## Coalgebraic Logic (1999)

Venue: | Annals of Pure and Applied Logic |

Citations: | 89 - 0 self |

### BibTeX

@ARTICLE{Moss99coalgebraiclogic,

author = {Lawrence S. Moss},

title = {Coalgebraic Logic},

journal = {Annals of Pure and Applied Logic},

year = {1999},

volume = {96}

}

### Years of Citing Articles

### OpenURL

### Abstract

We present a generalization of modal logic to logical systems which are interpreted on coalgebras of functors on sets. The leading idea is that infinitary modal logic contains characterizing formulas. That is, every model-world pair is characterized up to bisimulation by an infinitary formula. The point of our generalization is to understand this on a deeper level. We do this by studying a frangment of infinitary modal logic which contains the characterizing formulas and is closed under infinitary conjunction and an operation called 4. This fragment generalizes to a wide range of coalgebraic logics. We then apply the characterization result to get representation theorems for final coalgebras in terms of maximal elements of ordered algebras. The end result is that the formulas of coalgebraic logics can be viewed as approximations to the elements of the final coalgebra. Keywords: infinitary modal logic, characterization theorem, functor on sets, coalgebra, greatest fixed point. 1 Intr...

### Citations

298 | Universal coalgebra: a theory of systems
- Rutten
(Show Context)
Citation Context ...recent years, a number of authors have considered coalgebras of functors in connection with modeling issues in theoretical computer science; for example, see Aczel [A], Barwise and Moss [BM1], Rutten =-=[R96], and Turi-=- and Rutten [TR]. One important line of work concerns final coalgebra theorems. A final coalgebra for a functor often gives a "most abstract" model of a specified kind, and this is the inter... |

201 |
Non-Well-Founded Sets
- Aczel
- 1988
(Show Context)
Citation Context ... functor seems to be fairly new. In recent years, a number of authors have considered coalgebras of functors in connection with modeling issues in theoretical computer science; for example, see Aczel =-=[A], Barwise -=-and Moss [BM1], Rutten [R96], and Turi and Rutten [TR]. One important line of work concerns final coalgebra theorems. A final coalgebra for a functor often gives a "most abstract" model of a... |

162 | A nal coalgebra theorem - Aczel, Mendler - 1989 |

129 | Terminal coalgebras in well-founded set theory, Theoret - Barr - 1993 |

97 |
Vicious Circles
- Barwise, Moss
- 1996
(Show Context)
Citation Context ...airly new. In recent years, a number of authors have considered coalgebras of functors in connection with modeling issues in theoretical computer science; for example, see Aczel [A], Barwise and Moss =-=[BM1], Rutten [-=-R96], and Turi and Rutten [TR]. One important line of work concerns final coalgebra theorems. A final coalgebra for a functor often gives a "most abstract" model of a specified kind, and thi... |

53 |
Functorial Operational Semantics and its Denotational Dual
- Turi
- 1996
(Show Context)
Citation Context ...+ inr W Q;q (D) ] # : 20 Here f +idQ : C+Q ! D+Q is the natural map. Furthermore, we have ffl Q;q;C : W Q;q (C) ! V given by ffl Q;q;C = [[incl C ; q] + [ ] # : Definition [Turi and Rutten [TR], Turi =-=[T]-=-] F is uniform if there are a set Q, a map q : Q ! V and a natural transformation ae : F ! PW Q;q such that P ffl Q;q;V ffi ae FV = incl FV , where incl F(V ) is the inclusion of F(V ) in V . Our form... |

47 | On the Foundations of Final Semantics: Non-Standard Sets
- Rutten, Turi
- 1992
(Show Context)
Citation Context ...uthors have considered coalgebras of functors in connection with modeling issues in theoretical computer science; for example, see Aczel [A], Barwise and Moss [BM1], Rutten [R96], and Turi and Rutten =-=[TR]. One impo-=-rtant line of work concerns final coalgebra theorems. A final coalgebra for a functor often gives a "most abstract" model of a specified kind, and this is the interest of such a result. Now ... |

