A survey on applications of the semi‐tensor product method in Boolean control networks with time delays

As regulatory networks that simulate and establish gene interactions, Boolean networks (BNs) exhibit significant delays in state updates or trajectory tracking due to environmental factors in practical applications. This paper provides a comprehensive knowledge overview of Boolean control networks with time delays (TBCNs) via the semi‐tensor product theory (STP), which includes the history of BNs, the property of STP, several issues of TBCNs and future prospects. The related problems contain controllability and reachability, observability and detectability, stability and stabilization, synchronization, output tracking and regulation, and other issues. Finally, it also points out that some areas still need further research and promotion, such as disturbance decoupling, system decomposition, identification, deep reinforcement learning, and so forth.


INTRODUCTION
With the rapid development of systems biology, its research techniques are gradually transforming from phenotypic analysis to the modelling, analysis and control of complex biological systems.As regulatory networks that model and establish interactions between genes, Boolean networks (BNs) proposed by Kauffman in 1969, 1 have become the focus of research in bioinformatics.
In a Boolean network (BN), each gene has only two states, "active" and "inhibited", which can be quantified as "1" and "0", and only takes one of two states at each moment.At the same time, the state of each node is updated by a logical function where the variables are the states of neighbouring nodes.The topology of a BN determines the type of attractor of the network, which also corresponds to different states or processes of the cell.Thus, BNs have the advantage on mapping the complex connections among genes with simple binary logical relations, provide an effective support for characterizing changes in cell states, and describe qualitatively the most fundamental and subordinate relationships in gene regulatory networks.In addition, when external controls and observers are added, BNs can be transformed into Boolean control networks (BCNs).As an effective tool for dealing with high-dimensional array operations, the semi-tensor product (STP) theory is proposed by professor Cheng and his partners. 2Based on the established algebraic form under STP theory, innovative results on various areas were obtained, including controllability and reachability, [3][4][5] observability and detectability, [6][7][8] stability and stabilization, [9][10][11] output tracking, 12,13 optimal control, 14 disturbance decoupling problems, 15,16 synchronization, 17 decomposition, 18 state estimation, 19 and function perturbations, 20 identification, 21,22 reconstructability, 23 and so forth.As the research progresses, the applications of STP have also extended to algebra, 24,25 power systems, nonlinear control design, 26 game theory, 27,28 logical networks, 29,30 finite state machines, 31 fuzzy control, 32 cross-dimensional systems, 33 even internal combustion engines, 34 and so forth.
From the perspective of reality, due to environmental factors, gene mutations and drug effects, there are slow biochemical responses in state updating or trajectory tracking of BNs, such as time delays in gene transcription and translation.Some studies have shown that if the impact of time delay is not considered, it may lead to incorrect predictive behavior of BNs.Therefore, as the generalization of logical dynamic networks, Boolean networks with time delays (TBNs) can effectively model the various types of delayed phenomena in biological systems, and maintain important information about the system.However, the appearance of delay parameter leads to more complex conditions in the analysis of TBNs.On the one hand, as delay parameters are added, the initial conditions alter a single initial state into the initial state sequence, then the state trajectories evolving from initial state sequence are more complicated.On the other hand, using the expanded dimension method, TBCNs can be transformed into BCNs, and the relative methods can also be used directly, but the increase in delay parameters results in an exponential increase in computational complexity.Therefore, how to reduce the complexity regarding the analysis of TBCNs is a subject worthy of long-term research.In recent years, many scholars have adopted STP to model and study TBNs efficiently.Based on the support of existing theoretical results and broad application background, many significant results of TBNs were obtained, and further enrich the research framework of BNs.For example, Reference 35 investigated the controllability of TBCNs with two kinds of inputs. 36discussed the trajectory controllability of TBCNs using Perron-Frobenius theory. 37first gave the criteria of the synchronization for drive-response TBCNs.This paper collects and sorts out the results of research on TBCNs extensively, builds up a systematic knowledge vein, and also provides researchers with valuable topics for future research.
The rest of this paper is organized as follows.Section 2 provides some fundamental preliminaries about notations, STP, the logical and algebraic forms of TBCNs with different time delays.Section 3 reviews the main existing results, including controllability and reachability, observability and detectability, stability and stabilization, synchronization, output tracking and regulation, and other issues of TBCNs.Finally, the brief conclusions and prospects are drawn in Section 4.

