## From Set-theoretic Coinduction to Coalgebraic Coinduction: some results, some problems (1999)

Citations: | 15 - 0 self |

### BibTeX

@MISC{Lenisa99fromset-theoretic,

author = {Marina Lenisa},

title = { From Set-theoretic Coinduction to Coalgebraic Coinduction: some results, some problems},

year = {1999}

}

### OpenURL

### Abstract

### Citations

1000 | A calculus of mobile processes
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- 1992
(Show Context)
Citation Context ...h have a problematic coalgebraic description are those where "side-conditions" depending on the structure of the objects to be related appear. Both early and late bisimulations in Milner's -=-=calculus ([MPW92]-=-), are of this form. Also in this case, like in the previous example with quantifiers, we lack a uniform description. Luckily, in the -calculus case, it is still possible to get rid of the local side-... |

657 | Concurrency and Automata on Infinite Sequences - Park - 1981 |

315 |
Calculi for Synchrony and Asynchrony
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- 1983
(Show Context)
Citation Context ...st, we introduce various generalizations of the classical set-theoretical coinduction principle based on Tarski Fixpoint Theorem, which go in the direction of Milner's bisimulation "up-to" p=-=rinciple ([Mil83]). We call these pro-=-of principles coinduction principles "up-to". The practical interest of coinduction principles "up-to" lies in the fact that they allow to factorize the complexity of proofs by coi... |

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

201 | Lectures on Non-Well-Founded Sets - Aczel - 1987 |

169 | The category-theoretic solution of recursive domain equations - Smyth, Plotkin - 1982 |

162 |
A final coalgebra theorem
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Citation Context ..., in the sense that the two notions of F -bisimulations characterize the same equivalence (see Theorem 7 below), although in effect they give rise to different coinduction principles. 9 Definition 9 (=-=[AM89]-=-) Let F : C S ! C S . An F -bisimulation on the F -coalgebras (X; ff X ) and (Y; ff Y ) is a set-theoretic relation R ` X \Theta Y such that there exists an arrow of C, fl :R! F (R), making the follow... |

135 | G.: Towards a mathematical operational semantics
- Turi, Plotkin
- 1997
(Show Context)
Citation Context ...morphisms ([Acz88, AM89, RT93, RT94, Rut96, Tur96, TP97, Len98]). In this setting, the categorical counterparts of set-theoretic bisimulations are F -bisimulations, i.e. spans of coalgebra morphisms (=-=[TP97]-=-). One of the advantages of a categorical description is that we can deal uniformly with coinductively defined objects and coiterative morphisms. In fact, the latter arise naturally in a categorical c... |

128 | Terminal coalgebras in well-founded set theory - Barr - 1993 |

105 | Proving congruence of bisimulation in functional programming languages - Howe - 1996 |

84 | Infinite objects in type theory - Coquand - 1993 |

74 | Rutten: Bisimulation for Probabilistic Transition Systems: A Coalgebraic Approach - Vink, M - 1997 |

70 | Codifying guarded definitions with recursive schemes - Giménez - 1994 |

67 | Structural induction and coinduction in a fibrational setting
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Citation Context ...bras of the "component" functors of F , in such a way that the operator \Phi F (X;ff);(Y;fi) is obtained by "composing" the operators induced by the component functors. This is rel=-=ated to the work of [HJ98]-=- in the fibrational setting. We start by specifying a class Fun of covariant functors. The functors which we consider involve the constructors which are normally used for defining final semantics, i.e... |

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

55 | Initial Algebra and Final Coalgebra Semantics for Concurrency, Report CS-R9409 - Rutten, Turi - 1994 |

54 | Set theory with free construction principles. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 4 e série - Forti, Honsell - 1983 |

53 | Functorial Operational Semantics and its Denotational Dual - Turi - 1996 |

51 | Inductive and Coinductive types with Iteration and Recursion
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Citation Context ...g on equivalences induced by T -coiterative morphisms. I.e. morphisms into final coalgebras defined by the T -coiteration scheme. This generalizes both the coiteration and the corecursion scheme (see =-=[Geu92]-=-). The latter is dual to the (primitive) recursion scheme. The T -coiteration scheme allows to capture many interesting functions into final coalgebras, which escape the pure coiteration scheme (see S... |

