## Relating Two Approaches to Coinductive Solution of Recursive Equations (2004)

Venue: | Milius (Eds.), Proceedings of the 7th Workshop on Coalgebraic Methods in Computer Science, CMCS’04 (Barcelona, March 2004), Electron. Notes in Theoret. Comput. Sci |

Citations: | 3 - 2 self |

### BibTeX

@INPROCEEDINGS{Jacobs04relatingtwo,

author = {Bart Jacobs},

title = {Relating Two Approaches to Coinductive Solution of Recursive Equations},

booktitle = {Milius (Eds.), Proceedings of the 7th Workshop on Coalgebraic Methods in Computer Science, CMCS’04 (Barcelona, March 2004), Electron. Notes in Theoret. Comput. Sci},

year = {2004},

pages = {145--166},

publisher = {Elsevier}

}

### OpenURL

### Abstract

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.

### Citations

228 | A tutorial on (co)algebras and (co)induction
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- 1997
(Show Context)
Citation Context ...s on infinite terms is a special case of a more general recent approach of [4] using distributive laws. 1 Introduction The finality principle in the theory of coalgebras is usually called coinduction =-=[8]-=-. It involves the existence and uniqueness of suitable coalgebra homomorphisms to final coalgebras. It was realised early on (see [1,5]) that such coinductively obtained homomorphisms can be understoo... |

201 |
Non-Well-Founded Sets
- Aczel
- 1988
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Citation Context ...rinciple in the theory of coalgebras is usually called coinduction [8]. It involves the existence and uniqueness of suitable coalgebra homomorphisms to final coalgebras. It was realised early on (see =-=[1,5]-=-) that such coinductively obtained homomorphisms can be understood as solutions to recursive (or corecursive, if you like) equations. The equation itself is incorporated in the commuting square expres... |

134 | Toward a mathematical Operational Semantics
- Turi, Plotkin
- 1997
(Show Context)
Citation Context ...ce of the distribution-based approach. 2 Distributive laws and solutions of equations Distributive laws found their first serious application in the area of coalgebras in the work of Turi and Plotkin =-=[15]-=- (see also [14]), providing a joint treatment of operational and denotational semantics. In that setting a distributive laws provides a suitable form of compatibility between syntax and dynamics. It l... |

97 | Vicious Circles - Barwise, Moss - 1996 |

53 |
Functorial Operational Semantics and its Denotational Dual
- Turi
- 1996
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Citation Context ...ibution-based approach. 2 Distributive laws and solutions of equations Distributive laws found their first serious application in the area of coalgebras in the work of Turi and Plotkin [15] (see also =-=[14]-=-), providing a joint treatment of operational and denotational semantics. In that setting a distributive laws provides a suitable form of compatibility between syntax and dynamics. It leads to results... |

50 | Behavioural differential equations: a coinductive calculus of streams, automata, and power series, Theoret
- Rutten
- 2003
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Citation Context ...xample 3.3 Let Z = R N be the set of streams of real numbers. It is of course the final coalgebra of the functor F = R × (−), via the head and tail operations 〈hd, tl〉: Z ∼ = −→ R × Z. I=-=t is shown in [13] that on suc-=-h streams one can coinductively define binary operators ⊕ for sum and ⊗ for shuffle product satisfying the recursive equations: x ⊕ y = (hd(x) + hd(y)) · (tl(x) ⊕ tl(y)) x ⊗ y = (hd(x) × h... |

49 |
Monadic Computation and iterative algebraic theories
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Citation Context ...on. A systematic investigation of the solution of such equations first appeared in [12], followed by [2]. Their coalgebraic approach simplifies results on recursive equations with infinite terms from =-=[6,7]-=-. More recently, a general and abstract approach is proposed in [4], building on distributive laws. The contribution of this paper is that it shows how the approach of [2] for infinite terms fits in t... |

41 |
Infinite trees and completely iterative theories: a coalgebraic view. Theoretical Computer Science
- Aczel, Adámek, et al.
- 2003
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Citation Context ...s Department of Computer Science, University of Nijmegen P.O. Box 9010, 6500 GL Nijmegen, The Netherlands Email: bart@cs.kun.nl URL: http://www.cs.kun.nl/ ∼ bart This paper shows that the approach o=-=f [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. 1 Introduction The finality principle in the the... |

38 | On generalised coinduction and probabilistic specification formats
- Bartels
- 2004
(Show Context)
Citation Context ...RL: http://www.cs.kun.nl/ ∼ bart 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 o=-=f [4]-=- using distributive laws. 1 Introduction The finality principle in the theory of coalgebras is usually called coinduction [8]. It involves the existence and uniqueness of suitable coalgebra homomorphi... |

28 |
The algebraic structure of rooted trees
- Elgot, Bloom, et al.
- 1978
(Show Context)
Citation Context ...on. A systematic investigation of the solution of such equations first appeared in [12], followed by [2]. Their coalgebraic approach simplifies results on recursive equations with infinite terms from =-=[6,7]-=-. More recently, a general and abstract approach is proposed in [4], building on distributive laws. The contribution of this paper is that it shows how the approach of [2] for infinite terms fits in t... |

24 | Distributivity for endofunctors, pointed and copointed endofunctors, monads and comonads
- Lenisa, Power, et al.
(Show Context)
Citation Context ...ributive laws are natural transformations F G ⇒ GF between two endofunctors F, G: C → C on a category C. These F and G may have additional structure (of a point or copoint, or a monad or comonad, =-=see [10]),-=- that must then be preserved by the distributive law. We shall concentrate on the case of distribution of a monad over a functor, because it seems to be most common and natural—see the example in th... |

15 | From set-theoretic coinduction to coalgebraic coinduction: some results, some problems. ENTCS
- Lenisa
- 1999
(Show Context)
Citation Context ...e laws correspond to suitable rule formats for operators is further substantiated in [4]. The idea of using a distributive law in extended forms of coinduction (and hence equation solving) comes from =-=[9], an-=-d is further developed in [4]. In this section we present its essentials. Distributive laws are natural transformations F G ⇒ GF between two endofunctors F, G: C → C on a category C. These F and G... |

14 | Parametric corecursion
- Moss
(Show Context)
Citation Context ...s Department of Computer Science, University of Nijmegen P.O. Box 9010, 6500 GL Nijmegen, The Netherlands Email: bart@cs.kun.nl URL: http://www.cs.kun.nl/ ∼ bart This paper shows that the approach o=-=f [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. 1 Introduction The finality principle in the the... |

5 | On rational monads and free iterative theories - Adámek, Milius, et al. |

3 | On iteratable endofunctors
- Milius
- 2002
(Show Context)
Citation Context ...∞ (X) � � � � � � � � � � � � ��� F ∞ F (f) ∞ (Y ) F ∞ (f) ◦ ηX = ηY ◦ f F ∞ (f) ◦ τX = τY ◦ F (F ∞ (f)), i.e. that η: id ⇒ F ∞ and τ: F F �=-=�� ⇒ F ∞ are natural transformations. 9sJacobs It is shown in [11,3] th-=-at F ∞ is a monad 2 . The multiplication operation µ is rather complicated, and can best be introduced via substitution t[s/x]. What we mean is replacing all occurrences (if any) of the variable x ... |