Detectability vverification of probabilistic Boolean networks

doi:10.1016/j.ins.2020.10.019

Information Sciences

Volume 548, 16 February 2021, Pages 313-327

https://doi.org/10.1016/j.ins.2020.10.019 Get rights and content

Abstract

We in this paper study the current-state estimation problem of probabilistic Boolean networks with output observations. First, we propose the concepts of three fundamental categories of detectability in the context of PBNs based on the different purposes, which are periodic detectability, (periodic) k-detectability, and (periodic) d-detectability. Second, utilizing the semi-tensor product technique, we create a uniform methodology for the verification of all the aforementioned categories of detectability. Accordingly, several necessary and sufficient verification criteria are derived. All results obtained in this paper are numerically tractable since they avoid the graph-based symbolic manipulations. Finally, several examples modeled by the same PBN are provided to instantiate the correctness and effeteness of the obtained results.

Introduction

The human Genome Project has inspired the appearance of systems biology, which studies the global behavior of all cells and proteins rather than individual cells, proteins or genes [13]. Kauffman first introduced the Boolean networks (BNs) to model cellular networks for describing the relationship and behavior of cells or genes [14]. Compared with the deterministic BNs, the property of rule-based uncertainty for probabilistic Boolean networks (PBNs) was applied to gene regulatory networks [18]. Since then, BNs and PBNs have drawn considerable attention and have been investigated widely, see, e.g., [2], [3], [4], [5], [11], [12], [17], [24], [30], [31], [33] and the references therein.

State estimation in BNs is an interesting and important issue and has many practical applications in medical engineering. In the context of BNs, state estimation has been studied in the manners of observability [6], [15], [16], [22], [26], [32] and reconstructibility (also called detectability) [8], [27]. The former discusses whether or not we unambiguously capture the initial-state of a system based on the input and/or output observations. However, one, in some applications, may be only interested in capturing the current-state of a system, and it is dispensable to know all past states. For this reason, a weaker version of state estimate, called reconstructibility or detectability, appears to be more useful. It requires that we should utilize an input and/or output observation to ascertain the current-state of a system via a finite-length delay. For more introductions, see the recent monograph [28].

Recently, the concepts of two versions of detectability in the context of PBNs with output observations, called strong detectability and weak detectability, are proposed in [20]. Strong (resp., weak) detectability captures whether or not the current-state of a PBN can be exactly known at all times after a finite delay for all (resp., some) output observations. Meanwhile, the matrix-based criteria of verifying them are provided using the semi-tensor product (STP) technique. However, the detection requirements in [20] may be too strong in practice. For instance, in some practical applications, knowing the current-state of a PBN at certain times or within a small uncertainty set after finite observations may be sufficient. On the other hand, in diagnosis, sensor activation, and distributed control with communication problems, one only needs to distinguish certain state-pairs rather than the entire state space [21], [25]. To this end, inspired by [9], [19], we in this paper introduce the weaker notions of three categories of detectability in the context of PBNs, called periodic detectability, k-detectability and d-detectability, to capture the aforementioned requirements, respectively. Note that, each one of them can be seen as a generalization of detectability proposed in [20] (see Sections 4 Periodic detectability, 2 Preliminaries, 3 Matrix-based representation for PBNs and stochastic automata, 4 Periodic detectability, 5 , 6 in this paper). In the literature, to our knowledge there are no approaches for verifying periodic detectability, k-detectability, and d-detectability of PBNs.

In this paper, we create a uniform methodology based on the STP technique to tackle simultaneously the verification of these three categories of detectability in PBNs. Specifically, our contributions are summarized as follows.

•
The three new weaker categories of detectability in the context of PBNs, called periodic detectability, k-detectability, and d-detectability, are proposed and investigated.
•
We provide a matrix-based approach of how to transform a PBN into a stochastic automaton. Based on the observer automata presented in [19], we give the observer-based criteria for the verification of strong (resp., weak) periodic detectability of PBNs.
•
To avoid the graph-based symbolic manipulations, we further develop a uniform matrix-based methodology to check simultaneously periodic detectability, k-detectability, and d-detectability of PBNs. This leads to an advantage that all obtained results are tractable numerically.

We discuss the differences between this work and several relevant works in the literature. The authors in [20] only investigated the verification of non-periodic detectability for PBNs. While we study the problems of verifying periodic detectability, k-detectability, and d-detectability for PBNs, which have not been considered in [20]. In [9], [19], the notions of these three categories of non-probabilistic detectability are proposed in the setting of logical discrete event systems, but they are distinct to the stochastic setting modeled by a PBN in our work. Also, the verification approach developed in this paper is totally different from [9], [19]. Our approaches are the matrix-based techniques rather than the graph-based symbolic manipulations.