PRELIMINARIES
In this section, we first introduce some notations, definition and properties of STP, which will used in the following analysis.Then the processes of building the algebraic forms for logical forms of Boolean control networks with time delays are presented.

Notations
. Particularly, : = {0, 1}. 5. R w×v is the set of w × v real matrices.6.  i w is the i-th column of identity matrix I w .7.
10. 0 w is an w-dimensional zero column vector.11. ∼: two different expressions of the same thing.12.A T ∈ R v×w is the transpose of matrix A ∈ R w×v .13. Col i (A)(Row i (A)) is the i-th column (row) of matrix A. The set of columns (rows) of matrix A is denoted by Col(A)(Row(A)).In particular, [A] i,j : = Row i (Col j (A)). 14.

The semi-tensor product theory
This section provides a basic introduction to the definition and properties of STP.It is worth mentioning that as a successor and extension of the classical matrix product, STP overcomes the dimensional limitations of matrix product and has more excellent properties such as pseudo-commutativity.
Definition 1 (38).Let U ∈ R a×b , V ∈ R c×d .The semi-tensor product (STP) of U and V is where k = lcm(b, c) is the least common multiple of b and c.
Note that under the condition of b = c, U⋉V = UV.Hence, symbol "⋉" can be omitted without confusions in the following.Moreover, some nice properties of STP are introduced.

be the front-maintaining operator and rear-maintaining operator, respectively. Then one has D
2 n } be the n-valued power-reducing matrix, then one has Z 2 = Φ n Z.
To establish the matrix expression of finite-valued logic functions, we first construct a one-to-one mapping between  w and Δ w , that is for each i ∈  w , let  w−i w ∈ Δ w be its vector form.Under this framework, the structure matrix of each logical function is presented.
Lemma 2 (38).Let f (X 1 , X 2 , … , X n ) ∶  n w  →  w is an n-ary logical function with logical variables X i ∈  w , i ∈ [1 ∶ n].Using STP, there exists a unique structure matrix L f ∈  w×w n , such that the algebraic form of f According to Lemma 2, several kinds of monadic or binary logic operators in Reference 38 are presented as follows, Lemma 3 (38).
where y, z,

The algebraic representations of Boolean networks with different time delays
As the binary logical dynamic system, Boolean network can accurately depict the interaction relationships between genes and the expression process of genes, utilizing the concise dynamic evolution process.Its dynamic equation is represented as follows: where , k ∈ [1 ∶ l] be the vector forms of X i and Y k , respectively.Then let x(t) = ⋉ n i=1 x i (t) and y(t) = ⋉ l k=1 y l (t).According to Lemma 2, one has that for each and y k (t) = H k x(t).Based on Lemma 3, the algebraic form of BN ( 5) is shown as, where n are the state transition matrix and output matrix of BN (5), respectively.On the basis of BN (5), adding external input variables can obtain the corresponding BCN, the dynamic form of which is expressed as, where , j ∈ [1 ∶ m] and u(t) = ⋉ m j=1 u j (t).Similarly, according to Lemmas 2 and 3, we can get the algebraic form of BCN (7),

{
x(t + 1) = Fu(t)x(t), where F ∈  2 n ×2 m+n and Ĥ ∈  2 l ×2 n are the state transition matrix and output matrix of BCN (7), respectively.Furthermore, adding the influence of time delay factors on the dynamic behavior of BN (5) and BCN (7), the TBN and TBCN are obtained as follows, and Subsequently, using STP, the algebraic forms of TBN (9) and TBCN (10) are shown as follows, TA B L E 1 Several BCNs with delays, switched, probabilistic, impulsive, and other stochastic factors.

