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PSPACE bounds for rank 1 modal logics
 IN LICS’06
, 2006
"... 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 rank1 logics enjoy a sh ..."
Abstract

Cited by 25 (15 self)
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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 rank1 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 PSPACEbounds 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.
Majority logic
 In KR 2004, Principles of Knowledge Representation and Reasoning: Proceedings of the Ninth International Conference
, 2004
"... Graded modal logic, as presented in [5], extends propositional modal systems with a set of modal operators ✸n (n ∈ N) that express “there are more than n accessible worlds such that...”. We extend 1 GML with a modal operator W that can express “there are more than or equal to half of the accessible ..."
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Cited by 17 (0 self)
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Graded modal logic, as presented in [5], extends propositional modal systems with a set of modal operators ✸n (n ∈ N) that express “there are more than n accessible worlds such that...”. We extend 1 GML with a modal operator W that can express “there are more than or equal to half of the accessible worlds such that...”. The semantics of W is straightforward provided there are only finitely many accessible worlds; however if there are infinitely many accessible worlds the situation becomes much more complex. In order to deal with such situations, we introduce a majority space. A majority space is a set W together with a collection of subsets of W intended to be the weak majority (more than or equal to half) subsets of W. We then extend a standard Kripke structure with a function that assigns a majority space over the set of accessible states to each state. Given this extended Kripke semantics, majority logic is proved sound and complete. 1
PSPACE bounds for rank 1 modal logics
 In LICS’06
, 2006
"... 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 rank1 logics enjoy a sh ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
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 rank1 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 PSPACEbounds 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.