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Free iterative theories: a coalgebraic view

JIR ADMEK, STEFAN MILIUS, JIR VELEBIL
2003 Mathematical Structures in Computer Science  
Every finitary endofunctor of Set is proved to generate a free iterative theory in the sense of Elgot.  ...  This work is based on coalgebras, specifically on parametric corecursion, and the proof is presented for categories more general than just Set.  ...  In the joint paper Aczel et al. (2002) , whose extended abstract was published as Aczel et al. (2001) , we constructed a free completely iterative monad on every iteratable functor.  ... 
doi:10.1017/s0960129502003924 fatcat:ozymshakqfhi5kb3logrbfuini

From Iterative Algebras to Iterative Theories

Jiří Adámek, Stefan Milius, Jiří Velebil
2004 Electronical Notes in Theoretical Computer Science  
And our result is, nevertheless, much more general: we describe a free iterative theory on any finitary endofunctor of every locally presentable category A.  ...  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.  ...  Definition 2.10 The finitary monad on A formed by free iterative H-algebras is called the rational monad of H and is denoted by R = (R, η, µ).  ... 
doi:10.1016/j.entcs.2004.02.033 fatcat:qwbwa3klezhblkkxykp24gcj3i

Terminal coalgebras and free iterative theories

Jiřı´ Adámek, Stefan Milius
2006 Information and Computation  
Every finitary endofunctor H of Set can be represented via a finitary signature and a collection of equations called "basic".  ...  As an application we describe a free iterative theory on H (in the sense of Calvin Elgot) as the theory of all rational -trees modulo the analogous congruence.  ...  The concept of ideal and iterative theory of [17] then precisely correspond to ideal and iterative finitary monads, as explained in [9, 10] .  ... 
doi:10.1016/j.ic.2005.11.005 fatcat:2otsod2ld5bwritbx7ly6h6kva

On Rational Monads and Free Iterative Theories

Jiří Adámek, Stefan Milius, Jiří Velebil
2003 Electronical Notes in Theoretical Computer Science  
We present a coalgebraic proof that R is a free iterative theory on H for every finitary endofunctor H, which is substantially simpler than the previous proof by Elgot et al., as well as our previous proof  ...  For every finitary endofunctor H of Set a rational algebraic theory (or a rational finitary monad) R is defined by means of solving all finitary flat systems of recursive equations over H.  ...  of finitary equation morphism and iterative monad.  ... 
doi:10.1016/s1571-0661(04)80557-7 fatcat:bj23fzm5zfbg7mhobf26xwlvbq

Elgot Theories: A New Perspective of Iteration Theories (Extended Abstract)

Jiří Adámek, Stefan Milius, Jiří Velebil
2009 Electronical Notes in Theoretical Computer Science  
This defines a monad Rat on the category Sgn of signatures:  ...  T Σ ⊥ are all Σ ⊥ -trees (finite and infinite) on n variables.  ...  Here we prove that R X+1 is a free Elgot theory on X and that it is a quotient theory of the theory R Σ ⊥ for some Σ. This gives a monad Rat on the category Set F .  ... 
doi:10.1016/j.entcs.2009.07.100 fatcat:6pvk3dcndjgbpdualkssegd6cy

Corecursive Algebras, Corecursive Monads and Bloom Monads

Jiří Adámek, Mahdie Haddadi, Stefan Milius, Andrzej Tarlecki
2014 Logical Methods in Computer Science  
The monad of free corecursive algebras is proved to be the free corecursive monad, where the concept of corecursive monad is a generalization of Elgot's iterative monads, analogous to corecursive algebras  ...  We also characterize the Eilenberg-Moore algebras for the free corecursive monad and call them Bloom algebras.  ...  We are grateful to ZoltánÉsik for a substantial contribution to the discussion of equations in Bloom monads. And to Paul Levy who suggested that Proposition 3.4 holds.  ... 
doi:10.2168/lmcs-10(3:19)2014 fatcat:idq7yt4xaza5rmvrssemgn4abi

On a Description of Terminal Coalgebras and Iterative Theories

Jiří Adámek
2003 Electronical Notes in Theoretical Computer Science  
A concrete description of terminal coalgebras, T , of all finitary endofunctors of Set is presented: each such a functor is a quotient of the polynomial endofunctor of some finitary signature Σ modulo  ...  some "basic" equations.  ...  Given a finitary endofunctor H, presented as in 2.1 above, we denote by T Σ the free completely iterative monad on H Σ (of infinite Σ-trees) and by T H the free completely iterative monad on H.  ... 
doi:10.1016/s1571-0661(04)80629-7 fatcat:v7mzooorvffjtgcdiklbm6awue

Bases for parametrized iterativity

Jiří Adámek, Stefan Milius, Jiří Velebil
2008 Information and Computation  
For every base we prove that free base algebras and free iterative base algebras exist.  ...  Parametrized iterativity of an algebra means the existence of unique solutions of all finitary recursive systems of equations where recursion is allowed to use only some variables (chosen as a parameter  ...  In fact, each HX ⊗ − is a finitary monad on A and the functoriality is obvious.  ... 
doi:10.1016/j.ic.2008.05.002 fatcat:p42nvstz3zczfkavuknhce45l4