Literature Research model Type of delay Details
Time-variant delays in inputs and states: Multiple bound time-varying delays: Multiple bound time-invariant delays: where  is the time-invariant delay parameter, F ∈  2 n ×2 n and H ∈  2 l ×2 n are the state transition matrix and output matrix of TBN (11); F ∈  2 n ×2 n+m and Ĥ ∈  2 l ×2 n are the state transition matrix and output matrix of TBCN (12).Increasingly, more evidences have shown that neural networks broadly exist uncertainties and randomness, and closely followed by the diversity of models with different delay parameters, switched, probabilistic, impulsive, and other stochastic factors.Several scenarios of TBCNs are shown in Table 1.Next, based on but not limited to the network models presented above, we analyze the research status and applications of STP in TBCNs.

MAIN RESULTS
In this section, we summarize the applications of STP in TBCNs, in terms of controllability and reachability, observability and detectability, stability and stabilization, synchronization, output tracking, and other issues.

Controllability and reachability
In terms of physical intuition, the controllability of BCNs aims to portray the dynamic changes in the internal states of a system affected by inputs.Due to different conditions of inputs, the concepts of controllability also have diversity.Particularly, by adding delay parameters, the initial condition of the system changes from a single initial state to an initial state sequence.From this, the controllability of BCN with time delays becomes more complex, and the existing results of controllability of BCNs cannot be directly applied.Focusing on the urgent requirements of expanding practical applications and deepening theory, many scholars have focused on the controllability of TBCNs, and achieved some significant results.Moreover, since the state space of TBCNs is finite, the reachability is equivalent to the controllability. 35first investigate the controllability of TBCN with time-invariant delays in states, the algebraic form of which is formula (12).And there are two kinds of inputs considered.One is the networked input satisfying certain logical rules, the form of which is shown as where G ∈  2 m ×2 m .The other is free Boolean inputs u(t) ∈ Δ 2 m , ∀t ∈ N. Then the definition of controllability was given.
Definition 2 (35).Consider TBCN (12).Given initial state sequence (x(−), x(− + 1), … , x(0)) ∈ (Δ 2 n ) +1 , and destination state x d ∈ Δ 2 n .TBCN ( 12) is said to be controlled from x(i − ), i ∈ [0 ∶ ] to x d at s steps, if we can find control {u(t)} s+−1 t=0 , such that x(s + i) = x d .By iterating through algebraic form (12), the reachable sets at each step with the networked and free inputs are constructed.The details are shown as follows.For each s ∈ N and i ∈ 1.For networked input (13), one has where and its columns contain all reachable states at the s-th step, from initial state x(b − 1 − ) and initial input u(0).2. For free input sequence {u(t)} t∈N , let L = LW [2 n ,2 m ] , then we can get that and its columns express all reachable states at the s-th step, from initial state x(b − 1 − ) and any input sequence {u(k( + 1) Then we have the following results on controllability.
