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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 7 (2 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
On the construction of free algebras for equational systems
 IN: SPECIAL ISSUE FOR AUTOMATA, LANGUAGES AND PROGRAMMING (ICALP 2007). VOLUME 410 OF THEORETICAL COMPUTER SCIENCE
, 2009
"... The purpose of this paper is threefold: to present a general abstract, yet practical, notion of equational system; to investigate and develop the finitary and transfinite construction of free algebras for equational systems; and to illustrate the use of equational systems as needed in modern applica ..."
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Cited by 5 (4 self)
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The purpose of this paper is threefold: to present a general abstract, yet practical, notion of equational system; to investigate and develop the finitary and transfinite construction of free algebras for equational systems; and to illustrate the use of equational systems as needed in modern applications.
Generalizing Substitution
, 2003
"... It is well known that, given an endofunctor H on a category C, the initial (A + H−)algebras (if existing), i.e., the algebras of (wellfounded) Hterms over different variable supplies A, give rise to a monad with substitution as the extension operation (the free monad induced by the functor H). Mo ..."
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Cited by 4 (1 self)
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It is well known that, given an endofunctor H on a category C, the initial (A + H−)algebras (if existing), i.e., the algebras of (wellfounded) Hterms over different variable supplies A, give rise to a monad with substitution as the extension operation (the free monad induced by the functor H). Moss [17] and Aczel, Adámek, Milius and Velebil [2] have shown that a similar monad, which even enjoys the additional special property of having iterations for all guarded substitution rules (complete iterativeness), arises from the inverses of the final (A + H−)coalgebras (if existing), i.e., the algebras of nonwellfounded Hterms. We show that, upon an appropriate generalization of the notion of substitution, the same can more generally be said about the initial T ′ (A, −)algebras resp. the inverses of the final T ′ (A, −)coalgebras for any endobifunctor T ′ on any category C such that the functors T ′ (−,X) uniformly carry a monad structure.
COPRODUCTS OF IDEAL MONADS
, 2004
"... The question of how to combine monads arises naturally in many areas with much recent interest focusing on the coproduct of two monads. In general, the coproduct of arbitrary monads does not always exist. Although a rather general construction was given by ..."
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Cited by 4 (1 self)
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The question of how to combine monads arises naturally in many areas with much recent interest focusing on the coproduct of two monads. In general, the coproduct of arbitrary monads does not always exist. Although a rather general construction was given by
The Recursion Scheme from the Cofree Recursive Comonad
"... We instantiate the general comonadbased construction of recursion schemes for the initial algebra of a functor F to the cofree recursive comonad on F. Differently from the scheme based on the cofree comonad on F in a similar fashion, this scheme allows not only recursive calls on elements structura ..."
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We instantiate the general comonadbased construction of recursion schemes for the initial algebra of a functor F to the cofree recursive comonad on F. Differently from the scheme based on the cofree comonad on F in a similar fashion, this scheme allows not only recursive calls on elements structurally smaller than the given argument, but also subsidiary recursions. We develop a Mendler formulation of the scheme via a generalized Yoneda lemma for initial algebras involving strong dinaturality and hint a relation to circular proofs à la Cockett, Santocanale.
Finitary construction of free algebras for equational systems
, 2008
"... The purpose of this paper is threefold: to present a general abstract, yet practical, notion of equational system; to investigate and develop the finitary construction of free algebras for equational systems; and to illustrate the use of equational systems as needed in modern applications. Key words ..."
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The purpose of this paper is threefold: to present a general abstract, yet practical, notion of equational system; to investigate and develop the finitary construction of free algebras for equational systems; and to illustrate the use of equational systems as needed in modern applications. Key words: Equational system; algebra; free construction; monad. 1
Equational Systems and Free Constructions (Extended Abstract)
"... Abstract. The purpose of this paper is threefold: to present a general abstract, yet practical, notion of equational system; to investigate and develop a theory of free constructions for such equational systems; and to illustrate the use of equational systems as needed in modern applications, specif ..."
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Abstract. The purpose of this paper is threefold: to present a general abstract, yet practical, notion of equational system; to investigate and develop a theory of free constructions for such equational systems; and to illustrate the use of equational systems as needed in modern applications, specifically to the theory of substitution in the presence of variable binding and to models of namepassing process calculi. 1
Languages, Theory
"... Recently there has been a great deal of interest in higherorder syntax which seeks to extend standard initial algebra semantics to cover languages with variable binding by using functor categories. The canonical example studied in the literature is that of the untyped λcalculus which is handled as ..."
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Recently there has been a great deal of interest in higherorder syntax which seeks to extend standard initial algebra semantics to cover languages with variable binding by using functor categories. The canonical example studied in the literature is that of the untyped λcalculus which is handled as an instance of the general theory of binding algebras, cf. Fiore, Plotkin, Turi [8]. Another important syntactic construction is that of explicit substitutions. The syntax of a language with explicit substitutions does not form a binding algebra as an explicit substitution may bind an arbitrary number of variables. Nevertheless we show that the language given by a standard signature Σ and explicit substitutions is naturally modelled as the initial algebra of the endofunctor Id + FΣ ◦ + ◦ on a functor category. We also comment on the apparent lack of modularity in syntax with variable binding as compared to firstorder languages. Categories and Subject Descriptors
Substitution in nonwellfounded . . .
 ELECTRONIC NOTES IN THEORETICAL COMPUTER SCIENCE 82 NO. 1 (2003)
, 2003
"... Inspired from the recent developments in theories of nonwellfounded syntax (coinductively defined languages) and of syntax with binding operators, the structure of algebras of wellfounded and nonwellfounded terms is studied for a very general notion of signature permitting both simple variable bin ..."
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Inspired from the recent developments in theories of nonwellfounded syntax (coinductively defined languages) and of syntax with binding operators, the structure of algebras of wellfounded and nonwellfounded terms is studied for a very general notion of signature permitting both simple variable binding operators as well as operators of explicit substitution. This is done in an extensional mathematical setting of initial algebras and final coalgebras of endofunctors on a functor category. In the nonwellfounded case, the fundamental operation of substitution is more beneficially defined in terms of primitive corecursion than coiteration.