## A Structural Co-Induction Theorem (1993)

Venue: | PROC. MFPS '93, SPRINGER LNCS 802 |

Citations: | 7 - 1 self |

### BibTeX

@INPROCEEDINGS{Rutten93astructural,

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

title = {A Structural Co-Induction Theorem},

booktitle = {PROC. MFPS '93, SPRINGER LNCS 802},

year = {1993},

pages = {83--102},

publisher = {Springer-Verlag}

}

### Years of Citing Articles

### OpenURL

### Abstract

The Structural Induction Theorem (Lehmann and Smyth, 1981; Plotkin, 1981) characterizes initial F-algebras of locally continuous functors F on the category of cpo's with strict and continuous maps. Here a dual of that theorem is presented, giving a number of equivalent characterizations of final coalgebras of such functors. In particular, final coalgebras are order strongly-extensional (sometimes called internal full abstractness): the order is the union of all (ordered) F-bisimulations. (Since the initial fixed point for locally continuous functors is also final, both theorems apply.) Further a similar co-induction theorem is given for a category of complete metric spaces and locally contracting functors.

### Citations

3218 |
Communication and Concurrency
- Milner
- 1989
(Show Context)
Citation Context ...of the bisimilarity relation on a labelled transition system. It is defined as the greatest fixed point of a monotone function on the lattice of relations on the states of this transition system (see =-=[Mil89]-=-). An example of the above co-induction proof principle can be found in [MT91], where it is used to prove the consistency of the static and the dynamic semantics of a simple functional programming lan... |

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

219 | Categories for the working mathematician, volume 5 of Graduate Texts in Mathematics - Lane, S - 1998 |

169 |
The category-theoretic solution of recursive domain equations
- Smyth, Plotkin
- 1982
(Show Context)
Citation Context ...le mentioned above. In particular, locally continuous (endo-)functors on the category of complete partial orders will be investigated. These functors are well-known to have an initial F -algebra (see =-=[SP82]-=-), which is at the same time a final F -coalgebra. A structural co-induction theorem will be proved, giving a number of equivalent characterizations for such final F -coalgebras. Maybe the most surpri... |

162 |
A final coalgebra theorem
- Aczel, Mendler
- 1989
(Show Context)
Citation Context ...o-called order strong-extensionality, stating that two elements are ordered if and only if they are related by a so-called ordered bisimulation. Order-bisimulations generalize the F -bisimulations of =-=[AM89]-=-, which at their turn are categorical abstractions of the notion of bisimulation of [Par81, Mil89]. In the present paper, the definition of ordered bisimulation from [Fio93] is used, which generalizes... |

120 |
A domain equation for bisimulation
- Abramsky
- 1991
(Show Context)
Citation Context ...died.) The following definition generalizes the notion of strong extensionality used in [Acz88] (in the context of non-well-founded set theory). It is sometimes called internal full abstractness (cf. =-=[Abr91]-=-). 4. Ordered F -bisimulation 5 Definition 4.2 Let (A; ff) be an F -coalgebra, and let vA be the order on A. Let v F ` A \Theta A be defined by v F j [ fR ` A \Theta A j R is an ordered F -bisimulatio... |

77 |
Co-induction in relational semantics
- Milner, Tofte
- 1991
(Show Context)
Citation Context ...the greatest fixed point of a monotone function on the lattice of relations on the states of this transition system (see [Mil89]). An example of the above co-induction proof principle can be found in =-=[MT91]-=-, where it is used to prove the consistency of the static and the dynamic semantics of a simple functional programming language with recursive functions. By generalizing preorders to categories C and ... |

68 |
Solving reflexive domain equations in a category of complete metric spaces
- America, Rutten
- 1989
(Show Context)
Citation Context ...ctor F on CMS is locally contracting if there exists ffl with 0sffl ! 1 such that, for all D;E, the mapping FD;E is a contraction with factor ffl. In [RT93], it is shown (extending earlier results of =-=[AR89]-=-) that every locally contracting functor F has a unique fixed point which is both an initial F -algebra and a final F -coalgebra. A `metric version' of Theorem 5.1 is obtained by dropping---both in th... |

60 |
Post-graduate lecture notes in advanced domain theory
- Plotkin
- 1981
(Show Context)
Citation Context ...eralizes the original definition from [RT93] by the use of lax-homomorphisms. The co-induction theorem (Section 5) is presented as and named after a dualization of the structural induction theorem of =-=[Plo81]-=- (but see also [LS81]), which is repeated here in the Appendix. Part of this dualization is fairly straightforward; order strong-extensionality, however, does not arise as the dual of the structural i... |

54 |
Set theory with free construction principles. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 4 e série
- Forti, Honsell
- 1983
(Show Context)
Citation Context ...the collection of its compact subsets (note the presence of the `metric shrinker' id ffl , though), it is an instance of a hyperuniverse. (See [FH92] for a general construction of hyperuniverses, and =-=[FH83]-=- and [Acz88] for a hyperuniverse based on a non-standard collection of axioms. Cf. [Abr88, MMO89, Rut91].) By putting, for p; p 0 2 H, p 0 2H p j p 0 2 fl(p); H can be easily seen to contain all so-ca... |

47 | On the Foundations of Final Semantics: Non-Standard Sets
- Rutten, Turi
- 1992
(Show Context)
Citation Context ...gorical abstractions of the notion of bisimulation of [Par81, Mil89]. In the present paper, the definition of ordered bisimulation from [Fio93] is used, which generalizes the original definition from =-=[RT93]-=- by the use of lax-homomorphisms. The co-induction theorem (Section 5) is presented as and named after a dualization of the structural induction theorem of [Plo81] (but see also [LS81]), which is repe... |

