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We show a property of strings is expressible in the two-variable fragment of first-order logic if and only if it is express ible by both a 2 and a 2 sentence. We thereby establish: UTL = FO2 = 2 \ 2 = UL ; where UTL stands for the string properties expressible in th e temporal logic with ‘eventually in the future’ and ‘eventua lly 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 = FO2 involves a new combinatorial characterization of these two classes and introduce s a new method of playing Ehrenfeucht-Fraı̈ssé games to verif y identities in semigroups.

Over words, two variables are as powerful as one quantifier alternation