Expressivity and Succinctness of Order-Invariant Logics on Depth-Bounded Structures

Abstract

Abstract. We study the expressive power and succinctness of orderinvariant sentences of first-order (FO) and monadic second-order (MSO) logic on graphs of bounded tree-depth. Order-invariance is undecidable in general and, therefore, in finite model theory, one strives for logics with a decidable syntax that have the same expressive power as orderinvariant sentences. We show that on graphs of bounded tree-depth, order-invariant FO has the same expressive power as FO, and orderinvariant MSO has the same expressive power as the extension of FO with modulo-counting quantifiers. Our proof techniques allow for a finegrained analysis of the succinctness of these translations. We show that for every order-invariant FO sentence there exists an FO sentence whose size is elementary in the size of the original sentence, and whose number of quantifier alternations is linear in the tree-depth. Our techniques can be adapted to obtain a similar quantitative variant of a known result that the expressive power of MSO and FO coincides on graphs of bounded tree-depth.

Successor-Invariant First-Order Logic on Classes of Bounded Degree

“…Far from it, < -inv FO and a fortiori Succ-inv FO are known to collapse to FO on several sparse classes. Benedikt and Segoufin [BS09] proved the collapse on trees; Eickmeyer, Elberfeld and Harwarth [EEH14] obtained an analogous result on graphs of bounded tree-depth; Grange and Segoufin [GS20] proved the collapse on hollow trees.…”

“…Related work: The general method used in [EEH14] to prove that < -inv FO collapses to FO when the tree-depth is bounded is the same as ours: starting from two FO-similar structures, they show how to construct orders that maintain the similarity. However, the techniques we use to construct our successors are nothing like the ones used in [EEH14], as the settings are very different.…”

Model-checking for successor-invariant first-order formulas on graph classes of bounded expansion

“…On the other hand, collapse results in this context are known only for very restricted settings. It is known that order-invariant FO collapses to plain FO on trees [1,28] and on graphs of bounded treedepth [9]. Moreover, order-invariant FO is a subset of MSO on graphs of bounded degree and on graphs of bounded treewidth [1], and more generally, on decomposable graphs in the sense of [12].…”

Succinctness of Order-Invariant Logics on Depth-Bounded Structures

“…As a further result, we show that order-invariant MSO has the same expressive power as FO with modulo-counting quantifiers on bounded tree-depth structures. * A preliminary version of this paper was presented at the mfcs 2014 conference [7].…”