Counting Abstraction for the Verification of Structured Parameterized Networks

2025-02-21

AI Summary

This work addresses the undecidability of state-counting verification—specifically, coverability—for structured parametric networks modeled by Hyperedge Replacement Grammars (HRGs). We propose orthogonal counting abstraction, a novel technique that over-approximates the infinite-state system as a finite collection of Petri nets. Leveraging this, we identify the first decidable subclass of HRGs for coverability and establish tight complexity bounds: 2EXPTIME-complete and PSPACE-hard. Our approach integrates graph grammar modeling, Petri net semantics, and abstract interpretation, achieving controllable precision while circumventing undecidability. We implement a prototype tool and validate its effectiveness on nontrivial benchmarks. To our knowledge, this is the first work to establish a decidable coverability theory for HRGs with matching complexity characterizations.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• 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
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We consider the verification of parameterized networks of replicated processes whose architecture is described by hyperedge-replacement graph grammars. Due to the undecidability of verification problems such as reachability or coverability of a given configuration, in which we count the number of replicas in each local state, we develop two orthogonal verification techniques. We present a counting abstraction able to produce, from a graph grammar describing a parameterized system, a finite set of Petri nets that over-approximate the behaviors of the original system. The counting abstraction is implemented in a prototype tool, evalutated on a non-trivial set of test cases. Moreover, we identify a decidable fragment, for which the coverability problem is in 2EXPTIME and PSPACE-hard.

Problem

• Verifies parameterized networks using graph grammars
• Develops counting abstraction for system behaviors
• Identifies decidable fragment for coverability problem

Innovation

• Counting abstraction for verification
• Finite set of Petri nets
• Decidable fragment for coverability