## PSPACE bounds for rank 1 modal logics (2006)

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Venue: | IN LICS’06 |

Citations: | 27 - 16 self |

### BibTeX

@INPROCEEDINGS{Schröder06pspacebounds,

author = {Lutz Schröder and Dirk Pattinson},

title = {PSPACE bounds for rank 1 modal logics},

booktitle = {IN LICS’06},

year = {2006},

publisher = {}

}

### OpenURL

### Abstract

For lack of general algorithmic methods that apply to wide classes of logics, establishing a complexity bound for a given modal logic is often a laborious task. The present work is a step towards a general theory of the complexity of modal logics. Our main result is that all rank-1 logics enjoy a shallow model property and thus are, under mild assumptions on the format of their axiomatisation, in PSPACE. This leads to a unified derivation of tight PSPACE-bounds for a number of logics including K, KD, coalition logic, graded modal logic, majority logic, and probabilistic modal logic. Our generic algorithm moreover finds tableau proofs that witness pleasant prooftheoretic properties including a weak subformula property. This generality is made possible by a coalgebraic semantics, which conveniently abstracts from the details of a given model class and thus allows covering a broad range of logics in a uniform way.

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Citation Context ...mples of intractable rule sets (contrived examples are easy to construct, e.g. by using hard side conditions). The next lemma, which follows directly from size estimates in linear integer programming =-=[28]-=-, is crucial for establishing PSPACE-tractability in the examples. Following usual practice, we take the size |W | of a rational inequality W ≡ ( �n i=1 uixi op u0), op ∈ {<, ≤, >, ≥}, to be 1 + n + �... |

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Citation Context ...ed modal logic [31] and coalition logic [23]. Moreover, our method goes beyond re-proving known results in a uniform fashion: we obtain a previously unknown PSPACE-bound for probabilistic modal logic =-=[17, 13]-=-, with rational probabilities coded in binary. These logics are far from exotic: graded modal logic plays a role e.g. in decision support and knowledge representation [33, 19], and probabilistic modal... |

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Citation Context ...earing in [30]; this is the subject of further research, as is the extension of the theory beyond rank 1 by means ofscomonads and the treatment of iteration, possibly using automata theoretic methods =-=[35, 36]-=-. Ongoing work indicates that every modal logic can be equipped with a coalgebraic semantics, provided it is axiomatisable in rank 1 and satisfies the congruence rule. This means in particular that th... |

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Citation Context ...ions φ ∨ ψ, truth ⊤, and other boolean operations are defined as usual. The rank of a formula is its maximal nesting depth of modal operators; note however that the notion of rank-1 axiom employed in =-=[21, 8, 15, 30]-=-, replaced below by the notion of one-step rule, is stricter than ‘formula of rank 1’ in that it disallows non-trivial subformulae of rank 0. The satisfaction relation |=C between states x of a T - co... |

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Citation Context ...coinduction [2], corecursion [32], duality, and ultrafilter extensions [15]. Coalgebraic modal logic features in actual specification languages such as the object oriented specification language CCSL =-=[26]-=- and COCASL [18]. The coalgebraic study of computational aspects of modal logic was initiated in [30], where the finite model property and associated NEXPTIME -bounds were proved. Here, we push this f... |

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Citation Context ... (i.e. states identified by some pair of morphisms) have the same theory [22], and we can – in case T is accessible – always find enough (polyadic) modal operators to distinguish nonequivalent states =-=[29]-=-. In the interest of readability, we restrict our attention to unary modalities for the purpose of this work. However, we remark that our treatment extends to the polyadic case in a straightforward ma... |

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Citation Context ...otherwise, TX = ∅ for all X). Modal logic in the form considered here has been introduced as a specification logic for coalgebraically modelled reactive systems in [22], generalising previous results =-=[14, 25, 16, 20]-=-. The coalgebraic semantics is based on predicate liftings, which abstract from the concrete interpretation of modal operators in the same way that the signature functor abstracts from a concrete clas... |

