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Minimum observability of probabilistic Boolean networks

Jiayi Xu, Shihua Fu, Liyuan Xia, Jianjun Wang

TL;DR

This work tackles the minimum observability problem for probabilistic Boolean networks by formulating a STP-based algebraic representation and transforming observability into a robust set reachability issue on an augmented system. It then identifies minimal H-distinguishable state sets that must be separated and derives necessary and sufficient conditions to add the least number of measurements to achieve observability, selecting among feasible sensor configurations. A constructive algorithm is provided to compute the smallest sensor count via truth matrices and reachability analysis, and the approach is illustrated on a four-node apoptosis PBN. The results extend observability analysis to stochastic logical networks, offering practical observer-design options with reduced sensing costs and broad applicability to deterministic BNs as well. The methodology advances the theoretical and computational toolkit for monitoring and controlling gene regulatory-like networks under uncertainty.

Abstract

This paper studies the minimum observability of probabilistic Boolean networks (PBNs), the main objective of which is to add the fewest measurements to make an unobservable PBN become observable. First of all, the algebraic form of a PBN is established with the help of semi-tensor product (STP) of matrices. By combining the algebraic forms of two identical PBNs into a parallel system, a method to search the states that need to be H-distinguishable is proposed based on the robust set reachability technique. Secondly, a necessary and sufficient condition is given to find the minimum measurements such that a given set can be H-distinguishable. Moreover, by comparing the numbers of measurements for all the feasible H-distinguishable state sets, the least measurements that make the system observable are gained. Finally, an example is given to verify the validity of the obtained results.

Minimum observability of probabilistic Boolean networks

TL;DR

This work tackles the minimum observability problem for probabilistic Boolean networks by formulating a STP-based algebraic representation and transforming observability into a robust set reachability issue on an augmented system. It then identifies minimal H-distinguishable state sets that must be separated and derives necessary and sufficient conditions to add the least number of measurements to achieve observability, selecting among feasible sensor configurations. A constructive algorithm is provided to compute the smallest sensor count via truth matrices and reachability analysis, and the approach is illustrated on a four-node apoptosis PBN. The results extend observability analysis to stochastic logical networks, offering practical observer-design options with reduced sensing costs and broad applicability to deterministic BNs as well. The methodology advances the theoretical and computational toolkit for monitoring and controlling gene regulatory-like networks under uncertainty.

Abstract

This paper studies the minimum observability of probabilistic Boolean networks (PBNs), the main objective of which is to add the fewest measurements to make an unobservable PBN become observable. First of all, the algebraic form of a PBN is established with the help of semi-tensor product (STP) of matrices. By combining the algebraic forms of two identical PBNs into a parallel system, a method to search the states that need to be H-distinguishable is proposed based on the robust set reachability technique. Secondly, a necessary and sufficient condition is given to find the minimum measurements such that a given set can be H-distinguishable. Moreover, by comparing the numbers of measurements for all the feasible H-distinguishable state sets, the least measurements that make the system observable are gained. Finally, an example is given to verify the validity of the obtained results.
Paper Structure (12 sections, 6 theorems, 48 equations, 1 figure, 2 tables, 2 algorithms)

This paper contains 12 sections, 6 theorems, 48 equations, 1 figure, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Symbols and STP of matrices
  4. Algebraic representation of PBNs
  5. Observability of PBNs
  6. Main Results
  7. The method to make a PBN observable
  8. The minimum observability of PBNs
  9. An illustrative example
  10. Conclusion
  11. Acknowledgements
  12. Appendix

Key Result

Lemma 2.1

ALRD Let $f:\mathcal{D}^r\rightarrow \mathcal{D}$ be a Boolean mapping, then there exists a unique $M_f\in\mathcal{L}_{2\times 2^r}$ such that where $x_i\in \Delta,\ i=1, 2, \cdots, r$ is the vector form of $X_i\in \mathcal{D}$, and $M_f$ is the structure matrix of $f$.

Figures (1)

  • Figure 1: The state transfer graph of $\mathcal{S}_1$.

Theorems & Definitions (18)

  • Lemma 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • Lemma 3.1
  • Definition 3.2
  • Remark 3.3
  • Definition 3.4
  • Lemma 3.5
  • ...and 8 more