## Continuous functions on final coalgebras (2007)

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Citations: | 9 - 1 self |

### BibTeX

@MISC{Ghani07continuousfunctions,

author = {Neil Ghani and Peter Hancock and Dirk Pattinson},

title = {Continuous functions on final coalgebras},

year = {2007}

}

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### Abstract

In a previous paper we have given a representation of continuous functions on streams, both discrete-valued functions, and functions between streams. the topology on streams is the ‘Baire ’ topology induced by taking as a basic neighbourhood the set of streams that share a given finite prefix. We gave also a combinator on the representations of stream processing functions that reflects composition. Streams are the simplest example of a non-trivial final coalgebras, playing in the coalgebraic realm the same role as do the natural numbers in the algebraic realm. Here we extend our previous results to cover the case of final coalgebras for a broad class of functors generalising (×A). The functors we deal with are those that arise from countable signatures of finiteplace untyped operators. These have many applications. The topology we put on the final coalgebra for such a functor is that induced by taking for basic neighbourhoods the set of infinite objects which share a common prefix, according to the usual definition of the final coalgebra as the limit of a certain inverse chain starting at �. 1

### Citations

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(Show Context)
Citation Context ... covering structure of neighbourhoods in the final coalgebra. 4 One connection between streams and choice sequences may lie in the idea that (in the angel/demon terminology of the refinement calculus =-=[BvW98]-=-) inputs are chosen by the demon, while the outputs are chosen by the angel. However, choice sequences in intuitionism can involve choices to obtain further terms using a continuous function: this see... |

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Citation Context ...n this situation needs careful analysis based on the universal properties of initial algebras and final coalgebras. Provenance The basic idea for this analysis of continuity emerged from expositions (=-=[Dum00]-=-, [TvD88], [KT70], [ML65]) of Brouwer’s notion of choice sequence and associated principles of continuity and Bar induction. There are many notions akin to choice sequence (lawless sequence for exampl... |

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Citation Context ... Xn for some stream of sets C. . These functors include the normal functors of Girard [Gir88], but our representation does not (seem to) extend throughout the more general class of analytic functors (=-=[Joy86a]-=-, [Has02]). The essential points are that the final coalgebra of the functor should be obtained as the projective limit of the ω-sequence 1 ← F (1) ← F 2 (1) ← . . . starting with a terminal object 1.... |

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Citation Context ...) XP (s) for some family { P (s) } s∈S of finite sets. Such a functor is a prime example of a container in the sense of [AAG04]. We make heavy use of containers, and appeal to the results of [AAG03], =-=[AAG05]-=- and [], primarily the closure of containers under various operations, such as constants, +, ×, ·, Σ, Π, and 2sleast and greatest fixed points. The container technology seems to be much cleaner than t... |

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Citation Context ...at the second component of the operand family is used in the definition of the first component of the output family. For an explanation of the principles underlying induction recursion see the papers =-=[DS06]-=- [DS03] by Dybjer and Setzer and the references therein. The constructors and their associated decoding functions are as follows: • : S∗ P∗(•) ∆ = � (�) : (Π s : S∗) (P∗(s) → S) → S∗ P∗(s � σ) ∆ = (Σ ... |

40 | Categories of containers - Abbott, Altenkirch, et al. - 2003 |

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Citation Context ...second component of the operand family is used in the definition of the first component of the output family. For an explanation of the principles underlying induction recursion see the papers [DS06] =-=[DS03]-=- by Dybjer and Setzer and the references therein. The constructors and their associated decoding functions are as follows: • : S∗ P∗(•) ∆ = � (�) : (Π s : S∗) (P∗(s) → S) → S∗ P∗(s � σ) ∆ = (Σ p : P∗(... |

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Citation Context ...s and permutations), yet the finite power-set operator is not a (plain) container. (If there is quotienting, we cannot assume that the terminal sequence becomes an isomorphism at ω. As pointed out by =-=[Wor99]-=-, it may take ω + ω steps to reach a final coalgebra: while a weakly final coalgebra is obtained at stage ω, a further ω steps are may be required to whittle it down (by quotienting) to obtain a uniqu... |

