## Rank-1 modal logics are coalgebraic (2007)

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Venue: | IN STACS 2007, 24TH ANNUAL SYMPOSIUM ON THEORETICAL ASPECTS OF COMPUTER SCIENCE, PROCEEDINGS |

Citations: | 14 - 11 self |

### BibTeX

@INPROCEEDINGS{Schröder07rank-1modal,

author = {Lutz Schröder and Dirk Pattinson},

title = {Rank-1 modal logics are coalgebraic},

booktitle = {IN STACS 2007, 24TH ANNUAL SYMPOSIUM ON THEORETICAL ASPECTS OF COMPUTER SCIENCE, PROCEEDINGS},

year = {2007},

pages = {573--585},

publisher = {Springer}

}

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### Abstract

Coalgebras provide a unifying semantic framework for a wide variety of modal logics. It has previously been shown that the class of coalgebras for an endofunctor can always be axiomatised in rank 1. Here we establish the converse, i.e. every rank 1 modal logic has a sound and strongly complete coalgebraic semantics. As a consequence, recent results on coalgebraic modal logic, in particular generic decision procedures and upper complexity bounds, become applicable to arbitrary rank 1 modal logics, without regard to their semantic status; we thus obtain purely syntactic versions of these results. As an extended example, we apply our framework to recently defined deontic logics.

### Citations

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- 1991
(Show Context)
Citation Context ...es. Under this perspective, coalgebraic modal logic is a generic notion of modal logic that subsumes e.g. Hennessy-Milner logic, graded modal logic [4], majority logic [13], probabilistic modal logic =-=[12, 7]-=-, and coalition logic [16], but also modal operators of higher arity as e.g. in conditional logic [3]. It has been shown in [20] that every coalgebraic modal logic can be axiomatized by formulas of ra... |

298 | Universal coalgebra: a theory of systems
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(Show Context)
Citation Context ...mple, we apply our framework to recently defined deontic logics. Introduction In recent years, coalgebras have received a steadily growing amount of attention as general models of state-based systems =-=[18]-=-, encompassing such diverse systems as labelled transition systems, probabilistic systems, game frames, and neighborhood frames [21]. On the logical side, modal logic has emerged as the adequate speci... |

155 | Reasoning about knowledge and probability - Fagin, Halpern - 1994 |

148 |
Modal Logic
- Chellas
- 1980
(Show Context)
Citation Context ... Hennessy-Milner logic, graded modal logic [4], majority logic [13], probabilistic modal logic [12, 7], and coalition logic [16], but also modal operators of higher arity as e.g. in conditional logic =-=[3]-=-. It has been shown in [20] that every coalgebraic modal logic can be axiomatized by formulas of rank 1, i.e. with nesting depth of modal operators uniformly equal to 1 (logics of arbitrary rank are o... |

136 | A Modal Logic for Coalitional Power in Games
- Pauly
(Show Context)
Citation Context ...oalgebraic modal logic is a generic notion of modal logic that subsumes e.g. Hennessy-Milner logic, graded modal logic [4], majority logic [13], probabilistic modal logic [12, 7], and coalition logic =-=[16]-=-, but also modal operators of higher arity as e.g. in conditional logic [3]. It has been shown in [20] that every coalgebraic modal logic can be axiomatized by formulas of rank 1, i.e. with nesting de... |

129 |
Terminal coalgebras in well-founded set theory, Theoret
- Barr
- 1993
(Show Context)
Citation Context ... Set. We view coalgebras as generalised transition systems: the transition function maps states to a structured set of successors Assumption 3. We can assume w.l.o.g. that T preserves injective maps (=-=[2]-=-, proof of Theorem 3.2). For convenience, we will in fact sometimes assume that T X ⊆ T Y if X ⊆ Y . Moreover, we assume that T is non-trivial, i.e. T X = ∅ =⇒ X = ∅ (otherwise, T X = ∅ for all X). Re... |

112 | Deontic logic - Wright - 1951 |

89 | The Logic of - Baltag, Moss, et al. - 1998 |

79 | An Investigation of the Laws of Thought, on which are founded the mathematical theories of logic and probabilities - Boole - 1854 |