1. TBCN (12) with networked input (13) Following this, several kinds of delay parameter have been considered. 36introduced multiply time-invariant delays in TBCNs, where the algebraic from is As each state trajectory of TBCNs starts from an initial state sequence, the state controllability and trajectory controllability were presented in Reference 36.In contrast to state controllability, trajectory controllability means that there exists an input sequence, such that an initial trajectory The main method is that using STP, an augmented algebraic form of TBCN was equivalently proposed, which is y(t + 1) = Lu(t)y(t) with y(t) = ⋉ t i=t−+1 x(i).Then the criteria of trajectory controllability was obtained.
Theorem 2 (Theorem 3.2; 36).Consider the TBCN with x(t + 1) = Lu(t)x(t −  + 1) • • • x(t).The destination trajectory X d is trajectory reachable from initial state trajectory X(0) at the k-th step by a free Boolean input sequence {u(t)} k−1 t=0 , if and only if 2 m ] .Then Reference 36 also derived the number of input sequence that steer the state trajectory from X(0) to X(k), under avoiding undesirable trajectories.Moreover, it also used the Perron-Frobenius theory of nonnegative matrices to establish the equivalent relationship between the trajectory controllability and the irreducibility of matrix Q ∶= L1 2 m .Although the augmented system can completely reflect characters of TBCNs, the increase of delay parameters brings about an exponential growth in the computing complexity.To reduce the computing complexity, 39 introduced a non-augmented method based on graph theory, where the model is x(t + 1) = Lu(t)x(t − (t)).First, Reference 39 gave the concept of "constructed forest".Definition 3 (39).A directed graph G(V, E) is said to be the constructed forest of the TBCN, if the vertex set The constructed forest G(V, E) consists of (t 0 ) + 1 directed trees {T t 0 −(t 0 ) , T t 0 −(t 0 )+1 , … , T 0 }, where T t 0 −(t 0 )+i with its root t 0 − (t 0 ) + i, i ∈ [0 ∶ (t 0 ) − t 0 ].Then let P c and N c be any one given longest path in directed trees and the length of P c , respectively.Since describing the global controllability of the TBCN, P c is called the controllability constructed path (CCP).The main result shows that the TBCN is controllable, if and only if the subsystem that generates CCP P c is controllable.Moreover, as the obtained subsystem has no time delays, most methods for the controllability of traditional BCNs are now available, and the computational complexity is much less.
Subsequently, the authors in Reference 40 used the same method to analyze the controllability of probabilistic TBCNs with time-variant delays in states.The conclusion shows that only when the CCP is of infinite length, the controllability of the network is equivalent to the controllability of a CCP.In addition, many fundamental results on controllability and reachability of TBCNs with multiply delay and other stochastic factors have also been obtained.For example, References 45,46 studied the controllability of TBCNs with asynchronous stochastic update with time delay. 47,48considered BCNs with time-invariant delays both in states and inputs, and its optimal control problem.BCNs with multiple periodic delays and its controllability were described in Reference 42.Set controllability of BCNs with Markov jump time delays was investigated in References 49 and 50 presented an effective pinning control algorithm to steer autonomous BCNs from any given initial state to the desired state in the shortest time period. 51discussed the output controllability of temporal BCNs.Furthermore, the controllability of high-order BCNs were displayed in References 43,52-54.

