## The least fibred lifting and the expressivity of coalgebraic modal logic (2005)

Venue: | In Proc. CALCO 2005, volume 3629 of LNCS |

Citations: | 8 - 1 self |

### BibTeX

@INPROCEEDINGS{Klin05theleast,

author = {Bartek Klin},

title = {The least fibred lifting and the expressivity of coalgebraic modal logic},

booktitle = {In Proc. CALCO 2005, volume 3629 of LNCS},

year = {2005},

pages = {247--262}

}

### OpenURL

### Abstract

and relation-preserving functions. In this paper, the least (fibre-wise) of such liftings, L(B), is characterized for essentially any B. The lifting has all the useful properties of the relation lifting due to Jacobs, without the usual assumption of weak pullback preservation; if B preserves weak pullbacks, the two liftings coincide. Equivalence relations can be viewed as Boolean algebras of subsets (predicates, tests). This correspondence relates L(B) to the least test suite lifting T (B), which is defined in the spirit of predicate lifting as used in coalgebraic modal logic. Properties of T (B) translate to a general expressivity result for a modal logic for B-coalgebras. In the resulting logic, modal operators of any arity can appear. 1

### Citations

3500 |
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- 1989
(Show Context)
Citation Context ...valence coincides with bisimilarity. Usually, bisimilarity is defined as the greatest bisimulation on a transition system. A natural coalgebraic generalization of the classical notion of bisimulation =-=[13,11]-=- is based on spans of coalgebras [16]. There, a bisimulation is a relation that lifts to a span of coalgebra morphisms. If the behaviour functor B preserves weak pullbacks, the greatest bisimulation e... |

714 |
Concurrency and automata on infinite sequences
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(Show Context)
Citation Context ...valence coincides with bisimilarity. Usually, bisimilarity is defined as the greatest bisimulation on a transition system. A natural coalgebraic generalization of the classical notion of bisimulation =-=[13,11]-=- is based on spans of coalgebras [16]. There, a bisimulation is a relation that lifts to a span of coalgebra morphisms. If the behaviour functor B preserves weak pullbacks, the greatest bisimulation e... |

529 | Algebraic laws for nondeterminism and concurrency
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- 1985
(Show Context)
Citation Context ...ms; also, as in the span approach, the greatest bisimulation coincides with behavioural equivalence. In the case of labelled transition systems, bisimilarity is characterized by Hennessy-Milner logic =-=[3]-=-. The coalgebraic framework provides a more general perspective, where a modal logic can be derived for coalgebras for any functor. It is expected that coalgebraic modal logic is invariant under behav... |

340 | Universal coalgebra: A theory of systems
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- 2000
(Show Context)
Citation Context ...a coalgebra h : X → Pf(A × X), where A is a set of labels. Replacing the behaviour functor Pf(A × −) with various functors on the category Set of sets and functions, one models other kinds of systems =-=[16]-=-. Final coalgebras are abstract models of the behaviour of systems. For a coalgebra h : X → BX, two processes in X are considered behaviourally equivalent if they are identified by the unique morphism... |

167 |
A final coalgebra theorem
- Aczel, Mendler
- 1989
(Show Context)
Citation Context ...nal B-coalgebra is the equality relation. The definition of J(B) is elegant in that it does not depend on the structure of B. It is also closely related to the coalgebra span approach to bisimulation =-=[1,16]-=-. However, speaking in terms of fibrations, it does not really seem canonical: it is not immediately clear why one should choose this particular lifting from many available ones, and only the numerous... |

100 | Coalgebraic logic
- Moss
- 1999
(Show Context)
Citation Context ...oural equivalence. Moreover, it should be expressive: logically indistinguishable states should be behaviourally equivalent. Several approaches to deriving coalgebraic modal logics have been proposed =-=[6,10,12,14,15,17]-=-, based on different abstract notions of modal operators (modalities). In [12], a single modal operator is associated with every functor B. The resulting logic is expressive for essentially all functo... |

72 | Structural induction and coinduction in a fibrational setting
- Hermida, Jacobs
- 1998
(Show Context)
Citation Context ...sms. If the behaviour functor B preserves weak pullbacks, the greatest bisimulation exists and coincides with behavioural equivalence. Another, closely related approach was presented by Jacobs et al. =-=[4,7,9]-=-. There, the functor B is provided with a relation lifting J(B), which is a functor on the category of binary relations that behaves as B on the underlying sets. A bisimulation is then a J(B)-coalgebr... |

