## London, United Kingdom

### BibTeX

@MISC{_london,united,

author = {},

title = {London, United Kingdom},

year = {}

}

### OpenURL

### Abstract

The structure map turning a set into the carrier of a final coalgebra is not unique. This fact is well-known but commonly elided. In this paper we argue that any such concrete representation of a set as a final coalgebra is potentially interesting on its own. We discuss several examples, in particular, we consider different coalgebra structures that turn the set of infinite streams into the carrier of a final coalgebra. After that we focus on coalgebra structures that are made up using so-called cooperations. We say that a collection of cooperations is complete for a given set X if it gives rise to a coalgebra structure that turns X into the carrier set of a subcoalgebra of a final coalgebra. Any complete set of cooperations yields a coalgebraic proof and definition principle. We exploit this fact and devise a general definition scheme for constants and functions on a set X that is parametrically in the choice of the complete set of cooperations for X. Key words: Coalgebra, coinduction, infinite data structures, hidden algebra. 1

### Citations

298 | Universal coalgebra: a theory of systems - Rutten |

77 | Term Rewriting Systems - Terese - 2003 |

55 | The ubiquitious Prouhet-Thue-Morse sequence
- Allouche, Shallit
- 1999
(Show Context)
Citation Context ...ing our definition scheme we will be able to define functions of type (2 ω ) n → 2 ω for n ∈ IN (for n = 0 we obtain a constant). As a concrete example we treat the so-called Thue-Morse sequence (cf. =-=[1]-=-) TM = t0t1t2 . . . with tn = s2(n) mod 2, where s2(n) denotes the sum of the digits of the binary representation of n. In our scheme for the complete set of cooperations {head, even, odd} we define T... |

46 |
Context induction: a proof principle for behavioural abstractions and algebraic implementations, Formal Aspects of Computing 3 (4
- Hennicker
- 1991
(Show Context)
Citation Context ... more restrictive concerning the structure of the contexts that are allowed on the right hand side of a defining equation. In particular, we do not make use of any kind of context induction (see e.g. =-=[11]-=-). The latter is difficult to use in our setting as a simultaneous definition of several functions can easily lead to unwanted circularities. Different notions: Finally one should note that our notion... |

34 | Observer complete definitions are behaviourally coherent, in: OBJ/CafeOBJ/Maude at Formal Methods ’99
- Bidoit, Hennicker
- 1999
(Show Context)
Citation Context ...Our notion of a complete set of cooperations is inspired by the notion of a cobasis in hidden algebra (see e.g. [15, 14, 8]) and our definition scheme in Section 6 has some similarity with the one in =-=[3]-=-. Throughout the paper we will tell the reader precisely which of our notions and results are instances of similar ones in hidden algebra. At the same time, there are also many differences between our... |

30 | Elements of the general theory of coalgebras - Gumm - 1999 |

25 | A coinductive calculus of streams
- Rutten
- 2005
(Show Context)
Citation Context ...ly < h, t > is observational for IR ω . (6) We can also supply IR ω with an alternative coalgebra structure as follows. For σ ∈ IR ω we define ∆σ = (σ(1) − σ(0), σ(2) − σ(1), σ(3) − σ(2), . . .) (cf. =-=[13, 17]-=-). We claim that the coalgebra map < h, ∆ >: IR ω → IR × IR ω σ ↦→ < σ(0), ∆σ > is observational for IR ω . The unique morphism ϕ : (IR ω , < h, ∆ >) → (IR ω , < h, t >) 6is given by ϕ(σ) = ((∆ (0) σ... |

19 | Coalgebra semantics for hidden algebra: Parameterised objects and inheritance
- Cîrstea
- 1998
(Show Context)
Citation Context ... hand and existing work in hidden algebra and observational specification on the other hand. A strong link between coalgebra and hidden algebra has been established in a series of articles by Cîrstea =-=[4, 5]-=- in which it is shown that, under the assumption that any operation of a hidden algebra signature has at 2most one argument of hidden sort, hidden algebras can be seen as coalgebras. Our notion of a ... |