46 |
Non-well-founded sets. Number 14
- Aczel
- 1988
(Show Context)
Citation Context ...ion of its compact subsets (note the presence of the `metric shrinker' id ffl , though), it is an instance of a hyperuniverse. (See [FH92] for a general construction of hyperuniverses, and [FH83] and =-=[Acz88]-=- for a hyperuniverse based on a non-standard collection of axioms. Cf. [Abr88, MMO89, Rut91].) By putting, for p; p 0 2 H, p 0 2H p j p 0 2 fl(p); H can be easily seen to contain all so-called heredit... |

40 | A co-inducion principle for recursively defined domains
- Pitts
- 1994
(Show Context)
Citation Context ... such that fi 1sfi 2 and boths1 ands2 are coalgebra homomorphisms. The latter can be seen to be a special instance of the definition given above by taking fi j fi 2 . (Cf. the notion of simulation in =-=[Pit92]-=-; see also [Pit93], where proof principles that combine induction and co-induction are studied.) The following definition generalizes the notion of strong extensionality used in [Acz88] (in the contex... |

37 | A coinduction principle for recursive data types based on bisimulation
- Fiore
- 1996
(Show Context)
Citation Context ...ize the F -bisimulations of [AM89], which at their turn are categorical abstractions of the notion of bisimulation of [Par81, Mil89]. In the present paper, the definition of ordered bisimulation from =-=[Fio93]-=- is used, which generalizes the original definition from [RT93] by the use of lax-homomorphisms. The co-induction theorem (Section 5) is presented as and named after a dualization of the structural in... |

37 |
Algebraic Specification of Data Types: A Synthetic Approach
- Lehmann, Smyth
- 1981
(Show Context)
Citation Context ...definition from [RT93] by the use of lax-homomorphisms. The co-induction theorem (Section 5) is presented as and named after a dualization of the structural induction theorem of [Plo81] (but see also =-=[LS81]-=-), which is repeated here in the Appendix. Part of this dualization is fairly straightforward; order strong-extensionality, however, does not arise as the dual of the structural induction principle fo... |

26 |
General Topology, volume 6
- Engelking
- 1989
(Show Context)
Citation Context ...verse. Let CMS be the category with (1-bounded) complete metric spaces (D; dD ) as objects and nonexpansive (non-distance-increasing) functions as arrows. (For basic facts on metric spaces see, e.g., =-=[Eng89]-=-.) Hom sets in CMS are themselves complete metric spaces, using as a metric on arrows the usual pointwise extension. A functor F on CMS is locally contracting if there exists ffl with 0sffl ! 1 such t... |

21 | A cook’s tour of the finitary non-wellfounded sets
- Abramsky
(Show Context)
Citation Context ... easily seen to contain all so-called hereditarily finite sets and their limits (with respect to the metric on H). Note that these limits need not be hereditarily finite themselves. As pointed out in =-=[Abr88]-=-, the standard axioms of set theory hold in H, with topological versions of separation, replacement and choice. By (the metric version of) Theorem 5.1, strong extensionality can be added to these axio... |

15 | Relational properties of recursively defined domains
- Pitts
- 1993
(Show Context)
Citation Context ... 2 and boths1 ands2 are coalgebra homomorphisms. The latter can be seen to be a special instance of the definition given above by taking fi j fi 2 . (Cf. the notion of simulation in [Pit92]; see also =-=[Pit93]-=-, where proof principles that combine induction and co-induction are studied.) The following definition generalizes the notion of strong extensionality used in [Acz88] (in the context of non-well-foun... |

5 |
A general construction of hyperuniverses
- Forti, Honsell
- 1992
(Show Context)
Citation Context ...onclusion 14 Because the metric space H is isomorphic to the collection of its compact subsets (note the presence of the `metric shrinker' id ffl , though), it is an instance of a hyperuniverse. (See =-=[FH92]-=- for a general construction of hyperuniverses, and [FH83] and [Acz88] for a hyperuniverse based on a non-standard collection of axioms. Cf. [Abr88, MMO89, Rut91].) By putting, for p; p 0 2 H, p 0 2H p... |

5 |
I-categories and duality
- Smyth
- 1992
(Show Context)
Citation Context ...tors. For instance, let (A; ff) and (B; fi) be F -coalgebras, and suppose that (A; ff) is final. The following can be easily proved. For anys: (A; ff) ! (B; fi): ifsis epi thensis an isomorphism (cf. =-=[Smy92]-=-). Note that this generalizes the fact that for an ordered set (P; ) and a monotone function f : P ! P : if p; q 2 P , with p the greatest post-fixed point of f and qsp, then qsf(q) implies p = q---an... |

2 | Epis need not to be dense - Lehmann, Pasztor - 1982 |

2 |
Some notes on recursive domain equations. Handwritten notes for the Domain Theory PG Course
- Plotkin
- 1991
(Show Context)
Citation Context ...93]. (The proof given there makes explicit use of the way D is constructed (as the projective limit of its defining !-chain).) The equivalence of finality and maximal-finality ((1) and (5)) is due to =-=[Plo91]-=-. The main contribution of the above theorem is the proof of (3) ) (2), showing---for functors that weakly preserve ordered kernel pairs---that coalgebras are final if they are strongly extensional. M... |

1 | Non-well-founded sets obtained from ideal fixed points - Mislove, Moss, et al. - 1989 |

1 | Hereditarily-finite sets and complete metric spaces - Rutten - 1991 |