## Behavioural Differential Equations and Coinduction for Binary Trees

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Citations: | 5 - 1 self |

### BibTeX

@MISC{Silva_behaviouraldifferential,

author = {Ra Silva and Jan Rutten},

title = {Behavioural Differential Equations and Coinduction for Binary Trees},

year = {}

}

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### Abstract

Abstract. We study the set TA of infinite binary trees with nodes labelledinasemiringA from a coalgebraic perspective. We present coinductive definition and proof principles based on the fact that TA carries a final coalgebra structure. By viewing trees as formal power series, we develop a calculus where definitions are presented as behavioural differential equations. We present a general format for these equations that guarantees the existence and uniqueness of solutions. Although technically not very difficult, the resulting framework has surprisingly nice applications, which is illustrated by various concrete examples. 1

### Citations

115 |
Automatic Sequences: Theory, Applications, Generalizations
- Allouche, Shallit
- 2003
(Show Context)
Citation Context ...he closed formula that we have obtained for the (binary tree representing the) Thue-Morse sequence suggests a possible use of coinduction and differential equations in the area of automatic sequences =-=[2]-=-. Typically, automatic sequences are represented by automata. The present calculus seems an interesting alternative, in which properties such as algebraicity of sequences can be derived from the tree ... |

88 |
Algebraic Approaches to Program Semantics
- Manes, Arbib
- 1986
(Show Context)
Citation Context ...word w with the letter b. These definitions of the set TA and the respective coalgebra map may not seem obvious. The follow reasoning justifies its correctness: – It is well known from the literature =-=[4]-=- that the final system for the functor G(X) = A × X B is (A B∗ , π), where π : A B∗ → A × (A B∗ ) B π(φ) = 〈φ(ε), λb v. φ(bv)〉 – The functor F is isomorphic to H(X) = A × X 2 . – Therefore, the set A ... |

57 | The Ubiquitous Prouhet-Thue-Morse Sequence
- Allouche, Shallit
- 1999
(Show Context)
Citation Context ...× (nat + 2pow)) ⇔ (1 − L − R)nat = 1 + L(1 − 2L − 2R) −1 + 2R(1 − 2L − 2R) −1 ⇔ (1 − L − R)nat = (1 − L) × (1 − 2L − 2R) −1 ⇔ nat = (1 − L − R) −1 × (1 − L) × (1 − 2L − 2R) −1 The Thue-Morse sequence =-=[1]-=- can be obtained by taking the parities of the counts of 1’s in the binary representation of non-negative integers. Alternatively, it can be defined by the repeated application of the substitution map... |

52 | Behavioural differential equations: a coinductive calculus of streams, automata, and power series. Theoret
- Rutten
(Show Context)
Citation Context ...for them. Therefore, reasoning tools for such structures have become more and more relevant. Coalgebraic techniques turned out to be suited for proving and deriving properties of infinite systems. In =-=[6]-=-, a coinductive calculus of formal power series was developed. In close analogy to classical analysis, the definitions were presented as behavioural differential equations and properties were proved i... |

40 | Formal tree series
- Ésik, Kuich
(Show Context)
Citation Context ...eness of solutions of behavioural differential equations. Infinite trees arise in several forms in other areas. Formal tree series (functions from trees to an arbitrary semiring) have been studied in =-=[3]-=-, closely related to distributive Σ-algebras. The work presented in this paper is completely different since we are concerned with infinite binary trees and not with formal series over trees. In [5], ... |

27 | A coinductive calculus of streams
- Rutten
- 2005
(Show Context)
Citation Context ...ns were presented as behavioural differential equations and properties were proved in a calculational (and very natural) way. This approach has shown to be quite effective for reasoning about streams =-=[6,7]-=- and it seems worthwhile to explore its effectiveness for other data structures as well. In this paper, we shall take a coalgebraic perspective on a classical data structure – infinite binary trees, a... |

9 |
Infinite Words, volume 141
- Perrin, Pin
(Show Context)
Citation Context ...n [3], closely related to distributive Σ-algebras. The work presented in this paper is completely different since we are concerned with infinite binary trees and not with formal series over trees. In =-=[5]-=-, infinite trees appear as generalisations of infinite words and an extensive study of tree automata and topological aspects of trees is made. We have not yet addressed the relation of our work with a... |

3 |
M.A.: Algebraic Approaches to Program Semantics
- Manes, Arbib
- 1986
(Show Context)
Citation Context ...word w with the letter b. These definitions of the set TA and the respective coalgebra map may not seem obvious. The follow reasoning justifies its correctness: – It is well known from the literature =-=[4]-=- that the final system for the functor G(X) =A × XB is (AB∗ ,π), where π : A B∗ → A × (A B∗ ) B π(φ) =〈φ(ε),λbv. φ(bv)〉 – The functor F is isomorphic to H(X) =A × X 2 . – Therefore, the set A2∗ is the... |

3 |
Infinite Words. Pure and Applied Mathematics
- Perrin, Pin
- 2004
(Show Context)
Citation Context ...tion for Binary Trees 323 distributive Σ-algebras. The work presented in this paper is completely different since we are concerned with infinite binary trees and not with formal series over trees. In =-=[5]-=-, infinite trees appear as generalisations of infinite words and an extensive study of tree automata and topological aspects of trees is made. We have not yet addressed the relation of our work with a... |

1 |
A coinductive calculus of streams. Mathematical Structures in Computer Science 15(1), 93–147 (2005) A Proof of Theorem 2 Proof (Proof of theorem 2). Consider a well-formed system of differential equations for Σ, as defined above. We define a set T of term
- Rutten
(Show Context)
Citation Context ...ns were presented as behavioural differential equations and properties were proved in a calculational (and very natural) way. This approach has shown to be quite effective for reasoning about streams =-=[6,7]-=- and it seems worthwhile to explore its effectiveness for other data structures as well. In this paper, we shall take a coalgebraic perspective on a classical data structure – infinite binary trees, a... |