Set stability and synchronization of logical networks with probabilistic time delays

Using the algebraic state space representation (ASSR) method, this paper investigates the set stability and synchronization of Boolean networks with probabilistic time delays (PTDs). Firstly, an equivalent stochastic system is established for the Boolean network with PTDs by using the ASSR method. Secondly, based on the probabilistic state transition matrix of equivalent stochastic system, a necessary and sufficient condition is proposed for the set stability of Boolean networks with PTDs. Thirdly, as an application of set stability, the synchronization of coupled Boolean networks with PTDs is studied, and a necessary and sufficient condition is presented. Finally, an illustrative example is given to demonstrate the effectiveness of the obtained new results.     

State feedback design for set stabilization of probabilistic Boolean control networks

Set stabilization of probabilistic Boolean control networks (PBCNs) is investigated in this paper and some interesting results are derived. The main results consist of three parts. (1) A definition of set stabilizability with probability one by closed-loop control is proposed for PBCNs, which is not a natural extension from deterministic Boolean control networks to PBCNs due to the random feature of PBCNs. (2) A necessary and sufficient set stabilizability condition is provided for PBCNs. (3) An algorithm for designing a state feedback controller is developed. It is guaranteed that all designed controllers can stabilize a PBCN to a given subset with probability one. The design method is constructive, so it is convenient to use this method in practical application. The results derived above are fundamental and important, since based on them many problems about PBCNs can be solved, for example partial stabilization, synchronization, and so on. Finally, a practical example is employed to show the effectiveness of our results.     

Cascading state-space decomposition of Boolean control networks by nested method

Boolean control networks are a kind of discrete logical dynamical systems. They are recently attracting considerable interest as computational models for genetic and cellular networks. In this paper, we investigate the cascading state-space decomposition problem for Boolean control networks by nested method. Firstly, based on the semi-tensor product of matrices, we obtain some algebraic conditions for the cascading state-space decomposition. Secondly, the multi-layer nested block matrix is defined, and two necessary and sufficient conditions are put forward based on this kind of matrices. Besides, a method is given to design controllers. Finally, an example is given to display the effectiveness of the method provided in this paper.     

Finite horizon tracking control of probabilistic Boolean control networks

In this paper, the finite horizon tracking control problem of probabilistic Boolean control networks (PBCNs) is studied. For a given reference output trajectory, two trackability definitions are introduced according to whether the tracking probability is 1. Under the framework of the semi-tensor product, some necessary and sufficient conditions are obtained to determine whether the reference output trajectory is trackable with probability (probability one) by a PBCN starting from a given initial state. Based on this, two algorithms are proposed to determine the maximum tracking probability and the corresponding optimal control policy sequence. By determining the tracking error of the reference output trajectory, two related optimal control problems are considered: one is to minimize the expected value of the total tracking error, and the other is to minimize the maximum tracking error. Inspired by dynamic programming, corresponding algorithms are given to solve these two problems. Finally, two examples are given to verify the validity and correctness of the results.     

Stabilization of Boolean control networks with stochastic impulses

This paper studies the stabilization problem of Boolean control networks with stochastic impulses, where stochastic impulses model is described as a series of possible regulatory models with corresponding probabilities. The stochastic impulses model makes the research more realistic. The global stabilization problem is trying to drive all states to reach the predefined target with probability 1. A necessary and sufficient condition is presented to judge whether a given system is globally stabilizable. Meanwhile, an algorithm is proposed to stabilize the given system by designing a state feedback controller and different impulses strategies. As an extension, these results are applied to analyze the global stabilization to a fixed state of probability Boolean control networks with stochastic impulses. Finally, two examples are given to demonstrate the effectiveness of the obtained results.     

Optimal control of singular boolean control networks via Ledley solution method

This paper investigates the optimal control problem for a class of Boolean control networks, called singular Boolean control networks (SBCNs), which consist of two parts: difference equations and algebraic equations. By constructing the truth matrix of Ledley solution, necessary and sufficient conditions are provided for the solvability of SBNs (or SBCNs). Then an effective algorithm is presented to design an optimal control sequence by using the controllability matrix of normalized Boolean control networks.     