On second-order iterative monads

Jiří Adámek, Stefan Milius, Jiří Velebil
2011 Theoretical Computer Science  
We construct two second-order iterative monads: one, called the second-order rational monad, S H , is proved to be the initial secondorder iterative monad.  ...  The other one, called the context-free monad, C H , is a quotient of S H and in the original case of a polynomial endofunctor H of Set we prove that C H is the monad studied by B. Courcelle.  ...  , we proved that R H is iterative (and, in particular, ideal) in the sense of Calvin Elgot, and we characterized R H as the free iterative monad on H.  ... 
doi:10.1016/j.tcs.2011.04.027 fatcat:6kfrfynsdvdonffunsash6b6cu

A Description of Iterative Reflections of Monads (Extended Abstract) [chapter]

Jiří Adámek, Stefan Milius, Jiří Velebil
2009 Lecture Notes in Computer Science  
Examples: the monad of free commutative binary algebras has the monad of binary rational unordered trees as iterative reflection, and the finite list monad has the iterative reflection given by adding  ...  We prove now that this monad is iterative in the sense of Calvin Elgot, in fact, this is the iterative reflection of the given ideal monad.  ...  We are grateful to one referee who suggested several improvements of the presentation of our paper.  ... 
doi:10.1007/978-3-642-00596-1_12 fatcat:ayudpercezdx5gwfmxofciyvwq

Iterative algebras at work

JIRÍ ADÁMEK, STEFAN MILIUS, JIRÍ VELEBIL
2006 Mathematical Structures in Computer Science  
Despite this, our result is much more general: we describe a free iterative theory on any finitary endofunctor of every locally presentable category A.  ...  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.  ...  The finitary monad on A formed by free iterative H-algebras is called the rational monad of H and is denoted by R = (R, η, µ) .  ... 
doi:10.1017/s0960129506005706 fatcat:cb5becbpmjclzbdwj3pf3fayta

Iterative reflections of monads

JIŘÍ ADÁMEK, STEFAN MILIUS, JIŘÍ VELEBIL
2010 Mathematical Structures in Computer Science  
We also introduce the concept of iterativity for algebras for the monad M, following the footsteps of Evelyn Nelson and Jerzy Tiuryn, and we prove that M is iterative if and only if all free algebras for  ...  For every ideal monad M we prove that an iterative reflection, i.e., an embedding M −→ M into an iterative monad with the expected universal property, exists.  ...  Iterative Monads Assumption 2.1. Throughout the paper we work with finitary monads on an extensive, locally finitely presentable category A.  ... 
doi:10.1017/s0960129509990326 fatcat:nbojzjd7cjh6hjyswsewbrebhu

Recursive Program Schemes and Context-Free Monads

Jiří Adámek, Stefan Milius, Jiří Velebil
2010 Electronical Notes in Theoretical Computer Science  
Our aim is to generalize this to finitary endofunctors H of general categories: we construct a monad C H "generated" by solutions of recursive program schemes of type H, and prove that this monad is ideal  ...  In case of polynomial endofunctors of Set our construction precisely yields the monad of context-free Σ-trees of Courcelle. Our result builds on a result by N.  ...  We prove that the context-free monad C H Σ of the polynomial endofunctor H Σ of Set is indeed precisely the submonad C H Σ → T H Σ of the Σ-tree monad consisting of all contextfree Σ-trees of Definition  ... 
doi:10.1016/j.entcs.2010.07.011 fatcat:sgt2ujo2qnfq7pnawuse2erhpy

Algebras with parametrized iterativity

Jiří Adámek, Stefan Milius, Jiří Velebil
2007 Theoretical Computer Science  
Iterative algebras, as studied by Nelson and Tiuryn, are generalized to algebras whose iterativity is parametrized in the sense that only some variables can be used for iteration.  ...  In order to express such parametrized iterativity, we work with parametrized endofunctors of Set, i.e. finitary endofunctors H : Set × Set → Set, and introduce the concept of iterativity for algebras for  ...  A base on Set is a finitary functor from Set to FM(Set), the category of all finitary monads on Set.  ... 
doi:10.1016/j.tcs.2007.06.015 fatcat:puotjb4tavhv5furzxv7b5lnzi

A parameterized graph transformation calculus for finite graphs with monadic branches

Kazuyuki Asada, Soichiro Hidaka, Hiroyuki Kato, Zhenjiang Hu, Keisuke Nakano
2013 Proceedings of the 15th Symposium on Principles and Practice of Declarative Programming - PPDP '13  
For example, λ T FG can treat unordered graphs, ordered graphs, weighted graphs, probability graphs, and so on, by using the powerset monad, list monad, multiset monad, probability monad, respectively.  ...  A remarkable feature of UnCAL and λ T FG is structural recursion for graphs, which gives a systematic programming basis like that for trees.  ...  The research was supported in part by the Grand-Challenging Project on the "Linguistic Foundation for Bidirectional Model Transformation" of the National Institute of Informatics and KAK-ENHI No. 23700047  ... 
doi:10.1145/2505879.2505903 dblp:conf/ppdp/AsadaHKHN13 fatcat:y3n2hkv26bbmpgu3mjluoqdkxm
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