15 |
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Citation Context ...s on the category Set, including besides the functor (×A), such functors as X ↦→ 1 + 2 × X + X2 or X ↦→ � n∈ω Cn × Xn for some stream of sets C. . These functors include the normal functors of Girard =-=[Gir88]-=-, but our representation does not (seem to) extend throughout the more general class of analytic functors ([Joy86a], [Has02]). The essential points are that the final coalgebra of the functor should b... |

15 |
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Citation Context ...me stream of sets C. . These functors include the normal functors of Girard [Gir88], but our representation does not (seem to) extend throughout the more general class of analytic functors ([Joy86a], =-=[Has02]-=-). The essential points are that the final coalgebra of the functor should be obtained as the projective limit of the ω-sequence 1 ← F (1) ← F 2 (1) ← . . . starting with a terminal object 1. In that ... |

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Citation Context ...2X The finite powerset functor. 6 There is in fact an extended notion of container, which assigns to a shape s not only a set of positions P (s), but a sub-group of the group of permutations of P (s) =-=[AAGM04]-=-. 8sThe first example involves non-trivial quotienting. Regarding the second example, though the expression 22X is positive in X, it is not strictly positive. Although the functor which assigns to a s... |

11 |
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Citation Context ...eeds careful analysis based on the universal properties of initial algebras and final coalgebras. Provenance The basic idea for this analysis of continuity emerged from expositions ([Dum00], [TvD88], =-=[KT70]-=-, [ML65]) of Brouwer’s notion of choice sequence and associated principles of continuity and Bar induction. There are many notions akin to choice sequence (lawless sequence for example), and the conne... |

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(Show Context)
Citation Context ...at are of finite arity. In other words, that the functor be of the form (Σ s ∈ S) XP (s) for some family { P (s) } s∈S of finite sets. Such a functor is a prime example of a container in the sense of =-=[AAG04]-=-. We make heavy use of containers, and appeal to the results of [AAG03], [AAG05] and [], primarily the closure of containers under various operations, such as constants, +, ×, ·, Σ, Π, and 2sleast and... |

6 | On continuous normalization
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Citation Context ...h a sort-structure, the source of this aroma can be investigated. There may be connections here with continuous cut-elimination and continuous normalisation and the repetition rule of Mints: see also =-=[AJ02]-=-. related work There is an enormous amount of work pertaining to stream processing, which is the special case of our construction when the input and output functors have the form (A×) where A is a dis... |

5 |
Categories of containers
- Abbot, Altenkirch, et al.
- 2003
(Show Context)
Citation Context ... (Σ s ∈ S) XP (s) for some family { P (s) } s∈S of finite sets. Such a functor is a prime example of a container in the sense of [AAG04]. We make heavy use of containers, and appeal to the results of =-=[AAG03]-=-, [AAG05] and [], primarily the closure of containers under various operations, such as constants, +, ×, ·, Σ, Π, and 2sleast and greatest fixed points. The container technology seems to be much clean... |

5 |
A note on rewriting theory for uniqueness of iteration, Theory and Applications of Categories 6
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(Show Context)
Citation Context ...s available in the programming language Haskell. 3 To require in Gödel’s T that the natural numbers are a (strongly) initial algebra for the functor X ↦→ 1 + X would have disastrous consequences. See =-=[OS99]-=-. 3scomponents may fail to produce values. In such cases, one may need ‘potentially infinitely delayed’ entities (ν X) A + X, potentially finite streams (ν X) A + B × X, or other functors that involve... |

2 | for data: Differentiating data structures. Fundamenta Informaticae - Abbott, Altenkirch, et al. - 2004 |

1 |
Altenkirch and Conor McBride. δ is for data. Fundamenta Informaticae, 2005. Accepted For Publication
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(Show Context)
Citation Context ...e for representing the states of tree-structured storage devices, such as file-systems or linearly-structured stores such as the tape of a Turing machine. Equipped with a cursor (a ‘one-hole context’ =-=[MAM05]-=-) that navigates up and down the tree, one has an updatable store, that can be used for shared-memory communication. A storage device is a particularly well-behaved kind of state-machine, and indeed a... |