## Modular algorithms for heterogeneous modal logics (2007)

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Venue: | IN AUTOMATA, LANGUAGES AND PROGRAMMING, ICALP 07, VOL. 4596 OF LNCS |

Citations: | 16 - 11 self |

### BibTeX

@INPROCEEDINGS{Schröder07modularalgorithms,

author = {Lutz Schröder and Dirk Pattinson},

title = {Modular algorithms for heterogeneous modal logics},

booktitle = {IN AUTOMATA, LANGUAGES AND PROGRAMMING, ICALP 07, VOL. 4596 OF LNCS},

year = {2007},

pages = {459--471},

publisher = {Springer}

}

### Years of Citing Articles

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### Abstract

State-based systems and modal logics for reasoning about them often heterogeneously combine a number of features such as non-determinism and probabilities. Here, we show that the combination of features can be reflected algorithmically and develop modular decision procedures for heterogeneous modal logics. The modularity is achieved by formalising the underlying state-based systems as multi-sorted coalgebras and associating both a logical and an algorithmic description to a number of basic building blocks. Our main result is that logics arising as combinations of these building blocks can be decided in polynomial space provided that this is the case for the components. By instantiating the general framework to concrete cases, we obtain PSPACE decision procedures for a wide variety of structurally different logics, describing e.g. Segala systems and games with uncertain information.

### Citations

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Citation Context ...emantics of modal logics typically involves a notion of state and transition, which can take a number of different forms. Transitions can be probabilistic or weighted, as in probabilistic modal logic =-=[15,9]-=- and graded modal logic [6,5], induced by joint actions of agents as in coalition logic [18], or non-monotonically conditioned as in conditional logic [3]. An attractive aspect of many of these logics... |

300 | Universal coalgebra: a theory of systems
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Citation Context ...that can be instantiated to a large number of structurally different systems and logics. We recall some basic notions of multi-sorted coalgebra (cf. e.g. [16]), generalising the single-sorted setting =-=[19]-=-: 6sDefinition 5. We write Set for the category of sets and functions. Let Set S denote the category of S-sorted sets and S-sorted functions, with objects being families X = (Xa)a∈S (or just (Xa)) of ... |

239 |
Modelling and verification of randomized distributed real time systems
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Citation Context ...combined in different ways: E.g. in the alternating model of probabilistic transition systems [8], states may have either non-deterministic or probabilistic transitions, whereas simple Segala systems =-=[24]-=- have a two-layered structure where non-deterministic transitions lead to probability distributions over successor states. Bartels et al. [1] present 12 different types of probabilistic transition sys... |

224 |
Reasoning About Uncertainty
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- 2003
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Citation Context ...ey are the formalism of choice for reasoning about reactive systems and feature prominently in areas related to artificial intelligence such as knowledge representation and reasoning with uncertainty =-=[7]-=-. The semantics of modal logics typically involves a notion of state and transition, which can take a number of different forms. Transitions can be probabilistic or weighted, as in probabilistic modal... |

147 |
Modal Logic
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- 1980
(Show Context)
Citation Context ... weighted, as in probabilistic modal logic [15,9] and graded modal logic [6,5], induced by joint actions of agents as in coalition logic [18], or non-monotonically conditioned as in conditional logic =-=[3]-=-. An attractive aspect of many of these logics is that they admit shallow models and are decidable in low complexity classes, in the absence of fixpoint operators typically PSPACE (e.g. [25,18,23]), i... |

136 | A modal logic for coalitional power in games - Pauly - 2002 |

95 |
A calculus for communicating systems with time and probabilities
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Citation Context ...combined, leading to systems that incorporate more than one type of transition. Moreover, features can be combined in different ways: E.g. in the alternating model of probabilistic transition systems =-=[8]-=-, states may have either non-deterministic or probabilistic transitions, whereas simple Segala systems [24] have a two-layered structure where non-deterministic transitions lead to probability distrib... |

