Around Dot Depth Two

It is known that the languages definable by formulae of the logics FO2[<,S], ∆2[<,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 ∆2[<,S] is included in ∆3[<]; 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.) A rich set of correspondences has been worked out between diverse mechanisms for defining the first-order definable word languages and their subclasses (a recent survey is [DGK08]). In the following, CFA refers to counter-free automata, SFRE to star-free regular expressions and Ap refers to the variety of aperiodic monoids [Pin86]. CFA ≡ SFRE ≡ Ap ≡ FO[<] ≡ LTL[U,S] ≡ IT L Further, Thomas showed [Tho82] that by restricting the quantifier-alternation depth in the FO[<] formulae a strict dot-depth hierarchy of star-free languages is obtained, see the paper by Pin and Weil [PW97] for details. For example, B(Σ2)[<] is the class of languages defined by the boolean combination of Σ2[<] formulae, which are the ones which have one block of existential quantifiers followed by one block of universal quantifiers followed by a quantifierless formula. For the FO formulations below, given an alphabet A and a ∈ A, the unary predicate Qa(x) holds iff the letter at position x is a. The binary predicate S(x,y) denotes the successor relation on positions, and < is, as usual, its transitive closure. Example 1. Let A = {a,b} be the alphabet described by φA def = ∀x. Qa(x)∨Qb(x), which will be an additional conjunct below, not explicitly mentioned. – φ1 def = ∃x∃y. S(x,y)∧Qa(x)∧Qa(y) is a B(Σ1)[S] formula defining L1 = A∗aaA∗. – φ2 def = ∃x∃y. Qa(x)∧Qa(y)∧∀z. (x < z ⊃ y ≤ z) is a Σ2[<] formula defining L1. – Let φ3 def = (∀x. f irst(x) ⊃ Qa(x))∧ (∀x. last(x) ⊃ Qb(x))∧ (∀x,y. ((x < y)∧Qa(x)∧Qa(y) ⊃ ∃z. x < z∧ z < y∧Qb(z)))∧ (∀x,y. ((x < y)∧Qb(x)∧Qb(y) ⊃ ∃z. x < z∧ z < y∧Qa(z))) . Then, φ3 is a Π2[<] formula defining the language L2 = (ab)∗. ⊓⊔ More recently, Thérien and Wilke [TW98] showed that the 2-variable fragment FO2[<] [Mor75] (where only two variables occur, quantified any number of times), is expressively equivalent to the unambiguous languages and variety DA of Schützenberger [Sch76,TT02] and the subset ∆2[<] in the dot-depth hierarchy. Etessami, Vardi and Wilke [EVW02] identified the unary temporal logic LTL[F,P] and Schwentick, Thérien and Vollmer [STV02] identified partially-ordered 2-way deterministic finite automata (these are also called linear [LT00]) as equivalent formalisms. In [LPS08], we added to these correspondences a “deterministic” interval temporal logic called UITL. The papers [TW98,EVW02] also characterized FO2[<,S], which can define languages not definable in the logic FO2[<] such as those in Example 1. For a detailed study of these logics, see the recent papers of Weis and Immerman [WI07], and of Kufleitner and Weil [KW09]. PO2DFA ≡UL ≡ DA ≡ FO2[<] ≡ ∆2[<] ≡ LTL[F,P] ≡ UITL DA∗D ≡ FO2[<,S] ≡ ∆2[<,S] ≡ LTL[F,P,X,Y] It is clear that ∆2[<,S] ⊆ ∆3[<] since successor can be defined using < and one quantifier. In this paper we provide an automaton characterization and an interval logic characterization for this class of languages, and we separate it from B(Σ2)[<], the languages defined by the boolean combination of Σ2[<] formulae. This also shows that FO2[<,S] is a proper subset of ∆3[<], as diagrammatically depicted below.