines a uniform mechanism for defining such programs and shows that these programs satisfy generic theorems. These theorems are the basis for an automatic improvement algorithm. This algorithm is an improvement over the algorithm presented earlier [14] because, in addition to inducting over multiple structures, it can be incorporated into any algebraic language and is no longer restricted to a "safe" subset. 1 Introduction In previous work [14, 15, 6, 4, 5] we have shown how programming algebraically with generic recursion schemes provides a theory amenable to program calculation [13]. This theory provides a basis for automatic optimization techniques which capture many wellknown transformations. Unfortunately, these recursion sc...
Warm Fusion in Stratego: A Case Study in Generation of Program Transformation Systems
, 2000
"... Stratego is a domainspecic language for the specication of program transformation systems. The design of Stratego is based on the paradigm of rewriting strategies: userdenable programs in a little language of strategy operators determine where and in what order transformation rules are (automat ..."
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Cited by 23 (13 self)
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Stratego is a domainspecic language for the specication of program transformation systems. The design of Stratego is based on the paradigm of rewriting strategies: userdenable programs in a little language of strategy operators determine where and in what order transformation rules are (automatically) applied to a program. The separation of rules and strategies supports modularity of specications. Stratego also provides generic features for specication of program traversals. In this paper we present a case study of Stratego as applied to a nontrivial problem in program transformation. We demonstrate the use of Stratego in eliminating intermediate data structures from (also known as deforesting) functional programs via the warm fusion algorithm of Launchbury and Sheard. This algorithm has been specied in Stratego and embedded in a fully automatic transformation system for kernel Haskell. The entire system consists of about 2600 lines of specication code, which bre...
About Charity
, 1992
"... Charity is a categorical programming language based on distributive categories (in the sense of Schanuel and Lawvere) with strong datatypes (in the sense of Hagino). Distributive categories come with a term logic which can express most standard programs; and they are fundamental to computer science ..."
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Cited by 22 (0 self)
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Charity is a categorical programming language based on distributive categories (in the sense of Schanuel and Lawvere) with strong datatypes (in the sense of Hagino). Distributive categories come with a term logic which can express most standard programs; and they are fundamental to computer science because they permit proof by case analysis and, when strong datatypes are introduced, proof by structural induction. Charity is functional and polymorphic in style, and is strongly normalizing. As a categorical programming language it provides a unique marriage of computer science and mathematical thought. The above aspects are particularly important for the production of verified programs as the naturality of morphisms gives us "theorems for free", termination proofs are not required, and mathemathical specifications can be used. 1 Introduction Functional and logic programming languages have reduced the gap between theory and implementation by reducing the notational movement from mathema...