19 | Calculus in coinductive form
- Pavlović, Escardó
- 1998
(Show Context)
Citation Context ...ly < h, t > is observational for IR ω . (6) We can also supply IR ω with an alternative coalgebra structure as follows. For σ ∈ IR ω we define ∆σ = (σ(1) − σ(0), σ(2) − σ(1), σ(3) − σ(2), . . .) (cf. =-=[13, 17]-=-). We claim that the coalgebra map < h, ∆ >: IR ω → IR × IR ω σ ↦→ < σ(0), ∆σ > is observational for IR ω . The unique morphism ϕ : (IR ω , < h, ∆ >) → (IR ω , < h, t >) 6is given by ϕ(σ) = ((∆ (0) σ... |

18 | Conditional Circular Coinductive Rewriting with Case Analysis
- Goguen, Lin, et al.
- 2003
(Show Context)
Citation Context ...e has at 2most one argument of hidden sort, hidden algebras can be seen as coalgebras. Our notion of a complete set of cooperations is inspired by the notion of a cobasis in hidden algebra (see e.g. =-=[15, 14, 8]-=-) and our definition scheme in Section 6 has some similarity with the one in [3]. Throughout the paper we will tell the reader precisely which of our notions and results are instances of similar ones ... |

13 | Productivity of Stream Definitions
- Endrullis, Grabmayer, et al.
- 2007
(Show Context)
Citation Context ...ve been designed explicitly for these cases put less restrictions on the format of a “correct” definition. We are thinking, for example, of the recent work on defining streams and stream functions in =-=[7]-=- where techniques from (infinite) term rewriting are employed. At the moment we are working on making our definition scheme more liberal, mainly by using a refined induction argument for defining the ... |

7 |
A coalgebraic equational approach to specifying observational structures
- Cîrstea
- 2002
(Show Context)
Citation Context ...ional coalgebra and then motivate it with various examples. After that, in Section 4, we provide a simpler, syntactic version of the notion of an observational coalgebra by using the terminology from =-=[5]-=- of a cosignature and of a cooperation. We call a collection of cooperations complete for some set if it turns this set into an observational coalgebra. After having defined these notions we turn to t... |

5 | Behavioural differential equations and coinduction for binary trees
- Silva, Rutten
- 2007
(Show Context)
Citation Context ...ote by nat : 2 ∗ → IN the function that maps a binary code to the corresponding natural number. By convention we put nat(ɛ) := 0. The following is a well-known fact from universal coalgebra (see e.g. =-=[18]-=-). Fact 24 Define h : A2∗ → A by h(t) := t(ɛ), l : A2∗ → A2∗ by l(t) := t0 and r : A2∗ → A2∗ by r(t) := t1. The set A2∗ together with the map 〈h, l, r〉 : A2∗ → A × A2∗ × A2∗ form a final coalgebra for... |

4 | On Minimal Coalgebras
- Gumm
- 2008
(Show Context)
Citation Context ...gebra and, under the condition that the final coalgebra for the functor G : Set → Set exists, observational G-coalgebras are exactly the simple G-coalgebras from [16] or the minimal G-coalgebras from =-=[10]-=-. The novelty of our work lies in the fact, that we focus on the various observational coalgebra structures that turn a given set into an observational coalgebra. In order to motivate this definition ... |

3 | Observational coalgebras and complete sets of co-operations
- Kupke, Rutten
(Show Context)
Citation Context ...s. We conclude our paper in Section 7 by linking our research to related work, in particular, to the field of hidden algebra, and by the discussion of future work. The paper is an extended version of =-=[12]-=-. The main changes are the addition of more examples and a generalisation of the definition scheme from so-called basic cosignatures to arbitrary ones. Related work. There is a close connection betwee... |

2 |
Circular Coinduction. 2001. Short paper at
- Ro¸su, Goguen
(Show Context)
Citation Context ...e has at 2most one argument of hidden sort, hidden algebras can be seen as coalgebras. Our notion of a complete set of cooperations is inspired by the notion of a cobasis in hidden algebra (see e.g. =-=[15, 14, 8]-=-) and our definition scheme in Section 6 has some similarity with the one in [3]. Throughout the paper we will tell the reader precisely which of our notions and results are instances of similar ones ... |