First-Order Model Checking on Generalisations of Pushdown Graphs

TL;DR

This paper investigates the decidability of first-order model checking on generalized pushdown graphs, including nested and collapsible pushdown graphs, establishing new decidability results and interpretations within these classes.

Contribution

It proves decidability of first-order logic with reachability on nested pushdown trees and with regular reachability on level 2 collapsible pushdown graphs, extending the hierarchy of decidable classes.

Findings

First-order logic with reachability is decidable on nested pushdown trees.

Decidability of first-order logic without reachability in doubly exponential time.

First-order logic with regular reachability predicates is decidable on level 2 collapsible pushdown graphs.

Abstract

We study the first-order model checking problem on two generalisations of pushdown graphs. The first class is the class of nested pushdown trees. The other is the class of collapsible pushdown graphs. Our main results are the following. First-order logic with reachability is uniformly decidable on nested pushdown trees. Considering first-order logic without reachability, we prove decidability in doubly exponential alternating time with linearly many alternations. First-order logic with regular reachability predicates is uniformly decidable on level 2 collapsible pushdown graphs. Moreover, nested pushdown trees are first-order interpretable in collapsible pushdown graphs of level 2. This interpretation can be extended to an interpretation of the class of higher-order nested pushdown trees in the collapsible pushdown graph hierarchy. We prove that the second level of this new hierarchy of nested trees has decidable first-order model checking. Our decidability result for collapsible pushdown graph relies on the fact that level 2 collapsible pushdown graphs are uniform tree-automatic. Our last result concerns tree-automatic structures in general. We prove that first-order logic extended by Ramsey quantifiers is decidable on all tree-automatic structures.