Exptime Tableaux for ALC

We propose a tableaux calculus requiring simple expo nential time for satis ability of an ALC concept C wrt a TBox T containing general axioms of the form C v D From correspondences with Propositional Dynamic Logic PDL it is known that this problem is in EX PTIME Pratt Vardi and Wolper How ever an algorithm directly derived from the methods used to prove such a result would always require expo nential time and space even in simple cases e g when a simple model satisfying both T and C can be easily found On the other hand proposed tableaux methods Buch heit et al which explore a space of candidate models for T and C starting from simple ones can take advantage of such cases However there can be an ex ponential number of possibly exponential size candidate models Hence an algorithm based on tableaux methods requires doubly exponential time in the worst case We devise a re ned tableaux calculus that integrates the techniques used in PDL with tableaux thus achiev ing a tableaux based procedure working in simple expo nential time In a nutshell traditional tableaux methods close a branch only by rst principles atomic clashes whereas our enhanced tableau exploits previously proved inconsistencies as additional lemmata to decide that a branch can be closed without having to nd the same atomic clashes again For sake of brevity we assume wlog that all concepts are expressed in negation normal form i e we form con cepts in ALC by means of the following syntax A de notes a concept name C and D arbitrary concepts and R a role name