3 research outputs found
Model-checking for successor-invariant first-order formulas on graph classes of bounded expansion
A successor-invariant first-order formula is a formula that has access to an auxiliary successor relation on a structure's universe, but the model relation is independent of the particular interpretation of this relation. It is well known that successor-invariant formulas are more expressive on finite structures than plain first-order formulas without a successor relation. This naturally raises the question whether this increase in expressive power comes at an extra cost to solve the model-checking problem, that is, the problem to decide whether a given structure together with some (and hence every) successor relation is a model of a given formula. It was shown earlier that adding successor-invariance to first-order logic essentially comes at no extra cost for the model-checking problem on classes of finite structures whose underlying Gaifman graph is planar [1], excludes a fixed minor [2] or a fixed topological minor [3], [4]. In this work we show that the model-checking problem for successor-invariant formulas is fixed-parameter tractable on any class of finite structures whose underlying Gaifman graphs form a class of bounded expansion. Our result generalises all earlier results and comes close to the best tractability results on nowhere dense classes of graphs currently known for plain first-order logic
Model-Checking on Ordered Structures
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.Comment: arXiv admin note: substantial text overlap with arXiv:1701.0851