44 | Expressivity of coalgebraic modal logic: The limits and beyond
- Schröder
- 2005
(Show Context)
Citation Context ...oural equivalence. Moreover, it should be expressive: logically indistinguishable states should be behaviourally equivalent. Several approaches to deriving coalgebraic modal logics have been proposed =-=[6,10,12,14,15,17]-=-, based on different abstract notions of modal operators (modalities). In [12], a single modal operator is associated with every functor B. The resulting logic is expressive for essentially all functo... |

38 |
Categorical Logic and Type Theory. Number 141
- Jacobs
- 1999
(Show Context)
Citation Context ...e obvious forgetful functor p : Set ∗ → Set, mapping a pair 〈X, R〉 to its underlying set X, is then a fibration. This is only a special case of a more general definition used in fibration theory (see =-=[5]-=- for a detailed presentation), but only examples of this kind are considered in this paper. Slightly abusing the terminology, we will simply call functors (−) ∗ : Set op → Pos fibrations. The partial ... |

32 | Semantical principles in the modal logic of coalgebras
- Pattinson
- 2001
(Show Context)
Citation Context ...oural equivalence. Moreover, it should be expressive: logically indistinguishable states should be behaviourally equivalent. Several approaches to deriving coalgebraic modal logics have been proposed =-=[6,10,12,14,15,17]-=-, based on different abstract notions of modal operators (modalities). In [12], a single modal operator is associated with every functor B. The resulting logic is expressive for essentially all functo... |

26 |
Calculi for Higher-Order Communicating Systems
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- 1990
(Show Context)
Citation Context ...sy-Milner logic, except: [〈a〉 (φ1, φ2)](β) = tt iff 〈a, x, y〉 ∈ β s.t. [φ1 ]x = tt, [φ2 ]x = tt This logic has actually been used in the literature, to describe a simple calculus with process passing =-=[18]-=-. This example shows that the extension of coalgebraic modal logic to modalities of arbitrary arity, as done first in [17], is of practical importance. Note, however, that allowing multiple sorts of l... |

23 | Modular construction of modal logics
- Cîrstea, Pattinson
- 2004
(Show Context)
Citation Context ... to (−) ∗ . Given a functor B : Set → Set, a lifting of B to B ∗ is uniquely determined by its action on the elements of fibres, i.e., by a family (indexed by sets X) of functions BX : X ∗ → (BX) ∗ 5 =-=(2)-=-sFor any such family, the map defined by B ∗ 〈X, R〉 = 〈BX, BXR〉 B ∗ f = Bf is a functor on Set ∗ if and only if all functions BX are monotonic and, for any function f : X → Y and any S ∈ Y ∗ , BX(f ∗ ... |

21 | Towards a duality result in the modal logic of coalgebras
- Jacobs
- 2000
(Show Context)
Citation Context |

21 | Trace semantics for coalgebras
- Jacobs
- 2004
(Show Context)
Citation Context ...tion in this context, since behavioural equivalence, which one aims to model by bisimulations, is always an equivalence relation. (Note, however, that the lifting J(B) has been used in other contexts =-=[8]-=-, where arbitrary relations are essential.) 2sThe category ERel is isomorphic to the category BTS of test suites closed under union, intersection and complementation. This isomorphism relates L(B) cor... |

20 | A coalgebraic approach to process equivalence and a coinduction principle for traces - Klin |

18 | Exercises in coalgebraic specification
- Jacobs
- 2002
(Show Context)
Citation Context ...sms. If the behaviour functor B preserves weak pullbacks, the greatest bisimulation exists and coincides with behavioural equivalence. Another, closely related approach was presented by Jacobs et al. =-=[4,7,9]-=-. There, the functor B is provided with a relation lifting J(B), which is a functor on the category of binary relations that behaves as B on the underlying sets. A bisimulation is then a J(B)-coalgebr... |

6 |
Expressivity Results in the Modal Logic of Coalgebras
- Pattinson
- 2001
(Show Context)
Citation Context |