## Final coalgebras and the Hennessy-Milner property

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Venue: | Annals of Pure and Applied Logic |

Citations: | 1 - 1 self |

### BibTeX

@ARTICLE{Goldblatt_finalcoalgebras,

author = {Robert Goldblatt},

title = {Final coalgebras and the Hennessy-Milner property},

journal = {Annals of Pure and Applied Logic},

year = {},

volume = {138},

pages = {2005}

}

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### Abstract

The existence of a final coalgebra is equivalent to the existence of a formal logic with a set (small class) of formulas that has the Hennessy-Milner property of distinguishing coalgebraic states up to bisimilarity. This applies to coalgebras of any functor on the category of sets for which the bisimilarity relation is transitive. There are cases of functors that do have logics with the Hennessy-Milner property, but the only such logics have a proper class of formulas. The main theorem gives a representation of states of the final coalgebra as certain satisfiable sets of formulas. The key technical fact used is that any function between coalgebras that is truth-preserving and has a simple codomain must be a coalgebraic morphism.

### Citations

3217 |
Communication and Concurrency
- MILNER
- 1989
(Show Context)
Citation Context ...t the existence of a final coalgebra depends on the existence of a suitable logic whose formulas constitute a set. The Hennessy-Milner program has been applied to many species of process algebra (see =-=[8,9]-=-), and was extended to certain kinds of coalgebra over Set once it was recognised that such coalgebras can be used effectively to model data types and transition systems [1–5]. In that setting, final ... |

494 | Algebraic laws for nondeterminism and concurrency
- Hennessy, Milner
- 1985
(Show Context)
Citation Context ...lgebraic states is transitive. Such transitivity certainly holds in any process-algebraic context for which bisimilarity means observational (or behavioural) indistinguishability. Hennessy and Milner =-=[6,7]-=- introduced into the study of computational processes the seminal idea of associating with a given type of state-transition system a logic that has this fundamental property that two states are observ... |

298 | Universal coalgebra: A theory of systems
- Rutten
(Show Context)
Citation Context ...oalgebras are important both for providing operational semantics for certain languages [10] and for providing definitions of entities, and proofs of their properties, by the principle of co-induction =-=[11,5,4]-=-. Finitary modal languages have been devised that are expressive enough to distinguish coalgebraic elements up to bisimilarity when the functor T is poly2snomial [12,13]. A polynomial functor is one t... |

162 |
A final coalgebra theorem
- Aczel, Mendler
- 1989
(Show Context)
Citation Context ...ections from R to A and B are T -morphisms from ρ to α and β, i.e. the following diagram commutes: A �� α �� π1 R π2 �� B �� �� � ρ T π1 T π2 T A T R This definition of bisimulation was introduced in =-=[23]-=-. It gives a categorical formulation of a notion that has various manifestations in different kinds of state-transition system. The union of any collection of bisimulations from α to β is a bisimulati... |

128 |
Terminal coalgebras in well-founded set theory
- Barr
- 1993
(Show Context)
Citation Context ...{y : x i ↦→ y} is finite for all pairs (x, i). Such a system may be viewed as a (Pω−) I-coalgebra, where PωA is the set of all finite subsets of A. The category (Pω−) I-Coalg does have a final object =-=[19,4]-=-. ✷ The first comprehensive study of T -Coalg was made by Rutten [20,4], showing that many interesting results follow if T weakly preserves certain limits, 2 an assumption that is satisfied by most fu... |

92 |
An approach to object semantics based on terminal coalgebras
- Reichel
- 1995
(Show Context)
Citation Context ...h T -coalgebra α there is exactly one morphism from α to β. Such unique morphisms to a final coalgebra provide coinductive definitions of many operations of importance in the study of data structures =-=[1,5]-=-. If (A, α) is a final coalgebra, then α : A → T A is an isomorphism in Set, i.e. a bijection. This is Lambek’s Lemma [18]. Example 2.1 A non-deterministic state-transition system (A, I, τ) has a set ... |

75 |
On observing nondeterminism and concurrency
- HENNESSY, MILNER
- 1980
(Show Context)
Citation Context ...lgebraic states is transitive. Such transitivity certainly holds in any process-algebraic context for which bisimilarity means observational (or behavioural) indistinguishability. Hennessy and Milner =-=[6,7]-=- introduced into the study of computational processes the seminal idea of associating with a given type of state-transition system a logic that has this fundamental property that two states are observ... |

74 |
Advice on modal logic
- Scott
- 1970
(Show Context)
Citation Context ...e thought of as modelling transition systems that are deterministic, with the constant-valued functors corresponding to sets of “outputs”. The well known canonical model construction from modal logic =-=[14,15]-=- is used in [12,13] to build certain polynomial coalgebras. The essence of this method is to define a model whose elements are special sets of formulas with properties determined by the logic, and to ... |

