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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 ..."
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Cited by 27 (16 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 (1 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
Ma jority Logic (Draft) \Lambda
, 2004
"... Abstract We extend graded modal logic (GML) to a logic that captures the concept of majority. We provide an axiomatization for majority logic, MJL, and sketch soundness and completeness proofs. Along the way, we must answer the question what is a majority of an infinite set? Majority spaces are intr ..."
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Abstract We extend graded modal logic (GML) to a logic that captures the concept of majority. We provide an axiomatization for majority logic, MJL, and sketch soundness and completeness proofs. Along the way, we must answer the question what is a majority of an infinite set? Majority spaces are introduced as a solution to this question.
Majority Logic: Axiomatization and Completeness
"... Abstract. 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 ∗ GML with a modal operator W that can express “there are more than or equal to half of the a ..."
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Abstract. 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 ∗ 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.