43 | Control flow semantics - Bakker - 1995 |

37 | A coinduction principle for recursive data types based on bisimulation - Fiore - 1996 |

36 | From operational semantics to domain theory - Mason, Smith, et al. - 1996 |

16 |
Concurrent semantics for the -calculus
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Citation Context ...previous example with quantifiers, we lack a uniform description. Luckily, in the -calculus case, it is still possible to get rid of the local side-conditions in the definitions of bisimulations (see =-=[HLMP98]), thereby making po-=-ssible a coalgebraic description. This latter situation seems related to the difficulty of obtaining a "generalized minimal automata". 4.2.2 Coinduction "up-to". In this paper, we ... |

15 | Final semantics for a higher order concurrent language - Lenisa - 1996 |

11 | On the Foundations of Final Semantics - Rutten - 1993 |

10 | Final semantics for untyped lambdacalculus - Honsell, Lenisa - 1995 |

10 | Objects and classes, co-algebraically, Object-Orientation with Parallelism and Persistence - Jacobs - 1996 |

9 | Semantic Techniques for Deriving Coinductive Characterizations of Observational Equivalences for -calculi - Lenisa |

9 | A note on logical relations between semantics and syntax - Pitts - 1997 |

8 | Bisimulation in untyped lambda calculus: Böhm trees and bisimulation up to context - Lassen - 1999 |

6 | Behaviour-refinement of object-oriented specifications with coinductive correctness proofs - Jacobs - 1996 |

5 | and Classes, co-algebraically, Object-Orientation with Parallelism and Book Persistence, B.Freitag et - Objects - 1996 |

5 | On the Proof Method for Bisimulation - Sangiorgi - 1995 |

4 | of object-oriented specifications with coinductive correctness proofs - Behaviour-refinement - 1997 |

4 | A uniform syntactical method for proving coinduction principles in -calculi - Lenisa - 1997 |

4 | Types and Partial Functions, Post-Graduate Lecture Notes - Plotkin - 1985 |

3 | Coinductive Characterizations on Applicative Structures
- Honsell
(Show Context)
Citation Context ...retic coinduction principles induced by the operators \Phi i , it is not at all clear how to derive a coalgebraic description of the coinduction principle induced by \Phi. A similar example occurs in =-=[HL99] for the c-=-ase of a generalized applicative coinduction principle for -calculus. Other examples of bisimulations which have a problematic coalgebraic description are those where "side-conditions" depen... |

3 | Additions and corrections to "terminal coalgebras in well-founded set theory - Barr - 1994 |

3 | Themes in Final Semantics, Ph.D. Thesis TD-6/98, Dipartimento di Informatica, Universit`a di - Lenisa - 1998 |

2 | Coalgebraic Logic, to appear - Moss |

2 |
On the bisimulation proof method, Math. Struct
- Sangiorgi
- 1998
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Citation Context ...up-to-T ", and which subsumes the principle of Coinduction "up-to-[". We give two versions of the principle of Coinduction "up-to-T ". The first is an immediate modification o=-=f Sangiorgi's principle ([San98]-=-) for labelled transition systems, and it is a sound (but not in general complete) principle. The latter is obtained from the previous one by strengthening the conditions on the operator T , in order ... |

1 | A Complete Coinductive Logical System for Bisimulation Equivalence on Circular Objects - Lenisa |

1 | and Partial Functions, Post-Graduate Lecture Notes - Types - 1985 |

1 |
A Coalgebraic Foundation for Linear Time
- Power
- 1999
(Show Context)
Citation Context ...or an appropriate T . Similarly, also a trace semantics can be easily given by P( )-coiteration, where P( ) is the non-deterministic choice monad. This semantics has to be compared to that defined in =-=[PT99], where a -=-trace semantics is given by coiteration, by exploiting a distributivity property between an appropriate "behaviour functor" and the monad P( ). More generally, the investigation carried out ... |

1 | Final Semantics for the ss-calculus, PROCOMET'98 Conf - Honsell, Montanari - 1998 |

1 | for synchrony and asynchrony, TCS 25 - Calculi - 1983 |