Compositional Reasoning for Epistemic Logics

We provide decomposition and quotienting results for multi-modal logic with respect to a composition operator, traditionally used for epistemic models, due to van Eijck et al. (Journal of Applied NonClassical Logics 21(3–4):397–425, 2011), that involves sets of atomic propositions and valuation functions from Kripke models. While the composition operator was originally defined only for epistemic S5 models, our results apply to the composition of any pair of Kripke models. In particular, our quotienting result extends a specific result in the above mentioned paper by van Eijck et al. for the composition of epistemic models with disjoint sets of atomic propositions to compositions of any two Kripke models regardless of their sets of atomic propositions. We also explore the complexity of the formulas we construct in our decomposition result. Luca Aceto and Anna Ingólfsdóttir were partially supported by the project ‘Processes and Modal Logics’ (project nr. 100048021) of the Icelandic Research Fund. ICE-TCS, School of Computer Science, Reykjavik University, Reykjavik, Iceland. Email: [email protected] ICE-TCS, School of Computer Science, Reykjavik University, Reykjavik, Iceland. Email: [email protected] Dept. of Informatics – Univ. of Oslo, P.O. Box 1080 Blindern, N-0316 Oslo, Norway. E-mail: [email protected] Dept. of Mathematics and Statistics – California State Univ. Long Beach. E-mail: [email protected] 1 CONTENTS 2