95 | E-connections of Abstract Description Systems - Kutz, Lutz, et al. |

67 | Y i, Probabilistic extensions of process algebras - Jonsson, Larsen, et al. |

53 | Many-sorted coalgebraic modal logic: a model-theoretic study
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Citation Context ...se results imply PSPACE upper bounds for satisfiability w.r.t. general gluings, including the standard single-sorted semantics. Related Work. Our work is closely related to the framework presented in =-=[4,11]-=-, which focuses on completeness issues, with the main difference that our approach makes the multi-sorted nature of heterogeneous logics explicit by considering multisorted models. Our treatment of ty... |

44 | Fusions of modal logics revisited
- Wolter
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Citation Context ...22,23] generalise straightforwardly to the multi-sorted case. The multisorted approach to the complexity of composite modal logics complements transfer results obtained for the fusion of modal logics =-=[26,10]-=- in the sense that our framework is presently limited to logics axiomatised without nested modalities, but allows more flexible logic composition and covers also non-Kripke semantics. 2 Multisorted Mo... |

38 | Expressivity of coalgebraic modal logic: the limits and beyond. Theoret. Comput. Sci. In press
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Citation Context ...on function maps states to structured sets of observations and successor states, the latter taken from any of the available sorts. The interpretation of modal operators is based on predicate liftings =-=[17,20]-=-; in the multi-sorted setting, this takes the following shape. • Definition 6. A predicate lifting λ of profile λ : i1, . . . , ik → ∗ for a functor T : Set n → Set, where i1, . . . , ik ≤ n, is a nat... |

37 | A hierarchy of probabilistic system types. Theoretical Computer Science
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- 2004
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Citation Context ... probabilistic transitions, whereas simple Segala systems [24] have a two-layered structure where non-deterministic transitions lead to probability distributions over successor states. Bartels et al. =-=[1]-=- present 12 different types of probabilistic transition systems that arise as combinations of basic features. Here, we introduce a simple calculus that formalises the combination of features and estab... |

28 |
Probabilistic logic for type spaces
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(Show Context)
Citation Context ...emantics of modal logics typically involves a notion of state and transition, which can take a number of different forms. Transitions can be probabilistic or weighted, as in probabilistic modal logic =-=[15,9]-=- and graded modal logic [6,5], induced by joint actions of agents as in coalition logic [18], or non-monotonically conditioned as in conditional logic [3]. An attractive aspect of many of these logics... |

26 | Expressive logics for coalgebras via terminal sequence induction
- Pattinson
(Show Context)
Citation Context ...algorithmic template to realise decision procedures at the level of each individual feature. This is achieved by formalising the combined logics in a multi-sorted extension of coalgebraic modal logic =-=[17]-=- whose semantics is parametric in a set functor T ; models then appear as T -coalgebras. This pushes the generic PSPACE-decision procedure of [23], which works uniformly for such diverse logics as Hen... |

26 | PSPACE reasoning for graded modal logics
- Tobies
(Show Context)
Citation Context ...onal logic [3]. An attractive aspect of many of these logics is that they admit shallow models and are decidable in low complexity classes, in the absence of fixpoint operators typically PSPACE (e.g. =-=[25,18,23]-=-), i.e. the same as the standard modal logic K [2] and not dramatically worse than propositional logic. Features like non-determinism, probabilistic choice or joint actions are often combined, leading... |

24 | A finite model construction for coalgebraic modal logic
- Schröder
(Show Context)
Citation Context ...the next-operator of loc.cit. The main advantage of the new framework is that constructions such as cartesian product or disjoint union are no longer special cases and that the decision procedures of =-=[22,23]-=- generalise straightforwardly to the multi-sorted case. The multisorted approach to the complexity of composite modal logics complements transfer results obtained for the fusion of modal logics [26,10... |