Observability and detectability
As two common state estimation methods in control theory, observability and detectability of BCNs aim to estimate the internal state of dynamic networks using measurable input-output data.Observability emphasizes that the initial state of the network can be determined completely by the known data.However, it requires overly strict conditions to achieve for most networks.As a natural generalization of observability, detectability (reconstructability) of BCNs describes the fact that all unobservable states will achieve asymptotic stability.Compared with BCNs, the observability and detectability of TBCNs are more complex, and the types of time delay lead to many scenarios to consider.For this situation, 55 firstly analyzed the problem of observability for TBCN with x(t + 1) = Lu(t)x(t − ) and y(t) = Hx(t), and gave the rank criteria of observability via two kinds of inputs.First, the observability of TBCN was given.
On the one hand, the authors considered the networked inputs with u(t + 1) = Gu(t).By constructing a sequence of matrices Γ j ∈ L 2 l ×2 n+m , j = a( + 1) + b, where a ∈ {0, 1, 2, … } and b ∈ {1, 2, … ,  + 1}, as Then theorem 1 in Reference 55 were presented as follows.Theorem 3 (Theorem 1; 55).Consider the TBCN with networked inputs.Assume that u(0 The TBCN with networked inputs is observable, if and only if there exists finite time s = c( + 1), where c is a positive integer, such that On the other hand, for free input sequence, construct matrices And split Γj into 2 jm equal blocks as Then the criteria of observability was obtained by theorem 2 in Reference 56, which is shown as follows.
Theorem 4 (Theorem 2; 56).Consider the TBCN.Assume that there is a free input sequence as u(t) Then the TBCN is observable, if and only if there exists finite time s = d( + 1) ∈ N, such that where Row r ( Ôk Subsequently, Reference 57 considered the observability of temporal BCN with its algebraic form x(t + 1) = Lu(t)x(t) • • • x(t − ) and y(t) = Hx(t).Then Reference 57 obtained the rank criteria of observability under networked and free inputs.Moreover, Reference 41 investigated a TBCN with time varying delays in states and inputs, the algebraic form of which is For this model, Reference 41 used the method of expanding dimensions, let x(t − i), then the original TBCN was transformed into time-varying BCN with z(t + 1) = L t v(t)z(t).Based on this, the rank criteria of the observability are presented for a temporal BCN via two kinds of inputs.Due to the exponential increase in computational complexity caused by the increase in matrix dimension, 56 gave an equivalent condition for the observability of TBCN with time-variant delays, by using the graph-theoretic method, where the state transition equation is similar with Reference 39 and output equation is y(t) = Hx(t).By the constructed forest G(V, E), (t 0 ) + 1 directed trees {T t 0 −(t 0 ) , T t 0 −(t 0 )+1 , … , T 0 } are given, where T t 0 −(t 0 )+i with its root t 0 − (t 0 ) + i, i ∈ [0 ∶ (t 0 ) − t 0 ].Then let P o be any one given shortest path in directed trees and call it as the observability constructed path (a special subsystem without time delays).Then under the assumption "the TBCN has an observability constructed path P o of length +∞", the result is that "the TBCN is observable if and only if its observability constructed path P o is observable" in theorem 3.3 of Reference 56.On this basis, Reference 58 summarized the results of the reconstructability by STP, weighted pair graph, constructed forest, and finite automata.Different from linear system, the observability of BCNs has multiply concepts according to various input conditions, which are weak observability, 59 observability, 60 strong observability, 61 super-strong observability. 62As the requirement of input sequence to determine the initial states gradually upgrades from weak observability to super-strong observability, the relationship among four kinds of observability can be shown in Figure 1.Accordingly, there are four kinds of detectability of BCNs, which are weak detectability, detectability, strong detectability, and super-strong detectability.Moreover, for TBCNs, there are also four kinds of observability (detectability), respectively, which are weak observability (detectability), observability (detectability), strong observability (detectability), and super-strong observability (detectability).Compared with observability of BCNs, observability of TBCNs changes from determining the initial state to determining the initial state sequence.In Reference 63, weak detectability, strong detectability, and detectability of The relation among four kinds of observability.
TBCN were investigated by a full-order observer 64 studied the observability of singular BCNs with state delays.And in Reference 65, the observability of T-S fuzzy BN with time-varying delay.

