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Distributive laws for the coinductive solution of recursive equations
 Information and Computation
"... This paper illustrates the relevance of distributive laws for the solution of recursive equations, and shows that one approach for obtaining coinductive solutions of equations via infinite terms is in fact a special case of a more general approach using an extended form of coinduction via distributi ..."
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Cited by 13 (1 self)
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This paper illustrates the relevance of distributive laws for the solution of recursive equations, and shows that one approach for obtaining coinductive solutions of equations via infinite terms is in fact a special case of a more general approach using an extended form of coinduction via distributive laws. 1
The category theoretic solution of recursive program schemes
 Proc. First Internat. Conf. on Algebra and Coalgebra in Computer Science (CALCO 2005), Lecture Notes in Computer Science
, 2006
"... Abstract. This paper provides a general account of the notion of recursive program schemes, studying both uninterpreted and interpreted solutions. It can be regarded as the categorytheoretic version of the classical area of algebraic semantics. The overall assumptions needed are small indeed: worki ..."
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Cited by 11 (4 self)
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Abstract. This paper provides a general account of the notion of recursive program schemes, studying both uninterpreted and interpreted solutions. It can be regarded as the categorytheoretic version of the classical area of algebraic semantics. The overall assumptions needed are small indeed: working only in categories with “enough final coalgebras ” we show how to formulate, solve, and study recursive program schemes. Our general theory is algebraic and so avoids using ordered, or metric structures. Our work generalizes the previous approaches which do use this extra structure by isolating the key concepts needed to study substitution in infinite trees, including secondorder substitution. As special cases of our interpreted solutions we obtain the usual denotational semantics using complete partial orders, and the one using complete metric spaces. Our theory also encompasses implicitly defined objects which are not usually taken to be related to recursive program schemes. For example, the classical Cantor twothirds set falls out as an interpreted
Coalgebraic Monads
, 2002
"... This paper introduces coalgebraic monads as a unified model of term algebras covering fundamental examples such as initial algebras, final coalgebras, rational terms and term graphs. We develop a general method for obtaining finitary coalgebraic monads which allows us to generalise the notion of rat ..."
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Cited by 7 (5 self)
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This paper introduces coalgebraic monads as a unified model of term algebras covering fundamental examples such as initial algebras, final coalgebras, rational terms and term graphs. We develop a general method for obtaining finitary coalgebraic monads which allows us to generalise the notion of rational term and term graph to categories other than Set. As an application we sketch part of the correctness of the the term graph implementation of functional programming languages.
The Dual of Substitution is Redecoration
, 2002
"... It is well known that type constructors of incomplete trees (trees with variables) carry the structure of a monad with substitution as the extension operation. Less known are the facts that the same is true of type constructors of incomplete cotrees (=nonwellfounded trees) and that the correspondin ..."
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Cited by 7 (3 self)
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It is well known that type constructors of incomplete trees (trees with variables) carry the structure of a monad with substitution as the extension operation. Less known are the facts that the same is true of type constructors of incomplete cotrees (=nonwellfounded trees) and that the corresponding monads exhibit a special structure. We wish to draw attention to the dual facts which are as meaningful for functional programming: type constructors of decorated cotrees carry the structure of a comonad with redecoration as the coextension operation, and so doeven more interestinglytype constructors of decorated trees.
Under consideration for publication in Math. Struct. in Comp. Science An Algebraic Foundation and Implementation of Induction Recursion and Indexed Induction Recursion
, 2001
"... Induction recursion offers the possibility of a clean, simple and yet powerful metalanguage for the type system of a dependently typed programming language. At its crux, induction recursion allows us to defining a universe, that is a set U of codes and a decoding function T: U → D which assigns to ..."
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Induction recursion offers the possibility of a clean, simple and yet powerful metalanguage for the type system of a dependently typed programming language. At its crux, induction recursion allows us to defining a universe, that is a set U of codes and a decoding function T: U → D which assigns to every code u: U, a value T u of some type D. The key feature of induction recursion is that the codes in U are built up inductively at the same time as the recursive definition of their decoding function T. Despite its potential, induction recursion has not become as widely understood, or used, as it should be. We believe this is in part because: i) there is still scope for analysing the theoretical foundations of induction recursion; and ii) a presentation of induction recursion for the wider functional programming community still needs to be developed. The aim of this paper is to tackle exactly these two issues. That is, we aim to i) develop an algebraic foundation for induction recursion to complement the original typetheoretic one; and ii) use this foundation to construct a clean implementation of induction recursion which thereby broadens its accessibility to functional programmers. Theory and practice, hand in hand, as it should be! 1.