This paper is organized in seven parts. Some preliminaries including the basic notations, the STP of matrices and PBNs are provided to understand this work in Section 2. Section 3 focuses on stochastic automata-based and matrix-based representations for PBNs and the transition relationship between them. Section 4 provides the observer-based and the matrix-based approaches to verify periodic detectability for PBNs, respectively. In Section 5, we define four versions of k-detectability for PBNs, and then present the matrix-based approach to verify them. In Section 6, we focus on the verification of d-detectability of PBNs. Finally, Section 7 draws the conclusion.

Section snippets

Notations

We here give the primary notations employed in this paper.

•
$N$ : set of natural number.
•
$R_{m \times n}$ : set of $m \times n$ real matrices.
•
${Row}_{i} (L)$ : ith row vector of matrix L.
•
$Col (L)$ : set of all column vectors of matrix L.
•
$L_{(i, j)}$ : $(i, j)$ element of matrix L.
•
$D$ : set consisting of 0 and 1, i.e., $D = {0, 1}$ .
•
$δ_{n}^{k} : k$ th column vector of the unit matrix $I_{n}$ .
•
$δ_{n}^{0}$ : zero column vector of dimension n.
•
$Δ_{n} ≔ {δ_{n}^{k} | k = 1, 2, \dots, n}; {\tilde{Δ}}_{n} ≔ {δ_{n}^{k} | k = 0, 1, \dots, n}$ .
•
$L_{m \times n}$ (resp., ${\tilde{L}}_{m \times n}$ ): set of $m \times n$ logical matrices (resp., generalised logical matrix). $L \in L_{m \times n}$

Matrix-based representation for PBNs

In order to study the detectability of PBNs, a matrix-based representation, called the data form of PBNs, is obtained in [20] by means of Eq. (4), i.e., $E [x (t + 1)] = Ly (t + 1) E [x (t)], t \in N,$ where L is a $2^{n} \times 2^{l + n}$ stochastic matrix, which can be obtained from F and H presented in Eq. (4).

Using Lemma 2.1, Eq. (7) can be rewritten as follows: $E [x (t + 1)] = \overline{L} E [x (t)] y (t + 1), t \in N,$ where $\overline{L} = {LW}_{[2^{n}, 2^{l}]}$ is also a $2^{n} \times 2^{n + l}$ stochastic matrix. As mentioned in [20], Eqs. (4), (7), (8) can describe the same dynamics of PBNs,

Periodic detectability

For a given PBN (2), when the state-based output observations meet nondeterminism¹, knowing the current state may not imply knowing its all subsequent states. However, in some practical applications, it is sufficient to know its some subsequent states of PBN (2) at certain times by observing output sequence, and it is dispensable to know others. Such weaker detection requirement can be well-characterized by

k-Detectability

Sometimes, we do not need to accurate estimate the current state of a system, but only need to identify whether a system can (periodically) enter a specified subset of states or not. To this end, we introduce the concepts of four versions of k-detectability in this section, which relax the detection accuracy required in Section 4 to a given value of k. k-detectability of the PBN (2) is formalized as follows.

d-Detectability

In Section 4, we discuss standard detectability of PBNs, which captures whether or not the current state can be exactly ascertained by all or some output sequences. This detection requirement is useful in some practical systems but may be too restrictive in others. For this reason, we in this section propose the concept of d-detectability. It relaxes the detection condition and requires only that one should distinguish certain state-pairs after a finite delay for all or some output sequences.

Conclusion

In this paper, we developed respectively the observer-automata-based and matrix-based methodologies to check different categories of current-state detectability of PBNs from the different angles. Specifically, We first transformed a PBN into a stochastic automaton, then the observer proposed in [19] is employed to verify the periodic detectability of PBNs. To avoid the graph-based symbolic manipulations, using the STP technique, we further presented several matrix-based verification criteria

CRediT authorship contribution statement

Xiao-Guang Han: Conceptualization, Methodology. Wen-Dong Yang: Formal analysis, Writing - original draft. Xiao-Yan Chen: Software. Zhi-Wu Li: Writing - review & editing. Zeng-Qiang Chen: Investigation, Validation.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work is supported in part by National Natural Science Foundation of China under Grant 61903274 and the Tianjin Natural Science Foundation of China under Grant 18JCQNJC74000.

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Detectability vverification of probabilistic Boolean networks

Abstract

Introduction

Section snippets

Notations

Matrix-based representation for PBNs

Periodic detectability

k-Detectability

d-Detectability

Conclusion

CRediT authorship contribution statement

Declaration of Competing Interest

Acknowledgements

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