Counting in the two variable guarded logic with transitivity

Abstract

We show that the extension of the two-variable guarded fragment with transitive guards (GF+TG) by functionality statements is undecidable. This gives immediately undecidability of the extension of GF+TG by counting quantifiers. The result is optimal, since both the three-variable fragment of the guarded fragment with counting quantifiers and the two-variable guarded fragment with transitivity are undecidable. We also show that the extension of GF+TG with functionality, where functional predicate letters appear in guards only, is decidable and of the same complexity as GF+TG. This fragment captures many expressive modal and description logics. © Springer-Verlag Berlin Heidelberg 2005.