## State Based Systems Are Coalgebras (2003)

Venue: | Cubo - Matematica Educacional 5 |

Citations: | 2 - 0 self |

### BibTeX

@INPROCEEDINGS{Gumm03statebased,

author = {H. Peter Gumm},

title = {State Based Systems Are Coalgebras},

booktitle = {Cubo - Matematica Educacional 5},

year = {2003},

pages = {239--262}

}

### OpenURL

### Abstract

Universal coalgebra is a mathematical theory of state based systems, which in many respects is dual to universal algebra. Equality must be replaced by indistinguishability. Coinduction replaces induction as a proof principle and maps are defined by co-recursion. In this (entirely self-contained) paper we give a first glimpse at the general theory and focus on some applications in Computer Science. 1. State based systems State based systems can be found everywhere in our environment -- from simple appliances like alarm clocks and answering machines to sophisticated computing devices. Typically, such systems receive some input and, as a result, produce some output. In contrast to purely algebraic systems, however, the output is not only determined by the input received, but also by some modifiable "internal state". Internal states are usually not directly observable, so there may as well be di#erent states that cannot be distinguished from the input-output behavior of the system. A simple example of a state based system is a digital watch with several buttons and a display. Clearly, the buttons that are pressed do not by themselves determine the output - it also depends on the internal state, which might include the current time, the mode (time/alarm/stopwatch), and perhaps the information which buttons have been pressed previously. The user of a system is normally not interested in knowing precisely, what the internal states of the system are, nor how they are represented. Of course, he might try to infer all possible states by testing various input-output combinations and attribute di#erent behaviors to di#erent states. Some states might not be distinguishable by their outside behavior. It is therefore natural to define an appropriate indistinguishability relation "#...

### Citations

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Citation Context ...etical notion, using arbitrary Set-functors as types, was considered by Aczel and Mendler[AM89] and Barr[Bar93]. A comprehensive structure theory of universal coalgebra was formulated by J. Rutten in =-=[Rut00] for-=- type functors “weakly preserving pullbacks”. In [Gum99a] the theory was generalized and extended to work with arbitrary type functors. The structure theoretic effect of the (weak) preservation co... |

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Citation Context ...ded and, most of all, it was lacking any reasonable applications. The more useful category theoretical notion, using arbitrary Set-functors as types, was considered by Aczel and Mendler[AM89] and Barr=-=[Bar93]. A -=-comprehensive structure theory of universal coalgebra was formulated by J. Rutten in [Rut00] for type functors “weakly preserving pullbacks”. In [Gum99a] the theory was generalized and extended to... |

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Citation Context ...ϑ M (ε ∈ L ⇐⇒ ε ∈ M), ∀e ∈ Σ. (Le ϑ Me) Hence in order to show that two languages L1 and L2 are equal, we need to find a relation R, containing (L1, L2), and satisfying the above cond=-=ition. J. Rutten [Rut98] demo-=-nstrates how to prove regular language equations by coinduction. For instance, in order to show that for each language L, (1 + L · L ∗ ) = L ∗ ,sSTATE BASED SYSTEMS ARE COALGEBRAS 13 it suffices ... |

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Citation Context ... Le · L )e = ∗ , if ε /∈ L Le · L∗ + Le · L∗ , if ε ∈ L, = Le · L ∗ = (L ∗ )e. 3.4. Existence of terminal coalgebras. The terminal Φ-Kripke structure cannot exist due to the follo=-=wing lemma of Lambek [Lam68]. -=-Its base set T would have to be in bijective correspondence with P(Φ) × P(T ), which is impossible, since P(T ) has strictly larger cardinality than T for any set T : Lemma 3.4. If the terminal coal... |

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Citation Context ...nsidered by Aczel and Mendler[AM89] and Barr[Bar93]. A comprehensive structure theory of universal coalgebra was formulated by J. Rutten in [Rut00] for type functors “weakly preserving pullbacks”.=-= In [Gum99a]-=- the theory was generalized and extended to work with arbitrary type functors. The structure theoretic effect of the (weak) preservation conditions, as assumed in [AM89] and [Rut00], was characterized... |

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Citation Context ...fruit in computer science, remains to be seen. So far, it is well recognized that many data types are coalgebraic in nature and that co-recursive specification and verification methods and tools (see =-=[HHJT98]-=-) are appropriate to deal with them.s16 H. PETER GUMM 4.1. Historical note. The earliest papers on coalgebras defined them as straightforward dualizations of classical universal algebras [Drb69], i.e.... |

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Citation Context ...n69], one can also prove that subcoalgebras are closed under finite intersections, hence the (carrier sets of) all subcoalgebras of A are the open sets of a topology on A, see [GS00b]. Conversely, by =-=[Gum01b]-=-, every topology on a set A can be obtained this way. 2.5. Bisimulations. Bisimulations are the compatible relations between coalgebras. Their importance for computer science applications had been rea... |

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Citation Context ... theory was generalized and extended to work with arbitrary type functors. The structure theoretic effect of the (weak) preservation conditions, as assumed in [AM89] and [Rut00], was characterized in =-=[GS00a]. -=-L. Moss has introduced in [Mos99a], see also [Mos99b], a modal logic for coalgebras whose type functor weakly preserve pullbacks. The first Birkhoff characterization was given in [Gum99b] – the synt... |

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Citation Context ... a result of Trnkovà [Trn69], one can also prove that subcoalgebras are closed under finite intersections, hence the (carrier sets of) all subcoalgebras of A are the open sets of a topology on A, see=-= [GS00b]-=-. Conversely, by [Gum01b], every topology on a set A can be obtained this way. 2.5. Bisimulations. Bisimulations are the compatible relations between coalgebras. Their importance for computer science ... |

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Citation Context ...]). The elements of the final D × (−) M -coalgebra can be understood as infinite M-branching and D-labeled trees, so co-equations can actually be represented as equivalence classes of such trees (s=-=ee [Gum01a]-=-). Whether these further mathematical investigations will bear fruit in computer science, remains to be seen. So far, it is well recognized that many data types are coalgebraic in nature and that co-r... |

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Citation Context ...d to work with arbitrary type functors. The structure theoretic effect of the (weak) preservation conditions, as assumed in [AM89] and [Rut00], was characterized in [GS00a]. L. Moss has introduced in =-=[Mos99a], -=-see also [Mos99b], a modal logic for coalgebras whose type functor weakly preserve pullbacks. The first Birkhoff characterization was given in [Gum99b] – the syntactical side was added in [Gum01a]. ... |

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Citation Context ... long before coalgebras were introduced in this field. Intuitively, two states of a system are bisimilar, if they show the same behavior. The coalgebraic definition was introduced by Aczel and Mendler=-=[AM89]: Definition-=- 2.1. A bisimulation between coalgebras A and B is a binary relation R ⊆ A × B, on which a coalgebra structure ρ : R → F (R) can be defined, making the projections πA : R → A and πB : R → ... |

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1 |
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Citation Context ...(see [HHJT98]) are appropriate to deal with them.s16 H. PETER GUMM 4.1. Historical note. The earliest papers on coalgebras defined them as straightforward dualizations of classical universal algebras =-=[Drb69], i.-=-e. a coalgebra was a set A with a collection of maps αi : A → ni · A into the ni-fold direct sum of A. However, this notion was too simple minded and, most of all, it was lacking any reasonable ap... |