Two Variable FO on Two Successors

Extensions of the two variable fragment of FO (henceforth denoted as FO) with interpreted predicates have been of interest recently in the context of verification of infinite state systems and semi-structured data (in particular XML). We briefly discuss one specific problem, namely FO over datawords. Datawords are linear orders (denoted by ≺ where ≺ denotes the successor relation) labelled by a pair of elements, one from a finite set (Σ) and one from an infinite domain (∆). When operations on ∆ are limited to only equality checking, these structures can also be seen as words with an equivalence relation (denoted by ∼) on its positions. In a landmark paper, the authors of [1] show that FO(Σ,≺,≺+,∼) is decidable. However, when an ordering of elements of ∆ (namely ≺+∆) is allowed, the logic (FO 2(Σ,≺,≺,∼,≺+∆)) becomes undecidable. We look at the scenario when ∼ is trivial and show that FO(Σ,≺1,≺2) is decidable [2]. Our proof is automata theoretic and along the way we derive an earlier result in [3]. We also discuss another result in [4] which shows that FO2(Σ,≺1 ,≺ + 2 ) is decidable.