## Distributive laws for the coinductive solution of recursive equations

Venue: | Information and Computation |

Citations: | 13 - 1 self |

### BibTeX

@ARTICLE{Jacobs_distributivelaws,

author = {Bart Jacobs},

title = {Distributive laws for the coinductive solution of recursive equations},

journal = {Information and Computation},

year = {},

pages = {2006}

}

### Years of Citing Articles

### OpenURL

### Abstract

This paper illustrates the relevance of distributive laws for the solution of recursive equations, and shows that one approach for obtaining coinductive solutions of equations via infinite terms is in fact a special case of a more general approach using an extended form of coinduction via distributive laws. 1

### Citations

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Regular Algebra and Finite Machines
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- 1971
(Show Context)
Citation Context ...x + y satisfying distribution equations like (x + y) · z = x · z + y · z and z · (x + y) = z · x + z · y and 0 · x = 0 = x · 0. In the theorem we obtain algebras with arbitrary joins, such as used in =-=[9]-=-, under the name “standard Kleene algebras”. The associated iteration operation is obtained as x∗ = ∨ n∈N xn . Our L-algebras are also known as unital quantales, see [22]. 8The set of languages L(X) ... |

244 | A tutorial on (co)algebras and (co)induction
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Citation Context ...gebras. Hence also in this situation algebra and coalgebra meet, and appropriate distributive laws are to be expected. The finality principle in the theory of coalgebras is usually called coinduction =-=[15]-=-. It involves the existence and uniqueness of suitable coalgebra homomorphisms to final coalgebras. It was realised early on (see [1,6]) that such coinductively obtained homomorphisms can be understoo... |

213 |
Non-Well-Founded Sets
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Citation Context ...inciple in the theory of coalgebras is usually called coinduction [15]. It involves the existence and uniqueness of suitable coalgebra homomorphisms to final coalgebras. It was realised early on (see =-=[1,6]-=-) 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... |

201 | Toposes, Triples and Theories
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Citation Context ...g that the list monad (−) ⋆ is Cartesian. In order to investigate the consequence we use the following general result about distributive laws between monads. It is standard, and may be traced back to =-=[7,16,4]-=- or [25]. Proposition 8 Let π: ST ⇒ T S be a distributive law between monads S and T on a category C. Then: (1) T S is a monad, with unit and multiplication given as: η = ⎛ S �� �� T η � �� η� � S ���... |

116 | Weakly distributive categories
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Citation Context ...ution x(y + z) = xy + xz is common in many equational theories, such as vector spaces. It may also occur in so-called distributive categories, of the form X × (Y + Z) ∼ = (X × Y ) + (X × Z), see e.g. =-=[8]-=-, where one direction of the isomorphism is canonical and always exists. More generally, one can have distributions GF ⇒ F G between two endofunctors F, G on the same category, as first studied in [7]... |

105 | Vicious Circles - Barwise, Moss - 1996 |

92 |
Alternating automata on infinite trees
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Citation Context ...e formula in Theorem 10 we can describe it explicitly as union of intersections: ⊔ V = ⋃ {L1 ∩ · · · ∩ Ln | 〈L1, . . . , Ln〉 ∈ V }. These language automata X → L(X) A ×2 resemble alternating automata =-=[21]-=-. It is at this stage not clear how useful the additional expressive power is for solving more expressive differential equations (with λ L -coinduction). 4 Free monads and their distributive laws In t... |

69 |
Distributive laws
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(Show Context)
Citation Context ...[8], where one direction of the isomorphism is canonical and always exists. More generally, one can have distributions GF ⇒ F G between two endofunctors F, G on the same category, as first studied in =-=[7]-=-. This phenomenon is especially interesting when the functors F, G form signatures (or interfaces) for certain operations, either in algebraic or in coalgebraic form. Turi and Plotkin [25] first inves... |

54 | Behavioural Differential Equations: a Coinductive Calculus
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- 2003
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Citation Context ... D-coalgebra is given by the set of languages L(A) = P(A ∗ ) over the alphabet A, with coalgebra structure 〈δ, ε〉: L(A) → L(A) A ×2 given by the “derivative” function and “is nullable” predicate (see =-=[9,23]-=-): for L ∈ L(A) and a ∈ A, δ(L)(a) = La = {σ ∈ A⋆ | a · σ ∈ L} ε(L) = (1 ⊆ L) = (〈〉 ∈ L). For an arbitrary D-coalgebra X → X A × 2, the induced homomorphism to 9s�� �� �� �� �� �� �� �� �� �� �� 3.2.2... |

54 |
Functorial operational semantics and its denotational dual
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Citation Context ...her elaborated in [14]. 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 [25] (see also =-=[24]-=-), providing a joint treatment of operational and denotational semantics. In that setting a distributive law provides a suitable form of compatibility between syntax and dynamics. The claim of [25] th... |

53 |
Infinite Trees and Completely Iterative Theories: A Coalgebraic View, Theoret
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(Show Context)
Citation Context ...oalgebra, this source can also be identified with the recursive equation (see Section 3 for examples). A systematic investigation of the solution of such equations first appeared in [20], followed by =-=[2]-=-. Their coalgebraic approach simplifies results for recursive equations with infinite terms from [10,11]. More recently, a general and abstract approach is proposed in [5], using distributive laws. It... |

