Istituto per La Ricerca Scientifica E Tecnologica New Upper Bounds for Satisfiability in Modal Logics { the Case-study of Modal K New Upper Bounds for Satissability in Modal Logics { the Case-study of Modal K

limited distribution notice This report has been submitted for publication outside of ITC and will probably be copyrighted if accepted for publication. It has been issued as a Technical Report for early dissemination of its contents. In view of thetransfer of copyright to the outside publisher, its distribution outside of ITC prior to publication should be limited to peer communications and speciic requests. After outside publication, requests should be lled only by reprints or legally obtained copies of the article. Abstract Traditional results in modal logics state that, if a formula ' is satissable in K (K-satissable), then it has a Kripke model M s.t. jjM jj 2 j'j , jjM jj being the number of states of M and j'j the number of subformulas of '. A further result states a bound of jjM jj j'j d , d being the modal depth of '. In more recent papers, a decision procedure has been proposed which branches on the truth values of the distinct \atoms" { i.e., subformulas which cannot be decomposed propositionally. Following this approach, we propose here two new bounds based on the number of distinct atoms jjAtoms(')jj: rst, a K-satissable formula ' has a Kripke model T s.t. jjT jj jjAtoms(')jj d ; second, the K-satissability of ' can be checked in O(j'j 2 jjAtoms(')jj) time and O(j'j) writable memory. Notice that jjAtoms(')jj j'j and j'j is up to 2 O(jjAtoms(')jj) .