## Automata and coinduction (an exercise in coalgebra (1998)

Venue: | LNCS |

Citations: | 63 - 16 self |

### BibTeX

@ARTICLE{Rutten98automataand,

author = {J. J. M. M. Rutten},

title = {Automata and coinduction (an exercise in coalgebra},

journal = {LNCS},

year = {1998},

pages = {194--218}

}

### Years of Citing Articles

### OpenURL

### Abstract

The classical theory of deterministic automata is presented in terms of the notions of homomorphism and bisimulation, which are the cornerstones of the theory of (universal) coalgebra. This leads to a transparent and uniform presentation of automata theory and yields some new insights, amongst which coinduction proof methods for language equality and language inclusion. At the same time, the present treatment of automata theory may serve as an introduction to coalgebra.

### Citations

1379 | A Calculus of Communicating Systems - MILNER - 1980 |

677 | Concurrency and automata on infinite sequences - Park - 1981 |

381 | Representation of Events in Nerve Nets and Finite Automata - Kleene - 1956 |

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

280 |
Regular Algebra and Finite Machines
- Conway
- 1971
(Show Context)
Citation Context ...ew. The calculation rules for a-derivatives (Section 5) of regular combinations of languages are well-known, have been reinvented several times, and are originally due to Brzozowski [Brz64] (see also =-=[Con71]-=- and [BS86]). Both Brzozowski's paper [Brz64] and Conway's book ICon71] contain, more generally, many of the ingredients that have been used in the present paper. A well-known way of proving equality ... |

240 | Finite Automata and Their Decision Problems - Rabin, Scott - 1959 |

232 | An introduction to (co)algebra and (co)induction - Jacobs, Rutten - 2011 |

221 | Derivatives of regular expressions - Brzozowski - 1964 |

188 | A completeness theorem for Kleene algebras and the algebra of regular events
- Kozen
- 1994
(Show Context)
Citation Context ...be new. The definition of simulation up-to-similarity is a straightforward variation of Milner's notion of bisimulation up-to-bisimilarity [Mil80]. Some of the laws of Section 11 have been taken from =-=[Koz94]-=-, where a complete axiom system for equality of regular expressions is presented in terms of equational implications. The treatment of partial automata, which are coalgebras of the set functor D'(S) =... |

163 | A final coalgebra theorem - Aczel, Mendler - 1989 |

100 | From regular expressions to deterministic automata - Berry, Sethi - 1986 |

79 | Two complete axiom systems for the algebra of regular events - Salomaa - 1966 |

74 | Linear Automaton Transformations - Nerode - 1959 |

60 | Vicious Circles. On the Mathematics of non-well-founded phenomena - Barwise, Moss - 1996 |

50 | Taxonomies and toolkits of regular language algorithms - Watson - 1995 |

47 | On the Foundations of Final Semantics: Non-Standard Sets - Rutten, Turi - 1992 |

46 | Non-well-founded sets. Number 14 - Aczel - 1988 |

33 |
Realization is Universal
- Goguen
- 1972
(Show Context)
Citation Context ...at the end of Section 6). The question whether this can lead to (more) efficient algorithms is yet to be addressed. The connection between finality and minimality in Section 7 can already be found in =-=[Gog73]-=-. Our formulation of Kleene's theorem in Section 8 and its use as a criterion for nonregularity in Section 9 may be new, though the proofs involved are of course built from well-known ingredients. Cla... |

27 |
Control Flow Semantics. Foundations of Computing Series
- Bakker, Vink
- 1996
(Show Context)
Citation Context ... D ~, consists of metrically closed subsets. Such sets have been used at various places in the work of the French and Dutch schools of Nivat and De Bakker on metric semantics (cf. the recent textbook =-=[BV96]-=-). The notion of consistent language corresponds to the notion of reduced set in [dB91]. Automata theory has been and still is commonly understood as essentially algebraic. Cf. Ginzburg's Algebraic th... |

17 |
Comparative semantics for flow of control in logic programming without logic
- Bakker
- 1991
(Show Context)
Citation Context ... in the work of the French and Dutch schools of Nivat and De Bakker on metric semantics (cf. the recent textbook [BV96]). The notion of consistent language corresponds to the notion of reduced set in =-=[dB91]-=-. Automata theory has been and still is commonly understood as essentially algebraic. Cf. Ginzburg's Algebraic theory of automata [Gin68], Conway's Regular algebra and finite machines [Con711, and Koz... |

9 | A tutorial on (co)algebras and (co)induction. Bulletin of the European Association for Theoretical Computer Science 62 - Jacobs, Rutten - 1997 |

6 | Automata and computability. Undergraduate texts in computer science - Kozen - 1997 |

6 |
Non-Well-Founded Sets. Number 14 in csli Lecture Notes. csli
- aczel
- 1988
(Show Context)
Citation Context ...ulation rules for a-derivatives (Section 5) of regular combinations of languages are well-known, have been reinvented several times, and are originally due to Brzozowski [Brz64] (see also [Con71] and =-=[BS86]-=-). Both Brzozowski's paper [Brz64] and Conway's book ICon71] contain, more generally, many of the ingredients that have been used in the present paper. A well-known way of proving equality of regular ... |

4 |
Available at URL: www.elsevier.nl/locate/entcs
- Jacobs, Moss, et al.
- 1998
(Show Context)
Citation Context ...ons of the present paper are instances of far more general ones, belonging to a theory called (universal) coalgebra. See [Rut96, JR97] and the references therein for more information on coalgebra. In =-=[JMRR98]-=-, many recent developments in coalgebra are described. The coalgebraic definition of bisimulation is a categorical generalization, due to Aczel and Mendler [AM89], of Park's [Par81] and Milner's [Mil8... |

2 | Terminal metric spaces of finitely branching and image finite linear processes. Theoretical Computer Science - Breugel - 1998 |

1 | Products of automata, volume 7 - G'ecseg - 1986 |

1 |
Algebraic theory of automata. ACM Monograph series
- Ginzburg
- 1968
(Show Context)
Citation Context ...consistent language corresponds to the notion of reduced set in [dB91]. Automata theory has been and still is commonly understood as essentially algebraic. Cf. Ginzburg's Algebraic theory of automata =-=[Gin68]-=-, Conway's Regular algebra and finite machines [Con711, and Kozen's recent textbook Automata and computability, from which the following quotation is taken [Koz97, p. 112]: "It should be pretty appare... |

1 | Products of automata, volume 7 - Gcseg - 1986 |