Results 1  10
of
16
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 ..."
Abstract

Cited by 12 (1 self)
 Add to MetaCart
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
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 ..."
Abstract

Cited by 7 (5 self)
 Add to MetaCart
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 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 ..."
Abstract

Cited by 7 (2 self)
 Add to MetaCart
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
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 ..."
Abstract

Cited by 7 (3 self)
 Add to MetaCart
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.
From grammars and automata to algebras and coalgebras
, 2013
"... Abstract. The increasing application of notions and results from category theory, especially from algebra and coalgebra, has revealed that any formal software or hardware model is constructor or destructorbased, a whitebox or a blackbox model. A highlystructured system may involve both construct ..."
Abstract
 Add to MetaCart
Abstract. The increasing application of notions and results from category theory, especially from algebra and coalgebra, has revealed that any formal software or hardware model is constructor or destructorbased, a whitebox or a blackbox model. A highlystructured system may involve both constructor and destructorbased components. The two model classes and the respective ways of developing them and reasoning about them are dual to each other. Roughly said, algebras generalize the modeling with contextfree grammars, word languages and structural induction, while coalgebras generalize the modeling with automata, Kripke structures, streams, process trees and all other state or objectoriented formalisms. We summarize the basic concepts of co/algebra and illustrate them at a couple of signatures including those used in language or compiler construction like regular expressions or acceptors. 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 ..."
Abstract
 Add to MetaCart
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.
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 ..."
Abstract
 Add to MetaCart
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.
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 ..."
Abstract
 Add to MetaCart
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.
Formal Software Development: From Foundations to Tools
"... This exposé gives an overview of the author’s contributions to the area of formal software development. These range from foundational issues dealing with abstract models of computation to practical engineering issues concerned with tool integration and user interface design. We can distinguish three ..."
Abstract
 Add to MetaCart
This exposé gives an overview of the author’s contributions to the area of formal software development. These range from foundational issues dealing with abstract models of computation to practical engineering issues concerned with tool integration and user interface design. We can distinguish three lines of work: Firstly, there is foundational work, centred around categorical models of rewriting. A new semantics for rewriting is developed, which abstracts over the concrete term structure while still being able to express key concepts such as variable, layer and substitution. It is based on the concept of a monad, which is wellknown in category theory to model algebraic theories. We generalise this treatment to term rewriting systems, infinitary terms, term graphs, and other forms of rewriting. The semantics finds applications in functional programming, where monads are used to model computational features such as state, exceptions and I/O, and modularity proofs, where
Fundamental study The categorytheoretic solution of recursive program schemes
"... 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 ..."
Abstract
 Add to MetaCart
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 solution (in our sense) of a recursive program scheme.