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We study an extension of F O 2 [<], first-order logic interpreted in finite words, in which formulas are restricted to use only two variables. We adjoin to this language two-variable atomic formulas that say, 'the letter a appears between positions x and y'. This is, in a sense, the simplest property that is not expressible using only two variables.We present several logics, both first-order and temporal, that have the same expressive power, and find matching lower and upper bounds for the complexity of satisfiability for each of these formulations. We also give an effective necessary condition, in terms of the syntactic monoid of a regular language, for a property to be expressible in this logic. We show that this condition is also sufficient for words over a two-letter alphabet. This algebraic analysis allows us us to prove, among other things, that our new logic has strictly less expressive power than full first-order logic F O [<].

Interval Temporal Logic [11] is a highly expressive and succinct logic whose satisfiability over finite words is non-elementary in the number of alternations of chop and negation operators. All the sublogics of ITL with elementary decidability known to us restrict this alternation depth. In this paper, we define a sublogic of Interval Temporal Logic by replacing chops with marked chops but without any restriction on the alternation depth. We show that the resulting logic admits unique parsing of a word matching a formula, with the consequence that membership is in LOGDCFL and satisfiability is in PSPACE. As our first result, we give an effective model-preserving reduction from UITL to the partially ordered two-way deterministic finite automata of Schwentick, Thérien and Vollmer [14]. We show that the size of the resulting automaton is quadratic in the size of the formula. We also have an exponential converse reduction from po2dfa to UITL. It follows from the work of Schützenberger [13], Thérien and Wilke [19] that this unambiguous ITL has same expressive power as the first-order logic with two variables [10].