[Paper Review] Abstracting Asynchronous Multi-Valued Networks: An Initial Investigation

L. Jason Steggles|arXiv (Cornell University)|Aug 17, 2011

Gene Regulatory Network Analysis14 references3 citations

TL;DR

This paper presents a decision procedure for abstracting asynchronous multi-valued networks (MVNs) using under-approximation to reduce state space while preserving key dynamical properties. By leveraging finite state graphs and step terms to represent abstract states, the method enables sound analysis of reachability and attractors, successfully capturing two of three attractors in a tryptophan biosynthesis model.

ABSTRACT

Multi-valued networks provide a simple yet expressive qualitative state based modelling approach for biological systems. In this paper we develop an abstraction theory for asynchronous multi-valued network models that allows the state space of a model to be reduced while preserving key properties of the model. The abstraction theory therefore provides a mechanism for coping with the state space explosion problem and supports the analysis and comparison of multi-valued networks. We take as our starting point the abstraction theory for synchronous multi-valued networks which is based on the finite set of traces that represent the behaviour of such a model. The problem with extending this approach to the asynchronous case is that we can now have an infinite set of traces associated with a model making a simple trace inclusion test infeasible. To address this we develop a decision procedure for checking asynchronous abstractions based on using the finite state graph of an asynchronous multi-valued network to reason about its trace semantics. We illustrate the abstraction techniques developed by considering a detailed case study based on a multi-valued network model of the regulation of tryptophan biosynthesis in Escherichia coli.

Motivation & Objective

• To address the state space explosion problem in asynchronous multi-valued network (MVN) models.
• To develop a formal abstraction theory for asynchronous MVNs that preserves key dynamical properties such as attractors and reachability.
• To overcome the infeasibility of direct trace inclusion checks due to infinite trace sets in asynchronous systems.
• To provide a decision procedure based on finite state graphs and step terms for verifying asynchronous abstractions.
• To enable practical analysis and comparison of complex MVN models through sound abstraction.

Proposed method

• Extends synchronous MVN abstraction theory to asynchronous settings using under-approximation to ensure soundness in reachability analysis.
• Defines abstraction correctness via trace inclusion, where the abstraction's traces are a subset of the original model’s traces.
• Introduces 'step terms' to represent sets of concrete states as abstract states, enabling finite representation of potential abstractions.
• Develops an iterative pruning algorithm over step terms to find consistent abstractions or detect infeasibility.
• Uses the finite state graph of the MVN as the foundation for reasoning about trace semantics and verifying abstraction correctness.
• Provides a formal proof of correctness for the decision procedure, ensuring that all attractors in the abstraction correspond to those in the original model.

Experimental results

Research questions

• RQ1 How can asynchronous multi-valued networks be abstracted while preserving key dynamical properties such as attractors and reachability?
• RQ2 What is a feasible decision procedure for verifying asynchronous abstractions when trace sets are infinite?
• RQ3 Can under-approximation be effectively used in asynchronous MVNs without introducing false positives?
• RQ4 How can step terms be used to represent abstract states in a way that supports efficient abstraction checking?
• RQ5 To what extent can abstraction reduce model complexity while maintaining interpretability and correctness?

Key findings

• The decision procedure correctly identifies valid asynchronous abstractions by iteratively pruning step terms based on consistency with the finite state graph.
• All attractors in the abstraction correspond to attractors in the original MVN, ensuring soundness for attractor-based analysis.
• The method successfully captured two of the three known attractors in a Boolean abstraction of the tryptophan biosynthesis regulatory network in E. coli.
• The approach enables sound inference of positive reachability properties from the abstraction to the original model.
• Over-approximation is shown to be problematic within the MVN framework, justifying the use of under-approximation.
• The abstraction theory supports step-wise refinement and comparison of MVNs at different levels of abstraction.