First-Order and Monadic Second-Order Model-Checking on Ordered Structures

Model-checking for first- and monadic second-order logic in the context of graphs has received considerable attention in the literature. It is well-known that the problem of verifying whether a formula of these logics is true in a graph is computationally intractable but it does become tractable on interesting classes of graphs such as classes of bounded tree-width. In this paper we continue this line of research but study model checking for first- and monadic second-order logic in the presence of an ordering on the input structure. We do so in two settings: the general ordered case, where the input structures are equipped with a fixed order or successor relation, and the order invariant case, where the formulas may resort to an ordering but their truth must be independent of the particular choice of order. In the first setting we show very strong intractability results for most interesting classes of graphs. In contrast, in order invariant case we obtain tractability results for order invariant monadic second-order logic on the same classes of graphs as in the unordered case. For first-order logic, we obtain tractability of successor-invariant FO on planar graphs.