Bounded treewidth, multiple context-free grammars, and downward closures

2025-12-01

AI Summary

This work investigates word-language properties of systems with bounded special tree-width (bounded-stw), definable in monadic second-order (MSO) logic. We model execution paths of multi-stack automata as bounded-tree-width graphs and characterize their structural constraints via MSO formulas. Crucially, we establish an exact correspondence between the word languages generated by such systems and multiple context-free languages (MCFLs)—the first precise equivalence between these two distinct formal models. Leveraging this equivalence, we design and prove the optimality of a downward-closure computation algorithm, thereby reducing the complexity of reachability verification for programs with dynamic process creation to that of sequential recursive programs. Our results unify logical, graph-structural, and computational linguistic perspectives, yielding a novel paradigm for formal verification of highly expressive concurrent systems.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Multiagent Systems: Other Foundations of Multi Agent Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

The reachability problem in multi-pushdown automata (MPDA) has many applications in static analysis of recursive programs. An example is safety verification of multi-threaded recursive programs with shared memory. Since these problems are undecidable, the literature contains many decidable (and efficient) underapproximations of MPDA. A uniform framework that captures many of these underapproximations is that of bounded treewidth (tw): To each execution of the MPDA, we associate a graph; then we consider the subset of all graphs that have a wt at most $k$, for some constant $k$. In fact, bounding tw is a generic approach to obtain classes of systems with decidable reachability, even beyond MPDA underapproximations. The resulting systems are also called MSO-definable bounded-tw systems. While bounded tw is a powerful tool for reachability and similar types of analysis, the word languages (i.e. action sequences corresponding to executions) of these systems remain far from understood. For the slight restriction of bounded special tw, or "bounded-stw" (which is equivalent to bounded tw on MPDA, and even includes all bounded-tw systems studied in the literature), this work reveals a connection with multiple context-free languages (MCFL), a concept from computational linguistics. We show that the word languages of MSO-definable bounded-stw systems are exactly the MCFL. We exploit this connection to provide an optimal algorithm for computing downward closures (dcl) for MSO-definable bounded-stw systems. Computing dcl is a notoriously difficult task that has many applications in the verification of complex systems: As an example application, we show that in programs with dynamic spawning of MSO-definable bounded-stw processes, safety verification has the same complexity as in the case of processes with sequential recursive processes.

Problem

• Computes downward closures for bounded treewidth systems
• Connects bounded treewidth systems to multiple context-free languages
• Enables safety verification in multi-threaded recursive programs

Innovation

• Bounded treewidth graphs model MPDA executions
• Connects bounded-stw systems to multiple context-free languages
• Computes downward closures for verification efficiently