21 | Finality regained: A coalgebraic study of Scott-sets and multisets
- D’Agostino, Visser
(Show Context)
Citation Context ...ally involves a notion of state and transition, which can take a number of different forms. Transitions can be probabilistic or weighted, as in probabilistic modal logic [15,9] and graded modal logic =-=[6,5]-=-, induced by joint actions of agents as in coalition logic [18], or non-monotonically conditioned as in conditional logic [3]. An attractive aspect of many of these logics is that they admit shallow m... |

19 | Algebraiccoalgebraic specification in cocasl
- Mossakowski, Reichel, et al.
(Show Context)
Citation Context ...our framework that allows for uniform results that can be instantiated to a large number of structurally different systems and logics. We recall some basic notions of multi-sorted coalgebra (cf. e.g. =-=[16]-=-), generalising the single-sorted setting [19]: 6sDefinition 5. We write Set for the category of sets and functions. Let Set S denote the category of S-sorted sets and S-sorted functions, with objects... |

18 | Combining modal logics
- Kurucz
- 2007
(Show Context)
Citation Context ...operators with associated source and target sorts [π1] : a → f [π2] : b → f � : f → a ♥ : f → b. The [πi] are postulated to commute with all boolean connectives. The well-known fusion La⊗Lb (cf. e.g. =-=[13]-=-) disjointly combines the axioms and modalities of La and Lb. One can translate back and forth between the fusion and formulas of sort f, taking e.g. the operator � of the fusion to the composite oper... |

15 |
so many possible worlds
- In
- 1972
(Show Context)
Citation Context ...ally involves a notion of state and transition, which can take a number of different forms. Transitions can be probabilistic or weighted, as in probabilistic modal logic [15,9] and graded modal logic =-=[6,5]-=-, induced by joint actions of agents as in coalition logic [18], or non-monotonically conditioned as in conditional logic [3]. An attractive aspect of many of these logics is that they admit shallow m... |

14 |
PSPACE reasoning for rank-1 modal logics
- Schröder, Pattinson
- 2006
(Show Context)
Citation Context ...ion and whose branches are models of negated substitution instances of rule premises as in the statement. From Theorem 17, we obtain a multi-sorted version of the generic PSPACE decision procedure of =-=[23]-=-. This requires to compute matchings of given pseudovaluations, and we require that the rules associated with features are reduction closed, i.e. it suffices to consider matchings ((φ1, . . . , φn)/ψ,... |

10 |
Complexity transfer for modal logic
- Hemaspaandra
- 1994
(Show Context)
Citation Context ...22,23] generalise straightforwardly to the multi-sorted case. The multisorted approach to the complexity of composite modal logics complements transfer results obtained for the fusion of modal logics =-=[26,10]-=- in the sense that our framework is presently limited to logics axiomatised without nested modalities, but allows more flexible logic composition and covers also non-Kripke semantics. 2 Multisorted Mo... |

3 |
Modular construction of modal logics. Theoret. Copmut. Sci. To appear. Earlier version
- Cîrstea, Pattinson
- 2004
(Show Context)
Citation Context ...se results imply PSPACE upper bounds for satisfiability w.r.t. general gluings, including the standard single-sorted semantics. Related Work. Our work is closely related to the framework presented in =-=[4,11]-=-, which focuses on completeness issues, with the main difference that our approach makes the multi-sorted nature of heterogeneous logics explicit by considering multisorted models. Our treatment of ty... |

1 | A semantic PSPACE criterion for the next 700 rank 0-1 modal logics. Available at http://www.informatik.uni-bremen.de/∼lschrode/papers/rank01pspace.pdf
- Schröder
(Show Context)
Citation Context ... gluing, we can construct a flat gluing with an equivalent satisfiability problem. For flat gluings, one can generalise existing model constructions and complexity results for coalgebraic modal logic =-=[22,23,21]-=-, and the relevant criteria reduce to the component logics; for the shallow-model-based PSPACE algorithm of [23], this is discussed in more detail in Sect. 5. We thus obtain compositional algorithmic ... |