Over Words, Two Variables Are as Powerful as One Quantiier Alternation: Fo 2 = 2 \ 2

We show a property of strings is expressible in the two-variable fragment of rst-order logic if and only if it is expressible by both a 2 and a 2 sentence. We thereby establish: where UTL stands for the string properties expressible in the temporal logic withèventually in the future' andèventually in the past' as the only temporal operators and UL stands for the class of unambiguous languages. This enables us to show that the problem of determining whether or not a given temporal string property belongs to UTL is decidable (in exponential space), which settles a hitherto open problem. Our proof of 2 \ 2 = FO 2 involves a new combinatorial characterization of these two classes and introduces a new method of playing Ehrenfeucht-Fra ss e games to verify identities in semigroups. While the number of variables required to express a certain graph property in rst-order logic is an important measure for the descriptional complexity of the property in question Imm91], the measure to classify properties of strings that has received most attention is the number of quantiier alternations. In fact, the number of quantiier alternations induces a strict hierarchy on the class of all rst-order expressible string properties which is identical with the dot-depth hierarchy CB71, BK78, Tho82, PP86], the natural hierarchy on the star-free sets. The reason the number of variables has played a minor role in the realm of strings is that three variables are suf-cient to express any rst-order expressible string property Kam68, IK89]. Only recently, it was, however, shown that even in the restricted setting of strings the number of variables used is a relevant parameter EVW97]: the 1