3-SAT = SAT for a class of normal modal logics

International audienceFor the classical propositional logic SAT polynomially reduces to 3-SAT. In this note we show that for some propositional normal modal logics a similar result holds when modal clauses are considered. The proof given herein is a modal extension of Cook's proof for classical propositional logic. The difficulty of the proof rests on the use of a modal context for each reduction step. Improvements of the reduction algorithm for logics extending S4 are also given. Possible extensions of this work are also briefly discussed