On the complexity of the two-variable guarded fragment with transitive guards

We investigate the complexity of the satisfiability problem for the two-variable guarded fragment with transitive guards.We prove that the satisfiability problem for the monadic version of this logic without equality is 2EXPTIME-hard. It is in fact 2EXPTIME-complete, since as shown by Szwast and Tendera, the whole guarded fragment with transitive guards is in 2EXPTIME.We also introduce a new logic—the guarded fragment with one-way transitive guards and prove that the satisfiability problem for the two-variable version of this logic is EXPSPACE-complete. The two-variable guarded fragment with transitive guards can be seen as a counterpart of some branching temporal logics with both future and past operators, while the two-variable guarded fragment with one-way transitive guards corresponds to some branching temporal logics without past operators. Therefore, our results reveal the difference in the complexity of the reasoning about the future only and both the future and the past, in the two-variable guarded fragment with transitive guards. © 2006 Elsevier Inc. All rights reserved.