Scalable Counting of Minimal Trap Spaces and Fixed Points in Boolean Networks

TL;DR

This paper introduces efficient approximate methods for counting minimal trap spaces and fixed points in Boolean Networks, enabling analysis of large-scale systems in biology, logic, and AI.

Contribution

It presents novel approximate answer set counting techniques tailored for large Boolean Networks, improving computational feasibility over exhaustive enumeration.

Findings

Significant speed-up in counting minimal trap spaces and fixed points.

Effective approximation accuracy demonstrated on diverse benchmarks.

Enhanced scalability for large and complex Boolean Networks.

Abstract

Boolean Networks (BNs) serve as a fundamental modeling framework for capturing complex dynamical systems across various domains, including systems biology, computational logic, and artificial intelligence. A crucial property of BNs is the presence of trap spaces -- subspaces of the state space that, once entered, cannot be exited. Minimal trap spaces, in particular, play a significant role in analyzing the long-term behavior of BNs, making their efficient enumeration and counting essential. The fixed points in BNs are a special case of minimal trap spaces. In this work, we formulate several meaningful counting problems related to minimal trap spaces and fixed points in BNs. These problems provide valuable insights both within BN theory (e.g., in probabilistic reasoning and dynamical analysis) and in broader application areas, including systems biology, abstract argumentation, and logic programming. To address these computational challenges, we propose novel methods based on {\em approximate answer set counting}, leveraging techniques from answer set programming. Our approach efficiently approximates the number of minimal trap spaces and the number of fixed points without requiring exhaustive enumeration, making it particularly well-suited for large-scale BNs. Our experimental evaluation on an extensive and diverse set of benchmark instances shows that our methods significantly improve the feasibility of counting minimal trap spaces and fixed points, paving the way for new applications in BN analysis and beyond.