42 | On generalised coinduction and probabilistic specification formats: distributive laws
- Bartels
- 2004
(Show Context)
Citation Context ...ppeared in [20], followed by [2]. Their coalgebraic approach simplifies results for recursive equations with infinite terms from [10,11]. More recently, a general and abstract approach is proposed in =-=[5]-=-, using distributive laws. It builds on earlier work [17] and may also be described dually, for algebras, as developed independently in [26]. One of the main contributions of this paper is that it sho... |

37 |
On the algebraic structure of rooted trees
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- 1978
(Show Context)
Citation Context ...). A systematic investigation of the solution of such equations first appeared in [20], followed by [2]. Their coalgebraic approach simplifies results for recursive equations with infinite terms from =-=[10,11]-=-. More recently, a general and abstract approach is proposed in [5], using distributive laws. It builds on earlier work [17] and may also be described dually, for algebras, as developed independently ... |

26 | Distributivity for endofunctors, pointed and copointed endofunctors, monads and comonads
- Lenisa, Power, et al.
(Show Context)
Citation Context ...ributive laws are natural transformations GF ⇒ F G 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 =-=[18]-=-), 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 examples in t... |

24 |
Adjont lifting theorems for categories of algebras
- Johnstone
- 1975
(Show Context)
Citation Context ...g that the list monad (−) ⋆ is Cartesian. In order to investigate the consequence we use the following general result about distributive laws between monads. It is standard, and may be traced back to =-=[7,16,4]-=- or [25]. Proposition 8 Let π: ST ⇒ T S be a distributive law between monads S and T on a category C. Then: (1) T S is a monad, with unit and multiplication given as: η = ⎛ S �� �� T η � �� η� � S ���... |

23 | Recursion schemes from comonads
- Uustalu, Vene, et al.
(Show Context)
Citation Context ...ore recently, a general and abstract approach is proposed in [5], using distributive laws. It builds on earlier work [17] and may also be described dually, for algebras, as developed independently in =-=[26]-=-. One of the main contributions of this paper is that it shows how the approach of [2] for infinite terms fits in the general approach of [5] with distributive laws. This involves the identification o... |

21 | Trace semantics for coalgebras
- Jacobs
- 2004
(Show Context)
Citation Context ...Fi ⇒ FiT . (4) If our category C is Sets, and the functor T preserves weak pullbacks, then there is a distributive law of monads T P ⇒ PT , where P is the powerset monad. This construction comes from =-=[13]-=-, and is called the “power law”. Here we sketch the essentials. We associate the so-called “relation lifting” Rel(T ) with T . It is a functor that maps a relation 〈r1, r2〉: R ↣ X × Y to a relation Re... |

16 | A coalgebraic view of infinite trees and iteration - Aczel, Adámek, et al. - 2001 |

16 | From set-theoretic coinduction to coalgebraic coinduction: some results, some problems
- Lenisa
- 1999
(Show Context)
Citation Context ...oach simplifies results for recursive equations with infinite terms from [10,11]. More recently, a general and abstract approach is proposed in [5], using distributive laws. It builds on earlier work =-=[17]-=- and may also be described dually, for algebras, as developed independently in [26]. One of the main contributions of this paper is that it shows how the approach of [2] for infinite terms fits in the... |

15 | Parametric corecursion
- Moss
- 2001
(Show Context)
Citation Context ... from the source coalgebra, this source can also be identified with the recursive equation (see Section 3 for examples). A systematic investigation of the solution of such equations first appeared in =-=[20]-=-, followed by [2]. Their coalgebraic approach simplifies results for recursive equations with infinite terms from [10,11]. More recently, a general and abstract approach is proposed in [5], using dist... |

6 |
Quantales and their applications. Number 234
- Rosenthal
- 1990
(Show Context)
Citation Context ...bitrary joins, such as used in [9], under the name “standard Kleene algebras”. The associated iteration operation is obtained as x∗ = ∨ n∈N xn . Our L-algebras are also known as unital quantales, see =-=[22]-=-. 8The set of languages L(X) carries a free Kleene algebra structure µX: L 2 (X) → L(X), with the familiar structure induced by the multiplication µ: 0 = µX(∅) = ∅ 1 = µX({〈〉}) = {〈〉} L1 · L2 = µX({〈... |

4 | Relating two approaches to coinductive solution of recurisve equations
- Jacobs
- 2004
(Show Context)
Citation Context ...quations with infinite terms is then explained in Section 5. Finally, Section 6 shows that this approach is an instance of the distributionbased approach. An earlier version of this paper appeared as =-=[12]-=-. The present version extends [12] especially with Section 3 on distributive laws for languages and automata. This topic is further elaborated in [14]. 2 Distributive laws and solutions of equations D... |

3 | A bialgebraic review of regular expressions, deterministic automata and languages
- Jacobs
- 2005
(Show Context)
Citation Context ...ach. An earlier version of this paper appeared as [12]. The present version extends [12] especially with Section 3 on distributive laws for languages and automata. This topic is further elaborated in =-=[14]-=-. 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 [25] (see also [24]), providing a joi... |

3 | On iteratable endofunctors
- Milius
- 2002
(Show Context)
Citation Context ... � � � � This means that � � � � � � � � � 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. It is shown in =-=[3,19]-=- that F ∞ is a monad 1 . 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 ... |