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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 7 (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.
Explicit Substitutions and HigherOrder Syntax (Extended Abstract)
 in Proc. of 2nd ACM SIGPLAN Wksh. on Mechanized Reasoning about Languages with Variable Binding, MERLIN'03
, 2003
"... Neil Ghani Dept. of Math. and Comp. Sci. ..."
Relating Two Approaches to Coinductive Solution of Recursive Equations
 Milius (Eds.), Proceedings of the 7th Workshop on Coalgebraic Methods in Computer Science, CMCS’04 (Barcelona, March 2004), Electron. Notes in Theoret. Comput. Sci
, 2004
"... This paper shows that the approach of [2,12] for obtaining coinductive solutions of equations on infinite terms is a special case of a more general recent approach of [4] using distributive laws. ..."
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Cited by 4 (3 self)
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This paper shows that the approach of [2,12] for obtaining coinductive solutions of equations on infinite terms is a special case of a more general recent approach of [4] using distributive laws.
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
doi:10.1017/S0960129506005706 Printed in the United Kingdom Iterative algebras at work
, 2004
"... Iterative theories, which were introduced by Calvin Elgot, formalise potentially infinite computations as unique solutions of recursive equations. One of the main results of Elgot and his coauthors is a description of a free iterative theory as the theory of all rational trees. Their algebraic proof ..."
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Iterative theories, which were introduced by Calvin Elgot, formalise potentially infinite computations as unique solutions of recursive equations. One of the main results of Elgot and his coauthors is a description of a free iterative theory as the theory of all rational trees. Their algebraic proof of this fact is extremely complicated. In our paper we show that by starting with ‘iterative algebras’, that is, algebras admitting a unique solution of all systems of flat recursive equations, a free iterative theory is obtained as the theory of free iterative algebras. The (coalgebraic) proof we present is dramatically simpler than the original algebraic one. Despite this, our result is much more general: we describe a free iterative theory on any finitary endofunctor of every locally presentable category A. Reportedly, a blow from the welterweight boxer Norman Selby, also known as Kid McCoy, left one victim proclaiming, ‘It’s the real McCoy! ’.¶ 1.
found at the ENTCS Macro Home Page.
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found at the ENTCS Macro Home Page. The Coinductive Resumption Monad
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