Stabilization of dynamic-algebraic Boolean control networks via state feedback control

In this paper, we study the state feedback stabilization of dynamic-algebraic Boolean control networks (DABCNs). Using a novel normalization approach, we present necessary and sufficient conditions for the feedback stabilization of DABCNs, and a construction method for the corresponding feedback controllers is proposed. Reduced order feedback stabilization is also studied in this paper. Two examples are given to illustrate the obtained results.     

Matrix-based flipping method for improving the robustness of Boolean networks

This paper proposes a matrix-based flipping method for improving the robustness of Boolean networks (BNs). The robustness considered in this paper is to find a flip matrix such that the output of a given BN with the flip matrix is unaffected by disturbances, in which case the output of the given BN is said to be flipping-based robust. Firstly, a necessary and sufficient condition is presented to determine whether the output of a given BN is identity matrix flipping-based robust. Then a novel flipping method based on the Hamming distance is provided to find the flip matrix and realize that the output of a given BN is flipping-based robust. Finally, a biological example is used to validate the effectiveness of the proposed method.     

Input–output decoupling control design for switched Boolean control networks

We investigate the input–output decoupling problem of switched Boolean control networks (SBCNs) in this paper. Based on the matrix expression of Boolean functions, the dynamics of SBCNs are converted into an algebraic form via semi-tensor product of matrices first. Then, using the redundant variable separation technique, we give the necessary and sufficient conditions for the existence of three kinds of controllers to detect whether an SBCN can be input–output decomposed or not, respectively, including the open-loop controllers, the state feedback controllers, and the output feedback controllers. Meanwhile, a constructive procedure is presented to construct the open-loop controllers, as well as the state feedback controllers and output feedback controllers. Finally, an illustrative example is given to show that the new results obtained are effective.     

Sampled-data controllability and stabilizability of Boolean control networks: Nonuniform sampling

Derived from a simplified intelligent traffic control system, sampled-data controllability and stabilizability of Boolean control networks are considered. Compared with the existing case of uniform (periodic) sampling in Boolean control networks, the nonuniform one is more general. Using linear span with integral coefficients, the distribution of sampling points can be obtained. Then by constructing novel systems, some necessary and sufficient conditions are proposed to determine sampled-data controllability and stabilizability. Finally, two illustrative examples, which are on apoptosis networks and traffic control systems, respectively, are worked out to show the effectiveness of the obtained results.     

Lebesgue sampling approach to robust stabilization of boolean control networks with external disturbances

This paper introduces the Lebesgue sampling approach to the robust stabilization of Boolean control networks (BCNs) with external disturbances. Given a Lebesgue sampling region and a feedback control, a time aggregated system is obtained via the semi-tensor product method. Then, a new criterion is presented for the robust stabilization of time aggregated system. Furthermore, given a signal of Lebesgue sampling, a sequence of the Lebesgue type robust reachable sets is constructed. Based on these reachable sets, several algorithms are presented to design both Lebesgue sampling region and sampled-data state feedback control for the robust stabilization of BCNs.     

Optimal control algorithms for switched Boolean network

This paper investigates the time optimal control and optimal infinite-horizon control for a switched Boolean network. First, the switched Boolean network can be converted into a discrete switched system by using the semi-tensor product of matrices. Second, algorithms for time optimal control and optimal infinite-horizon control of the switched Boolean network are presented. Moreover, constrained optimal infinite-horizon control is studied. Finally, illustrative examples are given to show the efficiency of the obtained results.     

高阶布尔网络的结构

介绍高阶布尔（控制）网络,并研究了其拓扑结构.以矩阵的半张量积作为工具,把高阶布尔网络的动态过程转化为2种标准离散事件动态系统的代数形式.证明了高阶布尔网络和第1代数形式的一一对应关系,并由此得到其拓扑结构（不动点、极限圈以及暂态期等）.还研究了高阶布尔网络系统与它第2代数形式的关系.     

A mathematical model of a weighted boolean retrieval system

The use of weights to denote a query representation and/or the indexing of a document is analysed as a generalization of a Boolean retrieval system. Criteria are given for the functions used to evaluate the relevance of the records to a specific query, including self-consistency. Various mechanisms suggested in the literature for evaluating the relevance of records with regard to a given query are tested and found to be less than satisfactory. A new approach is suggested to avoid some of the perils of a weighted Boolean retrieval system.     

