# Model-Checking on Ordered Structures

## TL;DR

This paper investigates the complexity of model-checking for first- and monadic second-order logic on ordered structures, revealing intractability in general but tractability under certain order-invariant conditions on specific graph classes.

## Contribution

It extends understanding of model-checking complexity by analyzing order-invariant logic on classes of structures, showing tractability results where ordering does not affect truth.

## Key findings

- Order-invariant MSO model-checking is tractable on certain graph classes.
- First-order successor-invariant formulas are tractable on graphs with bounded expansion.
- Model-checking is tractable on coloured posets of bounded width.

## Abstract

We study the model-checking problem for first- and monadic second-order logic on finite relational structures. The problem of verifying whether a formula of these logics is true on a given structure is considered intractable in general, but it does become tractable on interesting classes of structures, such as on classes whose Gaifman graphs have bounded treewidth. In this paper we continue this line of research and 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 structures. In contrast, in the order invariant case we obtain tractability results for order-invariant monadic second-order formulas on the same classes of graphs as in the unordered case. For first-order logic, we obtain tractability of successor-invariant formulas on classes whose Gaifman graphs have bounded expansion. Furthermore, we show that model-checking for order-invariant first-order formulas is tractable on coloured posets of bounded width.