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Citation Context ...otherwise, TX = ∅ for all X). Modal logic in the form considered here has been introduced as a specification logic for coalgebraically modelled reactive systems in [22], generalising previous results =-=[14, 25, 16, 20]-=-. The coalgebraic semantics is based on predicate liftings, which abstract from the concrete interpretation of modal operators in the same way that the signature functor abstracts from a concrete clas... |

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Citation Context ...ed modal logic [31] and coalition logic [23]. Moreover, our method goes beyond re-proving known results in a uniform fashion: we obtain a previously unknown PSPACE-bound for probabilistic modal logic =-=[17, 13]-=-, with rational probabilities coded in binary. These logics are far from exotic: graded modal logic plays a role e.g. in decision support and knowledge representation [33, 19], and probabilistic modal... |

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Citation Context ... we arrive, due to minor differences of the rule sets, at a slight variant of Pauly’s PSPACE-algorithm [23]. 3. For graded modal logic, we obtain a new algorithm which confirms the known PSPACE bound =-=[31]-=-. One might claim that the new algorithm is not only nicely embedded into a unified framework, but also conceptually simpler than the constraint-based algorithm of [31] (which corrects an erroneous al... |

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Citation Context ...or the modal logic K [4], but also using markedly different methods such as in the constraintbased PSPACE-algorithm for graded modal logic [31]. A first glimpse of a generalisable method was given in =-=[34]-=-, where various epistemic logics, equipped with a neighborhood frame semantics, were shown to be in NP and PSPACE, respectively (with the K axiom being responsible for PSPACE-hardness; recent work [11... |

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Citation Context ...robabilistic modal logic [17, 13], with rational probabilities coded in binary. These logics are far from exotic: graded modal logic plays a role e.g. in decision support and knowledge representation =-=[33, 19]-=-, and probabilistic modal logic has appeared in connection with model checking [17] and in modelling economic behaviour [13]. The key to the generality is to parametrise the theory over the type of sy... |

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Citation Context ...features in actual specification languages such as the object oriented specification language CCSL [26] and COCASL [18]. The coalgebraic study of computational aspects of modal logic was initiated in =-=[30]-=-, where the finite model property and associated NEXPTIME -bounds were proved. Here, we push this further by proving a coalgebraic shallow model property. Our PSPACE-algorithm traverses a shallow mode... |

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Citation Context ...stracts from the details of a concrete class of models by encoding it as an endofunctor on the category of sets. As specific instances, one obtains e.g. Kripke frames, (monotone) neighbourhood frames =-=[12]-=-, game frames [23], probabilistic transition systems and automata [24, 3], weighted automata, linear automata [6], and multigraphs [9]. Despite the broad range of systems covered by the coalgebraic ap... |

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Citation Context ...ions φ ∨ ψ, truth ⊤, and other boolean operations are defined as usual. The rank of a formula is its maximal nesting depth of modal operators; note however that the notion of rank-1 axiom employed in =-=[21, 8, 15, 30]-=-, replaced below by the notion of one-step rule, is stricter than ‘formula of rank 1’ in that it disallows non-trivial subformulae of rank 0. The satisfaction relation |=C between states x of a T - co... |

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Citation Context ...e the broad range of systems covered by the coalgebraic approach, a substantial body of concepts and non-trivial results has emerged, encompassing e.g. generic notions of bisimilarity and coinduction =-=[2]-=-, corecursion [32], duality, and ultrafilter extensions [15]. Coalgebraic modal logic features in actual specification languages such as the object oriented specification language CCSL [26] and COCASL... |

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Citation Context ...earing in [30]; this is the subject of further research, as is the extension of the theory beyond rank 1 by means ofscomonads and the treatment of iteration, possibly using automata theoretic methods =-=[35, 36]-=-. Ongoing work indicates that every modal logic can be equipped with a coalgebraic semantics, provided it is axiomatisable in rank 1 and satisfies the congruence rule. This means in particular that th... |

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