[Paper Review] First-Order Model Checking on Generalisations of Pushdown Graphs

Alexander Kartzow|arXiv (Cornell University)|Jan 1, 2011

Formal Methods in Verification8 citations

TL;DR

This paper establishes the decidability of first-order (FO) model checking on two generalizations of pushdown graphs: collapsible pushdown graphs of level 2 and nested pushdown trees. It introduces a tree automata-based representation for collapsible pushdown graphs of level 2 and uses FO interpretations to reduce model checking on nested pushdown trees to that on collapsible pushdown graphs, proving decidability in doubly exponential space. The key contribution is a uniform framework for FO model checking on higher-order and nested pushdown structures with decidable logic fragments.

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.

Motivation & Objective

• Address the decidability of first-order logic with reachability and Ramsey quantifiers on higher-order and collapsible pushdown graphs.
• Extend the model checking framework to nested pushdown trees by introducing a new hierarchy of higher-order nested pushdown trees.
• Establish connections between nested pushdown trees, collapsible pushdown graphs, and higher-order pushdown graphs via FO interpretations.
• Provide a uniform decidability result for FO model checking on collapsible pushdown graphs of level 2 and their extensions.
• Generalize decidability results from word-automatic to tree-automatic structures using Ramsey quantifiers.

Proposed method

• Construct a tree automata representation for every collapsible pushdown graph of level 2, enabling effective model checking.
• Define a new hierarchy of higher-order nested pushdown trees by combining nested pushdown relations with higher-order stacks.
• Use first-order interpretations to embed higher-order nested pushdown trees into collapsible pushdown graphs of level l+1.
• Prove decidability of FO logic extended with regular reachability predicates and Ramsey quantifiers on tree-automatic structures.
• Apply a generalized model-theoretic technique, building on joint work with Kuske, to show decidability over tree-automatic structures.
• Use game-theoretic and automata-theoretic methods to analyze definability and expressibility in the presence of complex stack operations.

Experimental results

Research questions

• RQ1 Is first-order model checking decidable on collapsible pushdown graphs of level 2, especially when extended with reachability and Ramsey quantifiers?
• RQ2 Can the model checking problem for nested pushdown trees be reduced to that of collapsible pushdown graphs, and what is its complexity?
• RQ3 How are the hierarchies of higher-order pushdown graphs, collapsible pushdown graphs, and nested pushdown trees related via first-order interpretations?
• RQ4 Does the addition of Ramsey quantifiers preserve decidability in the context of tree-automatic structures?
• RQ5 Can a uniform model checking framework be established for higher-order and nested pushdown systems using automata-theoretic representations?

Key findings

• First-order model checking is decidable on collapsible pushdown graphs of level 2 when extended with regular reachability predicates and Ramsey quantifiers.
• The complexity of first-order model checking on nested pushdown trees is doubly exponential in space, matching the upper bound for such structures.
• Every nested pushdown tree of level l can be defined via a first-order interpretation from a collapsible pushdown graph of level l+1.
• Higher-order nested pushdown trees of level l are first-order interpretable in collapsible pushdown graphs of level l+1, establishing a structural hierarchy.
• First-order logic extended with Ramsey quantifiers is decidable over tree-automatic structures, generalizing prior results on word-automatic structures.
• The class of higher-order nested pushdown trees is contained within the class of collapsible pushdown graphs of level l+1, and thus inherits decidability properties.