Passive-active fault detection of Boolean control networks

This paper investigates the fault detectability of Boolean control networks. To deal with different circumstances, passive fault detectability and four types of active fault detectability are proposed for the first time. For passive fault detectability, in line with observed input-output data, several discriminant conditions are put forward. For active fault detectability, by resorting to an auxiliary system, the essence of active fault detectabilty is revealed via reachability. Subsequently, corresponding to different types of active fault detectability, algebraic criteria that facilitate verification are provided. At the same time, the relationships of four types of active fault detectability are discussed as well. In addition, an approach for calculating fault detection time is given by combining passive fault detectability with active fault detectability. Finally, the effectiveness of the results obtained is elaborated by several examples.     

Generalized cluster synchronization of Boolean control networks with delays in both the states and the inputs

In this paper, we study the generalized cluster synchronization problem for the Boolean control networks (BCNs) with delays in both the states and the inputs. First, by using the method of semi-tensor product of matrices, the original network is transformed into an equivalent extended system. Next, based on the updated iterative equation of the system, two types of generalized cluster synchronization are investigated: 1) generalized internal cluster synchronization within the BCN, and 2) generalized cluster synchronization between the BCN and the target reference network. Some necessary and sufficient conditions are proposed guaranteeing the realization of the generalized cluster synchronization. What is more, the gain matrices of the state-feedback controllers are explicitly designed. Numerical simulations are also given to illustrate effectiveness of the theoretical results obtained.     

Probabilistic topic modeling in multilingual settings: An overview of its methodology and applications

Probabilistic topic models are unsupervised generative models which model document content as a two-step generation process, that is, documents are observed as mixtures of latent concepts or topics, while topics are probability distributions over vocabulary words. Recently, a significant research effort has been invested into transferring the probabilistic topic modeling concept from monolingual to multilingual settings. Novel topic models have been designed to work with parallel and comparable texts. We define multilingual probabilistic topic modeling (MuPTM) and present the first full overview of the current research, methodology, advantages and limitations in MuPTM. As a representative example, we choose a natural extension of the omnipresent LDA model to multilingual settings called bilingual LDA (BiLDA). We provide a thorough overview of this representative multilingual model from its high-level modeling assumptions down to its mathematical foundations. We demonstrate how to use the data representation by means of output sets of (i) per-topic word distributions and (ii) per-document topic distributions coming from a multilingual probabilistic topic model in various real-life cross-lingual tasks involving different languages, without any external language pair dependent translation resource: (1) cross-lingual event-centered news clustering, (2) cross-lingual document classification, (3) cross-lingual semantic similarity, and (4) cross-lingual information retrieval. We also briefly review several other applications present in the relevant literature, and introduce and illustrate two related modeling concepts: topic smoothing and topic pruning. In summary, this article encompasses the current research in multilingual probabilistic topic modeling. By presenting a series of potential applications, we reveal the importance of the language-independent and language pair independent data representations by means of MuPTM. We provide clear directions for future research in the field by providing a systematic overview of how to link and transfer aspect knowledge across corpora written in different languages via the shared space of latent cross-lingual topics, that is, how to effectively employ learned per-topic word distributions and per-document topic distributions of any multilingual probabilistic topic model in various cross-lingual applications.     

Observability of singular Boolean control networks with state delays

This paper investigates the observability of singular Boolean control networks (SBCNs) with state delays. First, via Cheng product, the considered system is converted into its algebraic form. Then, under the uniqueness of the solution, two methods are proposed to transform the system into a general Boolean control network (BCN). Therefore, the considered system’s observability is equivalent to the BCN’s observability. Subsequently, the restricted input-state incidence matrix is introduced and some necessary and sufficient conditions are presented for the observability of BCNs with state delays. Finally, two illustrative examples are presented to show that the main results obtained are effective in analyzing the observability of SBCNs with state delays.     

Observability of Boolean control networks with stochastic disturbances

In this paper, the observability problem of Boolean control networks (BCNs) with stochastic disturbances is investigated via two kinds of control schemes: deterministic control and state feedback control. Firstly, based on the proposed indicator matrix, a simplified system of the original augmented Boolean system is constructed. Based on the analysis of the auxiliary system, observability of the original BCN is converted to determine whether an observable set can be reached from another unobservable set. After that, some necessary and sufficient conditions are obtained to judge the observability of BCNs. At the same time, two algorithms are proposed for designing these two types of control sequences. Finally, numerical simulations are also provided to demonstrate feasibility of the theoretical results.