Stability and stabilization
Stability and stabilization of BCNs both describe that starting from any initial state, the state trajectory enters a stable state after finite steps.The difference is that the former has no control effect, while the latter has Reference 66 firstly investigated the stabilization of TBCN with its algebraic form x(t And it designed the control algorithm to select a fraction of key nodes, such that the considered TBCN can achieve stability. Definition 5 (66).The TBCN is globally stabilized to the fixed point  a 2 n , if for arbitrary initial state sequence (x(0), … x(−)) ∈ (Δ 2 n ) +1 , there exist an input sequence and T ∈ N, such that 2 n ) the set consisting of all the states that can be steered to 2 n ) contains all states can be controlled to state sequence 2 n .Then by changing the columns of the state transition matrix, the BN achieved the stabilization.Subsequently, the corresponding algorithms are provided to select pinning nodes.
From a cost saving perspective, Reference 67 presented the stability and stabilization of TBCNs by the periodic sampling method.Then Reference 68 studied the stabilization for TBCNs using state feedback controllers, and presented the conclusion for the stabilization of TBCN.

Theorem 5 (Theorem 1; 68). Consider TBCN with x(t
Then, the TBCN can be globally stabilized to x d =  𝜃 2 n under the state feedback controller, for any initial state sequence (x(−), 2 n ]} be the set consisting all the states that can be steered to From the above procedure, the following result are obtained.Theorem 6 (Theorem 3; 68).The TBCN can be globally stabilized to state x d =  𝜃 2 n by a state feedback controller, if and only if the following conditions are satisfied: 2) Furthermore, using reachable set S i , i ∈ [1 ∶ M], the feedback controllers were designed to successfully stabilize the TBCN. 69adopted the method of expanding dimensions to equivalently convert the stabilization of TBCNs into the stabilization of augment system without delay.Reference 44 discussed the stability and stabilization of TBNs with stochastic delays, and obtained some results of stochastic stability based on stability results of positive systems.Reference 70 gave the conclusions on topological structure and set stability of TBN with state-dependent delay.In addition, the stability and stabilization of TBCNs have been studied under many stochastic factors, such as probabilistic, [71][72][73] impulsive, 74 switched, 75 disturbance. 76From the perspective of selecting control methods, sampled-data control, [77][78][79] event-triggered control, 80 pinning control 81 were considered to increase efficiency and reduce waste.Moreover, the researches on stability and stabilization of TBCNs also used barrier Lyapunov function method, 82 Ledley antecedence solution method, 83 and so forth.

Synchronization
As a common group behavior, the synchronization phenomenon can be traced back to Huygens' observation of pendulum synchronous oscillation in 1665.It emphasizes that the dynamic behaviors of two or more networks gradually evolve to a unified state.The study of synchronization has received wide attention from physics, chemistry, biology, psychology, and engineering fields.Recently, synchronization of TBCNs have been studied and achieved some significant results.Based on the drive-response configuration, 37 first analyzed the synchronization of two deterministic TBCNs, which are And the concept of synchronization were shown in definition 2 of Reference 37.
Then the subsequent analysis was divided into two situations: One is that there exists a ∈ N, such that  ′ = a( + 1) + .Under this condition, the criteria of synchronization were obtained by theorem 1 of Reference 37.
The other is that  ′ ≠ (mod  + 1).Then the result is shown as follows.From the results above, it revealed that the inherent delay in state and the unidirectional coupling delay may lead to quite distinct synchronization phenomena for drive-response BNs.Later, to further extended the conclusions of Reference 37, complete synchronization of two temporal BNs were investigated in Reference 84, where the model of drive BN and response BN are The authors adopted the extended dimension approach, which sets Then the original temporal BNs are equivalently transformed into the form without time-delay: Based on this equation, the necessary and sufficient condition for complete synchronization of the drive-response temporal BNs are established by theorem 3.1 of Reference 84.
On the basis of this, the authors in Reference 85 further designed the state feedback control to make the response BCN synchronize with the drive BN.Subsequently, References 86-88 discussed the synchronization in an array of output-coupled BNs with time delays.Furthermore, delay synchronization, 89,90 inner synchronization, 91 intermittent synchronization, 92 cluster synchronization 93 of TBNs were successively studied.
Remark 1.It is worth noting that in synchronization problems, the research objects are often multiple coupling forms, such as state (output) coupling, master-slave, interconnection, cascading, and so forth.With these coupling relationships, information communication can be achieved between multiple networks.However, as information transmission exists time delay, it is often laborious to achieve synchronous among multiple networks.For example, in the UAV formation, when multiple drones have a queue order, a common synchronization problem is that drones perform the same operation one by one to maintain the orderliness of the team.In addition, on the field, fans receive instructions to raise their hands, stand up, and set down, forming wave after wave of crowds.In nature, the leader of migratory geese guides subsequent geese to keep consistency in group flight.These natural and artificial phenomena reflect the synchronous trajectories of multiple networks.But compared to leaders, there is time delay when responders reach the same state, which is the application scope of time delay in synchronization.Hence, the impact of time delay factors on synchronization problems is very important, and studying delay phenomenon in synchronization has both theoretical and practical significance.

