[Paper Review] Verification of infinite-step and K-step opacity Using Petri Nets

Hao Lan, Yin Tong+2 more|arXiv (Cornell University)|Sep 9, 2019

Petri Nets in System Modeling22 references4 citations

TL;DR

This paper proposes efficient verification methods for infinite-step and K-step opacity in bounded labeled Petri nets using basis reachability graphs (BRG) and a modified two-way observer (TW-observer). By leveraging structural analysis and a refined observer construction, the approach reduces computational complexity compared to prior methods, enabling scalable opacity verification in discrete event systems with secret state hiding guarantees.

ABSTRACT

This paper addresses the problem of infinite-step opacity and K-step opacity of discrete event systems modeled with Petri nets. A Petri net system is said to be infinite-step/K-step opaque if all its secret states remains opaque to an intruder for any instant within infinite/K steps. In other words, the intruder is never able to ascertain that the system used to be in a secrete state within infinite/K steps based on its observation of the systems evolution. Based on the notion of basis reachability and the twoway observer, an efficient approach to verify infinite-step opacity and K-step opacity is proposed.

Motivation & Objective

• To formalize infinite-step and K-step opacity in bounded labeled Petri net (LPN) systems.
• To address the challenge of verifying opacity in LPNs, where traditional reachability graph methods are computationally expensive.
• To propose a more efficient verification method by exploiting the basis reachability graph (BRG) instead of the full reachability graph.
• To reduce computational complexity through a modified TW-observer structure for infinite-step opacity verification.
• To establish necessary and sufficient conditions for both opacity properties using BRG-based analysis under a reasonable assumption on secret markings.

Proposed method

• Define secret states as a subset of reachable markings in a bounded LPN, with the assumption that all markings reachable via unobservable transitions from a secret basis marking are also considered secret.
• Construct the basis reachability graph (BRG) of the LPN to represent the system’s behavior compactly, significantly reducing state space compared to the full reachability graph.
• Introduce a two-way observer (TW-observer) built from concurrent composition of a forward and reverse observer on the BRG to track observable behavior and detect potential secret state inferences.
• Propose a modified TW-observer for infinite-step opacity that reduces transition count by avoiding redundant observer composition, lowering complexity from O(|Eₒ|×2^|X|×2^|X|) to the same asymptotic bound but with improved constant factors.
• Use the K-reduced TW-observer for K-step opacity verification, which limits the search depth to K steps in the reverse observer component to bound complexity.
• Apply Theorems V.5 and V.7 to check opacity by verifying that no state in the TW-observer has a non-empty intersection of forward and reverse estimates that is fully contained in the secret set.

Experimental results

Research questions

• RQ1 How can infinite-step and K-step opacity be formally defined in the context of labeled Petri nets?
• RQ2 Can the basis reachability graph (BRG) be used to verify opacity more efficiently than the full reachability graph?
• RQ3 What is the computational complexity of verifying infinite-step and K-step opacity using the BRG and TW-observer framework?
• RQ4 How can a modified TW-observer reduce complexity compared to the standard TW-observer construction in prior work?
• RQ5 Under what conditions is a labeled Petri net system guaranteed to be infinite-step or K-step opaque?

Key findings

• The proposed method verifies infinite-step and K-step opacity using the basis reachability graph (BRG), which is significantly smaller than the full reachability graph, leading to reduced computational cost.
• The modified TW-observer for infinite-step opacity reduces the number of transitions compared to the original TW-observer construction, resulting in lower computational complexity.
• The computational complexity of the modified TW-observer is O(|Eₒ|×2^|X|×2^|X|), matching the best-known bound but with improved constant factors due to optimized construction.
• For K-step opacity, the K-reduced TW-observer enables bounded verification within K steps, ensuring scalability for small K values.
• Theoretical conditions are established: an LPN is infinite-step opaque iff no state in the modified TW-observer has a non-empty intersection of forward and reverse estimates fully contained in the secret set.
• The method is proven correct under the assumption that all markings reachable via unobservable transitions from a secret basis marking are also considered secret, which is shown to be reasonable and practical.