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Free iterative theories: a coalgebraic view

2003
*
Mathematical Structures in Computer Science
*

Every

doi:10.1017/s0960129502003924
fatcat:ozymshakqfhi5kb3logrbfuini
*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. ...##
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From Iterative Algebras to Iterative Theories

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). ...

##
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Terminal coalgebras and free iterative theories

2006
*
Information and Computation
*

Every

doi:10.1016/j.ic.2005.11.005
fatcat:2otsod2ld5bwritbx7ly6h6kva
*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] . ...##
###
On Rational Monads and Free Iterative Theories

2003
*
Electronical Notes in Theoretical Computer Science
*

We present a coalgebraic proof that R is a free

doi:10.1016/s1571-0661(04)80557-7
fatcat:bj23fzm5zfbg7mhobf26xwlvbq
*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*. ...##
###
Elgot Theories: A New Perspective of Iteration Theories (Extended Abstract)

2009
*
Electronical Notes in Theoretical Computer Science
*

This defines a

doi:10.1016/j.entcs.2009.07.100
fatcat:6pvk3dcndjgbpdualkssegd6cy
*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 . ...##
###
Corecursive Algebras, Corecursive Monads and Bloom Monads

2014
*
Logical Methods in Computer Science
*

The

doi:10.2168/lmcs-10(3:19)2014
fatcat:idq7yt4xaza5rmvrssemgn4abi
*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. ...##
###
On a Description of Terminal Coalgebras and Iterative Theories

2003
*
Electronical Notes in Theoretical Computer Science
*

A concrete description of terminal coalgebras, T , of all

doi:10.1016/s1571-0661(04)80629-7
fatcat:v7mzooorvffjtgcdiklbm6awue
*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. ...##
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Bases for parametrized iterativity

2008
*
Information and Computation
*

For every base we prove that free base algebras

doi:10.1016/j.ic.2008.05.002
fatcat:p42nvstz3zczfkavuknhce45l4
*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. ...##
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On second-order iterative monads

2011
*
Theoretical Computer Science
*

We construct two second-order

doi:10.1016/j.tcs.2011.04.027
fatcat:6kfrfynsdvdonffunsash6b6cu
*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. ...##
###
A Description of Iterative Reflections of Monads (Extended Abstract)
[chapter]

2009
*
Lecture Notes in Computer Science
*

Examples: the

doi:10.1007/978-3-642-00596-1_12
fatcat:ayudpercezdx5gwfmxofciyvwq
*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. ...##
###
Iterative algebras at work

2006
*
Mathematical Structures in Computer Science
*

Despite this, our result is much more general: we describe a free

doi:10.1017/s0960129506005706
fatcat:cb5becbpmjclzbdwj3pf3fayta
*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) . ...##
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Iterative reflections of monads

2010
*
Mathematical Structures in Computer Science
*

We also introduce the concept of

doi:10.1017/s0960129509990326
fatcat:nbojzjd7cjh6hjyswsewbrebhu
*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. ...##
###
Recursive Program Schemes and Context-Free Monads

2010
*
Electronical Notes in Theoretical Computer Science
*

Our aim is to generalize this to

doi:10.1016/j.entcs.2010.07.011
fatcat:sgt2ujo2qnfq7pnawuse2erhpy
*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 ...##
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Algebras with parametrized iterativity

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. ...

##
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A parameterized graph transformation calculus for finite graphs with monadic branches

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,

doi:10.1145/2505879.2505903
dblp:conf/ppdp/AsadaHKHN13
fatcat:y3n2hkv26bbmpgu3mjluoqdkxm
*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 ...
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