61 |
Specifying Coalgebras with Modal Logic
- Kurz
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(Show Context)
Citation Context ...al Logic We briefly recapitulate the basics of the coalgebraic semantics of modal logic. Coalgebraic modal logic in the form considered here has been introduced in [15], generalising previous results =-=[9, 17, 11, 14]-=-. For the sake of readability, we restrict the exposition to unary modalities. However, we emphasize that all our results extend in a straightforward way to polyadic operators, found e.g. in condition... |

53 | Many-sorted coalgebraic modal logic: a model-theoretic study - Jacobs |

51 | Coalgebraic modal logic: Soundness, completeness and decidability of local consequence - Pattinson |

39 | Expressivity of coalgebraic modal logic: The limits and beyond
- Schröder
(Show Context)
Citation Context ...ven when the latter are given purely syntactically or equipped with a semantics that fails to be, or has not yet been recognized as, coalgebraic. This includes results on the Hennessy-Milner property =-=[19]-=- and bisimulation-somewhere-else [10], and most notably generic decidability and complexity results [20, 21], of which we now obtain purely syntactic versions. As an extended example, we discuss appli... |

33 | Coalgebras and modal logic
- Rößiger
(Show Context)
Citation Context ...al Logic We briefly recapitulate the basics of the coalgebraic semantics of modal logic. Coalgebraic modal logic in the form considered here has been introduced in [15], generalising previous results =-=[9, 17, 11, 14]-=-. For the sake of readability, we restrict the exposition to unary modalities. However, we emphasize that all our results extend in a straightforward way to polyadic operators, found e.g. in condition... |

31 | A.: Duality for Logics of Transition Systems - Bonsangue, Kurz - 2005 |

30 | Semantical principles in the modal logic of coalgebras
- Pattinson
- 2001
(Show Context)
Citation Context ...al Logic We briefly recapitulate the basics of the coalgebraic semantics of modal logic. Coalgebraic modal logic in the form considered here has been introduced in [15], generalising previous results =-=[9, 17, 11, 14]-=-. For the sake of readability, we restrict the exposition to unary modalities. However, we emphasize that all our results extend in a straightforward way to polyadic operators, found e.g. in condition... |

28 |
Probability Logic for Type Spaces
- Heifetz, Mongin
(Show Context)
Citation Context ...es. Under this perspective, coalgebraic modal logic is a generic notion of modal logic that subsumes e.g. Hennessy-Milner logic, graded modal logic [4], majority logic [13], probabilistic modal logic =-=[12, 7]-=-, and coalition logic [16], but also modal operators of higher arity as e.g. in conditional logic [3]. It has been shown in [20] that every coalgebraic modal logic can be axiomatized by formulas of ra... |

26 | Expressive logics for coalgebras via terminal sequence induction
- Pattinson
(Show Context)
Citation Context ..., modal logic has emerged as the adequate specification language for coalgebraically modelled systems. A variety of different frameworks have been proposed; here, we work with coalgebraic modal logic =-=[15]-=-, which allows for a high level of generality while retaining a close relationship to the established syntactic and semantic tradition of modal logic. In fact, one can reverse the viewpoint that coalg... |

24 | A finite model construction for coalgebraic modal logic
- Schröder
(Show Context)
Citation Context ...aded modal logic [4], majority logic [13], probabilistic modal logic [12, 7], and coalition logic [16], but also modal operators of higher arity as e.g. in conditional logic [3]. It has been shown in =-=[20]-=- that every coalgebraic modal logic can be axiomatized by formulas of rank 1, i.e. with nesting depth of modal operators uniformly equal to 1 (logics of arbitrary rank are obtained by restricting the ... |

22 | Modular construction of modal logics - Cîrstea, Pattinson - 2004 |

21 | Finality regained: A coalgebraic study of Scott-sets and multisets - D’Agostino, Visser |

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

20 |
Deontic Logic
- Hilpinen
- 2001
(Show Context)
Citation Context ...ic logics, which are often defined either without any reference to semantics at all or with a neighbourhood semantics essentially equivalent to the canonical semantics described above. Deontic logics =-=[8]-=-, which have received much recent interest in computer science as logics for obligations of agents, are moreover often axiomatised in rank 1. Standard deontic logic [3] is just the modal logic KD. Thi... |