Output tracking and regulation
Output tracking and regulation are the basic research areas of TBCNs.Its objective is to design controller to achieve the output trajectory tracking the given reference outputs.Motivated by its widely applied background, 13 first studied the output trajectory of TBCNs with the algebraic form x(t Let z(t) = ⋉ t i=t−+1 x(i) and given a constant reference signal y * =   2 l .Then the output tracking problem is to design a state feedback control in the form of U(t) = Hx(t −  + 1) • • • x(t), under which there exists an integer T ≥ 0, such that y(t; z(0); u(t)) = y * holds, for any initial trajectory z(0) ∈ Δ 2 n and any integer t ≥ T. Construct Γ() = {a|Col a (G) =  𝛾 2 l , 1 ≤ a ≤ 2 n }.Using augment system with z(t + 1) = Lu(t)z(t) and y(t) = Gz(t), the output tracking problem is converted into set controllability.Subsequently, 12 considered the output regulation problem of TBCN, where the algebraic form is and the reference TBN is Then the objective of output regulation is defined as follows.
Definition 7 (12).For any given initial state sequence (x(0), … , x(−)) ∈ (Δ ]}, then the ORP was transformed into designing the controller to stabilize the augmented system (27) into the largest control-depending invariant subset of Λ.Then the conclusions of solving the event-based ORP was obtained.
Therefore, the existing results often transformed TBCNs into augmented systems, and then gave some methods of the ORP by set controllability and set stabilization theory.

Other issues
In other issues, the relationship among states, outputs and random factors have been attracted widespread attention.For example, in the scope of analyzing the relationship between states and outputs, besides the issues of observability and detectability, invertibility of TBCNs was considered in Reference 94, which means that for any initial state sequence x 0 and input sequence u 0 , mapping HL x 0 ,u 0 is bijective.Moreover, as random factors always exist in practical applications, it is very important to explore the impact of external interference on states and outputs of the system.Reference 95 considered two types of random delay in measurements, and discussed the state estimation of TBCNs with stochastic disturbances coming from measurements with random delay.Reference 96 investigated the disturbance decoupling problem of TBCNs, and provided the algorithm to select the minimum pinning nodes.

CONCLUSIONS AND PROSPECTS
This paper comprehensively summarizes and analyzes the research status on TBCNs based on the semi-tensor product theory, which contains controllability and reachability, observability and detectability, stability and stabilization, synchronization, output tracking and regulation, and other issues.STP provides a unified research framework for the analysis and control of TBCNs, and its application has gradually expanded to game theory, coding theory, finite automata, fault diagnosis, fuzzy control and other fields.It demonstrates tremendous vitality and theoretical value in the future.Here lists several key issues: 1. Due to the existence of time delay, the dimension of the algebraic forms of TBCNs increases exponentially with the growth of delay parameter.How to reduce computational complexity is an important challenge in research.
The existing results have adopted some methods to solve this problem, such as pinning control, block decoupling, model decomposition, and so forth.Moreover, from the prospective of practical application, the study of Boolean networks using big data and artificial intelligence has lower complexity and broader application prospects.The reinforcement learning, including deep reinforcement learning (DRL), double deep-Q Learning-based, has emerged to solve the controllability, 97 output tracking, 98 stabilization, 99 synchronization 100 of probabilistic Boolean networks (PBNs).Hence, the reinforcement learning method for studying TBCNs deserves further analysis.2. Some issues regarding TBCNs have not been fully resolved yet, for instance, disturbance decoupling, system decomposition, function perturbations, and so forth.Further establishing and improving the research scopes of TBCNs can broaden the research framework of BNs. 3. Another key focus in the future is to find application areas for TBCNs and attempt to establish connections with other research fields.
The purpose of this paper is to provide a systematic and comprehensive knowledge overview for researchers interested in the development of TBCNs, and to inspire researchers to continuously improve the relevant knowledge framework.

Theorem 7 (
Theorem 1; 37).Consider the drive and response BNs.Let Θ = (F ⊗ G)(Φ n ⊗ I 2 n ).Then the following statements are equivalent: (a) The drive BN and the response BN are synchronized.(b) There exists a positive integer n, such that Col((