16 | A Concrete Final Coalgebra Theorem for ZF Set Theory
- Paulson
- 1994
(Show Context)
Citation Context ...tation one has of the final coalgebra, the better. Final coalgebra theorems of various types may be found in the works cite above, and also in Barr [Bar93, Bar94], Moss and Danner[MD], and in Paulson =-=[P]-=-. The results of this paper also give a final coalgebra: the maximal formulas of L F in the semantic preorder. This connects coalgebraic logic with the issue of getting representations of the final co... |

13 | On the Foundations of Corecursion
- Moss, Danner
- 1997
(Show Context)
Citation Context ...e concrete a representation one has of the final coalgebra, the better. Final coalgebra theorems of various types may be found in the works cite above, and also in Barr [Bar93, Bar94], Moss and Danner=-=[MD]-=-, and in Paulson [P]. The results of this paper also give a final coalgebra: the maximal formulas of L F in the semantic preorder. This connects coalgebraic logic with the issue of getting representat... |

11 | A Quantitative Analysis of Modal Logic
- Fagin
- 1994
(Show Context)
Citation Context ...and F which relates x to b. The proof of Theorem 2.4 may be found in Chapter 11 of [BM1]. Applications are discussed there, in [BM2], and in [Bal]. Both Baltag [Bal] and Gerbrandy [G] note that Fagin =-=[F]-=- also 5 obtains characterization results for semantics of infintary modal logic based on ff-worlds. As it happens, those worlds are essentially what we are calling the canonical formulas. Having intro... |

8 |
Non-well-founded sets modelled as ideal fixed points
- Mislove, Oles, et al.
- 1991
(Show Context)
Citation Context ...ndeed, not long after the appearance of Aczel's book[A], a number of papers appeared on the subject of getting domain-theoretic representations of the non-wellfounded sets. For example, Mislove et al =-=[MMO]-=- show how to obtain a domain-theoretic representation of the set HF of hereditarily finite sets in terms of initial ordered algebras of a certain type. In other work, Barr[Bar93, Bar94] considered end... |

6 |
Modal correspondence for models
- Barwise, Moss
- 1998
(Show Context)
Citation Context ... b 2 B such that b j= E 0 ` x , then there is a bisimulation R between E and F which relates x to b. The proof of Theorem 2.4 may be found in Chapter 11 of [BM1]. Applications are discussed there, in =-=[BM2]-=-, and in [Bal]. Both Baltag [Bal] and Gerbrandy [G] note that Fagin [F] also 5 obtains characterization results for semantics of infintary modal logic based on ff-worlds. As it happens, those worlds a... |

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

1 |
Modal charactersation for sets and Kripke models. Unpublished manuscript
- Baltag
- 1996
(Show Context)
Citation Context ...at b j= E 0 ` x , then there is a bisimulation R between E and F which relates x to b. The proof of Theorem 2.4 may be found in Chapter 11 of [BM1]. Applications are discussed there, in [BM2], and in =-=[Bal]-=-. Both Baltag [Bal] and Gerbrandy [G] note that Fagin [F] also 5 obtains characterization results for semantics of infintary modal logic based on ff-worlds. As it happens, those worlds are essentially... |

1 |
Modal Correspondence Theory
- Bethem
- 1976
(Show Context)
Citation Context ...s are essentially what we are calling the canonical formulas. Having introduced L(4) as a fragment of infinitary modal logic, we would like to pose one open question about it. As shown by van Benthem =-=[Ben]-=-, infinitary modal logic is equivalent to the fragment of infinitary first-order logic which contains the formulas invariant under bisimulation of models. Is there a similar result for L(4), isolating... |

1 |
Characterizations of bisimulation and bounded bisimulations
- Gerbrandy
- 1996
(Show Context)
Citation Context ...l ff with the following property: if hF ; bi is any model-world pair and b j= F ' a ff , then there is a bisimulation of E and F relating a to b. Proof We sketch Gerbrandy's proof of this result from =-=[G]-=-. First, consider the submodel E 0 of E generated by a. This submodel is easily seen to be a set. The approximations ' a ff have several important properties: if ffsfi, then ' a fi ! ' a ff ; and for ... |