19 | Functors for coalgebras - Gumm |

18 | Intensional logics without iterative axioms - Lewis |

17 | Majority logic
- Pacuit, Salame
- 2004
(Show Context)
Citation Context ...ng non-normal and nonmonotone ones. Under this perspective, coalgebraic modal logic is a generic notion of modal logic that subsumes e.g. Hennessy-Milner logic, graded modal logic [4], majority logic =-=[13]-=-, probabilistic modal logic [12, 7], and coalition logic [16], but also modal operators of higher arity as e.g. in conditional logic [3]. It has been shown in [20] that every coalgebraic modal logic c... |

16 | Presburger modal logic is only PSPACE-complete - Demri, Lugiez - 2006 |

16 | Algebraic semantics for coalgebraic logics - Kupke, Kurz, et al. - 2004 |

16 | Modular algorithms for heterogeneous modal logics - Schröder, Pattinson - 2007 |

15 |
so many possible worlds
- In
- 1972
(Show Context)
Citation Context ...rary nature, including non-normal and nonmonotone ones. Under this perspective, coalgebraic modal logic is a generic notion of modal logic that subsumes e.g. Hennessy-Milner logic, graded modal logic =-=[4]-=-, majority logic [13], probabilistic modal logic [12, 7], and coalition logic [16], but also modal operators of higher arity as e.g. in conditional logic [3]. It has been shown in [20] that every coal... |

15 |
PSPACE reasoning for rank-1 modal logics
- Schröder, Pattinson
- 2006
(Show Context)
Citation Context ...] and an additional copy of [0, 1]∩Q, where the second copy of q ∈ [0, 1]∩Q is infinitesimally greater than the first. The distributions P ∈ M fin L (X) are required to obey the axiomatization of PML =-=[21]-=- w.r.t. the canonical semantics; it is presently unclear whether this requirement can be replaced by a simpler condition. 3 An Adjunction between Syntax and Semantics We now set up an adjoint correspo... |

14 | From modal logic to terminal coalgebras - Rößiger |

13 | Ultrafilter extensions of coalgebras
- Kupke, Kurz, et al.
- 2005
(Show Context)
Citation Context ...syntactically or equipped with a semantics that fails to be, or has not yet been recognized as, coalgebraic. This includes results on the Hennessy-Milner property [19] and bisimulation-somewhere-else =-=[10]-=-, and most notably generic decidability and complexity results [20, 21], of which we now obtain purely syntactic versions. As an extended example, we discuss applications of these results to recently ... |

10 | Strongly complete logics for coalgebras - Kurz, Rosick´y - 2006 |

7 | On tree coalgebras and coalgebra presentations
- Adámek, Porst
(Show Context)
Citation Context ... X ⊆ T Y if X ⊆ Y . Moreover, we assume that T is non-trivial, i.e. T X = ∅ =⇒ X = ∅ (otherwise, T X = ∅ for all X). Recall that a functor is ω-accessible if it preserves directed colimits. Lemma 4. (=-=[1]-=-, Proposition 5.2) For a set functor T , the following are equivalent: 1. T is ω-accessible 2. T preserves directed unions 3. For every set X, T X = ⋃ Y ⊆X finite T Y (recall Assumption 3).4 The coal... |

7 | Beyond rank 1: Algebraic semantics and finite models for coalgebraic logics - Pattinson, Schröder - 2008 |

6 |
A proposal for dealing with deontic dilemmas
- Goble
- 2004
(Show Context)
Citation Context ...ility and complexity results [20, 21], of which we now obtain purely syntactic versions. As an extended example, we discuss applications of these results to recently defined variants of deontic logic =-=[5]-=-. 1 Coalgebraic Modal Logic We briefly recapitulate the basics of the coalgebraic semantics of modal logic. Coalgebraic modal logic in the form considered here has been introduced in [15], generalisin... |

2 | Solution of equations in Boolean algebra - Levchenkov - 2000 |

2 | Generic strong completeness results in modal logic - Schröder, Pattinson |

2 | Shallow models for non-iterative modal logics. Available as e-print arXiv:0802.0116 - Schröder, Pattinson - 2008 |