## How to Cover without Lifting Relations

### BibTeX

@MISC{Pattinson_howto,

author = {Dirk Pattinson and Luigi Santocanale},

title = {How to Cover without Lifting Relations},

year = {}

}

### OpenURL

### Abstract

It has by now been recognised that coalgebras of various different endofunctors can be used to give semantics to a wide range of modal logics [5] and the last decade has seen an intensive study of modal

### Citations

148 |
Modal Logic
- Chellas
- 1980
(Show Context)
Citation Context ...that defines the frame classes preserves weak pullbacks which precludes the instantiation of the generic theory to important examples such as (monotone) neighbourhood frames [6] or conditional frames =-=[4]-=-. The conceptual reason for requiring that the underlying endofunctor preserve weak pullbacks lies at the very heart of the definition of the semantics of the cover modality: the endofunctor needs to ... |

89 | The Logic of
- Baltag, Moss, et al.
- 1998
(Show Context)
Citation Context ...f modal logics [5] and the last decade has seen an intensive study of modal languages based on predicate liftings [10] and logical languages based on the so-called cover modality, first introduced in =-=[8]-=-. Given an endofunctor T ∶ Set → Set on the category of sets and functions, the role of frames is played by T -coalgebras (C, γ) where C is a set (the carrier) and γ ∶ C → T C is a (transition) functi... |

51 | Coalgebraic modal logic: Soundness, completeness and decidability of local consequence
- Pattinson
(Show Context)
Citation Context ...bras of various different endofunctors can be used to give semantics to a wide range of modal logics [5] and the last decade has seen an intensive study of modal languages based on predicate liftings =-=[10]-=- and logical languages based on the so-called cover modality, first introduced in [8]. Given an endofunctor T ∶ Set → Set on the category of sets and functions, the role of frames is played by T -coal... |

43 |
V.: Automata and algebras in categories
- Adámek, Trnková
- 1990
(Show Context)
Citation Context ..., γ, π) where (C, γ) is a T -frame and π ∶ C → P(V) is a valuation (of propositional variables V). We may assume without loss of generality that T preserves set-theoretic inclusions and intersections =-=[1, 2]-=- (including empty intersections). 1Definition 1. The language induced by T is given by the grammar LV(T ) ∋ φ, ψ ∶∶= p ∣ φ ∧ ψ ∣ φ ∨ ψ ∣ ¬φ ∣ ▼Φ where p ∈ V and Φ ∈ T P(Σ) for some finite set Σ ⊆ LV(... |

15 | Automata and fixed point logic: a coalgebraic perspective
- Venema
(Show Context)
Citation Context ...applied to the satisfaction relation ⊧⊆ C × L(T ) in a meaningful way. Languages arising in this way are by now welldeveloped: Venema presents automata for fixpoint logics based on the cover modality =-=[12]-=-, complete axiomatisations may be found in [3, 7] and Moss’ original paper [8] establishes the Hennessy-Milner property. While the cover modality provides a generic treatment of many different classes... |

8 |
Terminal coalgebras in well-fonded set theory. 114:229–315
- Barr
- 1993
(Show Context)
Citation Context ..., γ, π) where (C, γ) is a T -frame and π ∶ C → P(V) is a valuation (of propositional variables V). We may assume without loss of generality that T preserves set-theoretic inclusions and intersections =-=[1, 2]-=- (including empty intersections). 1Definition 1. The language induced by T is given by the grammar LV(T ) ∋ φ, ψ ∶∶= p ∣ φ ∧ ψ ∣ φ ∨ ψ ∣ ¬φ ∣ ▼Φ where p ∈ V and Φ ∈ T P(Σ) for some finite set Σ ⊆ LV(... |

7 |
Modal Logics are Coalgebraic, The Computer Journal
- Cirstea, Kurz, et al.
(Show Context)
Citation Context ...don Luigi Santocanale Université de Provence Introduction It has by now been recognised that coalgebras of various different endofunctors can be used to give semantics to a wide range of modal logics =-=[5]-=- and the last decade has seen an intensive study of modal languages based on predicate liftings [10] and logical languages based on the so-called cover modality, first introduced in [8]. Given an endo... |

6 | Proof systems for the coalgebraic cover modality
- Bílková, Palmigiano, et al.
(Show Context)
Citation Context ... ) in a meaningful way. Languages arising in this way are by now welldeveloped: Venema presents automata for fixpoint logics based on the cover modality [12], complete axiomatisations may be found in =-=[3, 7]-=- and Moss’ original paper [8] establishes the Hennessy-Milner property. While the cover modality provides a generic treatment of many different classes of frames, all of the above results rely on the ... |

4 | Neighbourhood structures: Bisimilarity and basic model theory
- Hansen, Kupke, et al.
(Show Context)
Citation Context ...sumption that the functor that defines the frame classes preserves weak pullbacks which precludes the instantiation of the generic theory to important examples such as (monotone) neighbourhood frames =-=[6]-=- or conditional frames [4]. The conceptual reason for requiring that the underlying endofunctor preserve weak pullbacks lies at the very heart of the definition of the semantics of the cover modality:... |

2 | Completeness of the finitary Moss Logic
- Kupke, Kurz, et al.
- 2008
(Show Context)
Citation Context ... ) in a meaningful way. Languages arising in this way are by now welldeveloped: Venema presents automata for fixpoint logics based on the cover modality [12], complete axiomatisations may be found in =-=[3, 7]-=- and Moss’ original paper [8] establishes the Hennessy-Milner property. While the cover modality provides a generic treatment of many different classes of frames, all of the above results rely on the ... |

1 |
A note on expressive coalgebraic logics for finitary set functors
- Moss
(Show Context)
Citation Context ... the preservation of weak pullbacks has so far remained open. Taking formulas to be elements of the final sequence of the underlying endofunctor removes the requirement for weak pullback preservation =-=[9]-=-, but does not provide a syntactial notion of modal operator. Other advances have been made for monotone neighbourhood frames [11] (where weak pullback preservation fails) by means of changing the def... |

1 |
Uniform interpolation for monotone modal logic
- Santocanale, Venema
- 2010
(Show Context)
Citation Context ...ing endofunctor removes the requirement for weak pullback preservation [9], but does not provide a syntactial notion of modal operator. Other advances have been made for monotone neighbourhood frames =-=[11]-=- (where weak pullback preservation fails) by means of changing the definition of relation lifting, however it seems difficult to obtain a generalisation to arbitrary classes of frames. Here, we take a... |

1 |
A modal distributive law (abstract
- Venema
- 2007
(Show Context)
Citation Context ...ally equivalent points c ∈ C and d ∈ D have the same logical theory. If T is finitary then the converse holds as well. Similarly to the situation with modal languages in terms of Moss’ cover modality =-=[13]-=-, one can establish that – if T maps finite sets to finite sets – conjunction can be eliminated and negation can be pushed to atomic propositions. For example ¬ ▼ Φ can be replaced by the disjunction ... |