# Conservative extensions in modal logic

Ghilardi, S. and Lutz, C. and Wolter, F. and Zakharyaschev, Michael
(2006)
Conservative extensions in modal logic.
In:
Governatori, G. and Hodkinson, I.M. and Venema, Y. (eds.)
*Advances in Modal Logic 6.*
College Publications, pp. 187-207.
ISBN 9781904987208.

## Abstract

Every normal modal logic L gives rise to the consequence relation ' |=L which holds if, and only if, is true in a world of an L-model whenever ' is true in that world. We consider the following al- gorithmic problem for L. Given two modal formulas '1 and '2, decide whether '1^'2 is a conservative extension of'1 in the sense that whenever '1 ^'2 |=L and does not contain propositional variables not occurring in '1, then '1 |=L. We first prove that the conservativeness problem is coNExpTime-hard for all modal logics of unbounded width (which have rooted frames with more than N successors of the root, for any N < !). Then we show that this problem is (i) coNExpTime-complete for S5 and K, (ii) in ExpSpace for S4 and (iii) ExpSpace-complete for GL.3 (the logic of finite strict linear orders). The proofs for S5 and K use the fact that these logics have uniform interpolants of exponential size.

## Metadata

Item Type: | Book Section |
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School: | School of Business, Economics & Informatics > Computer Science and Information Systems |

Depositing User: | Sarah Hall |

Date Deposited: | 25 Oct 2021 17:18 |

Last Modified: | 25 Oct 2021 17:30 |

URI: | https://eprints.bbk.ac.uk/id/eprint/46481 |

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