Abstract

. We show a property of strings is expressible in the twovariable fragment of first-order logic if and only if it is expressible by both a \Sigma 2 and a \Pi 2 sentence. We thereby establish: UTL = FO 2 = \Sigma 2 " \Pi 2 = UL ; where UTL stands for the string properties expressible in the temporal logic with `eventually in the future' and `eventually 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 \Sigma 2 " \Pi 2 = FO 2 involves a new combinatorial characterization of these two classes and introduces a new method of playing Ehrenfeucht-Fraiss'e games to verify identities in semigroups. While the number of variables required to express a certain graph property in first-order logic is an important measure for the descriptional...

Citations