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A sound and complete tableau calculus for reasoning about only knowing and knowing at most

Riccardo Rosati.

*Studia Logica*, volume 69, pages 171-191, Kluwer Academic Publisher, 2001. ISSN 0039-3215.

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Abstract:

We define a tableau calculus for the logic of only knowing and knowing at most ONL, which is an extension of Levesque's logic of only knowing OL. The method is based on the possible-world semantics of the logic ONL, and can be considered as an extension of known tableau calculi for modal logic K45. From the technical viewpoint, the main features of such an extension are the explicit representation of "unreachable" worlds in the tableau, and an additional branch closure condition implementing the property that each world must be either reachable or unreachable. The calculus allows for establishing the computational complexity of reasoning about only knowing and knowing at most. Moreover, we prove that the method matches the worst-case complexity lower bound of the satisfiability problem for both ONL and OL.

### Bibtex entry:

@String{SL = "Studia Logica"}
@String{KAP = "Kluwer Academic Publisher"}
@Article{Rosa00d,
author = "Riccardo Rosati",
title = "A sound and complete tableau calculus for reasoning about only knowing and knowing at most",
journal = SL,
volume = 69,
pages = "171--191",
publisher = KAP,
year = 2001,
issn = "0039-3215",
}

Link to electronic version of published paper