Around Dot Depth Two

It is known that the languages definable by formulae of the logics FO2[<,S], Δ2[<,S], LTL[F,P,X,Y] are exactly the variety DA*D. Automata for this class are not known, nor is its precise placement within the dot-depth hierarchy of starfree languages. It is easy to argue that Δ2[<,S] is included in Δ3[<]; in this paper we show that it is incomparable with B(Σ2)[<], the boolean combination of Σ2[<] formulae. Using ideas from Straubing's "delay theorem", we extend our earlier work [LPS08] to propose partially-ordered two-way deterministic finite automata with look-around (po2dla) and a new interval temporal logic called LITL and show that they also characterize the variety DA*D. We give effective reductions from LITL to equivalent po2dla and from po2dla to equivalent FO2[<,S]. The po2dla automata admit efficient operations of boolean closure and the language non-emptiness of po2dla is NP-complete. Using this, we show that satisfiability of LITL remains NP-complete assuming a fixed look-around length. (Recall that for LTL[F,X], it is PSPACE-hard.)

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