60 |
Specifying coalgebras with modal logic
- Kurz
(Show Context)
Citation Context ...the principle of co-induction [11,5,4]. Finitary modal languages have been devised that are expressive enough to distinguish coalgebraic elements up to bisimilarity when the functor T is poly2snomial =-=[12,13]-=-. A polynomial functor is one that is constructed from constantvalued functors and the identity functor by forming binary products and coproducts, and exponentials with constant exponent. In the polyn... |

55 | Initial Algebra and Final Coalgebra Semantics for Concurrency, Report CS-R9409
- Rutten, Turi
- 1994
(Show Context)
Citation Context ...uch coalgebras can be used effectively to model data types and transition systems [1–5]. In that setting, final coalgebras are important both for providing operational semantics for certain languages =-=[10]-=- and for providing definitions of entities, and proofs of their properties, by the principle of co-induction [11,5,4]. Finitary modal languages have been devised that are expressive enough to distingu... |

55 |
An essay in classical modal logic
- Segerberg
- 1971
(Show Context)
Citation Context ...e thought of as modelling transition systems that are deterministic, with the constant-valued functors corresponding to sets of “outputs”. The well known canonical model construction from modal logic =-=[14,15]-=- is used in [12,13] to build certain polynomial coalgebras. The essence of this method is to define a model whose elements are special sets of formulas with properties determined by the logic, and to ... |

53 | Many-sorted coalgebraic modal logic: a model-theoretic study
- Jacobs
(Show Context)
Citation Context ...finite output-set as its constant value. To model non-determinism, the class of polynomial functors must be extended to allow powerset formation. Modal languages are also available for this extension =-=[16,17]-=-, but the full use of the powerset functor P prevents there being any final coalgebra at all. On the other hand, we can model finitely-branching non-determinism by using the finitary powerset functor ... |

33 | Coalgebras and modal logic
- Rößiger
(Show Context)
Citation Context ...finite output-set as its constant value. To model non-determinism, the class of polynomial functors must be extended to allow powerset formation. Modal languages are also available for this extension =-=[16,17]-=-, but the full use of the powerset functor P prevents there being any final coalgebra at all. On the other hand, we can model finitely-branching non-determinism by using the finitary powerset functor ... |

25 | A calculus of transition systems (towards universal co-algebra - Rutten - 1995 |

13 | Coalgebraic structure from weak limit preserving functors
- Gumm, Schröder
- 2000
(Show Context)
Citation Context ...ly preserves certain limits, 2 an assumption that is satisfied by most functors used in practice to describe data structures. The theory without this assumption has been explored by Gumm and Schröder =-=[21,22]-=- and we will call on many results from these references. The situation is subtle, and needs careful attention, particularly in relation to the distinction between “congruences” and “bisimulations”, as... |

11 |
Elements of the general theory of coalgebras. LUATCS’99
- Gumm
- 1999
(Show Context)
Citation Context ...ly preserves certain limits, 2 an assumption that is satisfied by most functors used in practice to describe data structures. The theory without this assumption has been explored by Gumm and Schröder =-=[21,22]-=- and we will call on many results from these references. The situation is subtle, and needs careful attention, particularly in relation to the distinction between “congruences” and “bisimulations”, as... |

9 |
A tutorial on (co)algebras and (co)induction. Bulletin of the European Association for Theoretical Computer Science 62
- Jacobs, Rutten
- 1997
(Show Context)
Citation Context ...oalgebras are important both for providing operational semantics for certain languages [10] and for providing definitions of entities, and proofs of their properties, by the principle of co-induction =-=[11,5,4]-=-. Finitary modal languages have been devised that are expressive enough to distinguish coalgebraic elements up to bisimilarity when the functor T is poly2snomial [12,13]. A polynomial functor is one t... |

7 |
Co-induction in relational semantics, Theoretical Computer Science 87
- Milner, Tofte
- 1991
(Show Context)
Citation Context ...oalgebras are important both for providing operational semantics for certain languages [10] and for providing definitions of entities, and proofs of their properties, by the principle of co-induction =-=[11,5,4]-=-. Finitary modal languages have been devised that are expressive enough to distinguish coalgebraic elements up to bisimilarity when the functor T is poly2snomial [12,13]. A polynomial functor is one t... |

4 |
From modal logic to terminal coalgebras Theoret
- Rößiger
- 2001
(Show Context)
Citation Context ...the principle of co-induction [11,5,4]. Finitary modal languages have been devised that are expressive enough to distinguish coalgebraic elements up to bisimilarity when the functor T is poly2snomial =-=[12,13]-=-. A polynomial functor is one that is constructed from constantvalued functors and the identity functor by forming binary products and coproducts, and exponentials with constant exponent. In the polyn... |