Under consideration for publication in Math. Struct. in Comp. Science Monads of Coalgebras: Rational Terms and Term Graphs
, 2004
"... This paper introduces guarded and strongly guarded monads as a unified model of a variety of different term algebras covering fundamental examples such as initial algebras, final coalgebras, rational terms and term graphs. We develop a general method for obtaining finitary guarded monads which allow ..."
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This paper introduces guarded and strongly guarded monads as a unified model of a variety of different term algebras covering fundamental examples such as initial algebras, final coalgebras, rational terms and term graphs. We develop a general method for obtaining finitary guarded monads which allows us to define and prove properties of the rational and term graph monads. Furthermore, our treatment of rational equations extends the traditional approach to allow righthand sides of equations to be infinite terms, term graphs or other such coalgebraic structures. As an application, we use these generalised rational equations to sketch part of the correctness of the the term graph implementation of functional programming languages. 1.
From Parity Games to Circular Proofs
"... Abstract We survey on the ongoing research that relates the combinatorics of parity games to the algebra of categories with finite products, finite coproducts, initial algebras and final coalgebras of definable functors, i.e. _bicomplete categories. We argue that parity games with a given starting ..."
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Abstract We survey on the ongoing research that relates the combinatorics of parity games to the algebra of categories with finite products, finite coproducts, initial algebras and final coalgebras of definable functors, i.e. _bicomplete categories. We argue that parity games with a given starting position play the role of terms for the theory of _bicomplete categories. We show that the interpretation of a parity game in the category of sets and functions is the set of deterministic winning strategies for one player in the game. We discuss bounded memory communication strategies between two parity games and their computational significance. We describe how an attempt to formalize them within the algebra of _bicomplete categories leads to develop a calculus of proofs that are allowed to contain cycles.
Abstract From Parity Games to Circular Proofs
"... We survey on the ongoing research that relates the combinatorics of parity games to the algebra of categories with finite products, finite coproducts, initial algebras and final coalgebras of definable functors, i.e. µbicomplete categories. We argue that parity games with a given starting position ..."
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We survey on the ongoing research that relates the combinatorics of parity games to the algebra of categories with finite products, finite coproducts, initial algebras and final coalgebras of definable functors, i.e. µbicomplete categories. We argue that parity games with a given starting position play the role of terms for the theory of µbicomplete categories. We show that the interpretation of a parity game in the category of sets and functions is the set of deterministic winning strategies for one player in the game. We discuss bounded memory communication strategies between two parity games and their computational significance. We describe how an attempt to formalize them within the algebra of µbicomplete categories leads to develop a calculus of proofs that are allowed to contain cycles. This paper is a survey on our recent work lifting results on free µlattices [1,2] to a categorical setting. A µlattice is a lattice with enough least and greatest fixed points to interpret formal µterms. A generalization of this notion leads to consider categories with finite products, finite coproducts, and enough initial algebras and final coalgebras of functors. We call these categories µbicomplete. The outcome of this research is so far described in [3,4,5]. A main goal for us is to understand how the algebra of µbicomplete categories describes a computational situation through the combinatorics of games; when attempting to achieve this goal, computational logic and prooftheory become unavoidable ingredients. It is the aim of this note to give insights on how these four worlds – categories, games, computation and logic – relate in this context. As the need of a mathematical formalization has too often hidden these relationships, we shall present here only informal arguments. The reader will find formal proofs of the statements in the references cited above.
DOI: 10.1051/ita:2003021 SOLVING ALGEBRAIC EQUATIONS USING COALGEBRA
"... Abstract. Algebraic systems of equations define functions using recursion where parameter passing is permitted. This generalizes the notion of a rational system of equations where parameter passing is prohibited. It has been known for some time that algebraic systems in Greibach Normal Form have uni ..."
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Abstract. Algebraic systems of equations define functions using recursion where parameter passing is permitted. This generalizes the notion of a rational system of equations where parameter passing is prohibited. It has been known for some time that algebraic systems in Greibach Normal Form have unique solutions. This paper presents a categorical approach to algebraic systems of equations which generalizes the traditional approach in two ways i) we define algebraic equations for locally finitely presentable categories rather than just Set; and ii) we define algebraic equations to allow righthand sides which need not consist of finite terms. We show these generalized algebraic systems of equations have unique solutions by replacing the traditional metrictheoretic arguments with coalgebraic arguments. Mathematics Subject Classification. 18C10, 18C35, 18C50. 1.