Single-Experiment Reconstructibility of Boolean Control Networks

Revisited

Guisen Wu

1 a

and Jun Pang

2 b

1

School of Computer and Information Science, Southwest University, China

2

Faculty of Science, Technology and Medicine & Interdisciplinary Centre for Security, Reliability and Trust, University of

Luxembourg, Luxembourg

Keywords:

Boolean Control Networks, Reconstructibility, Observability, Veriﬁcation Algorithms.

Abstract:

We ﬁrst demonstrate that, BCNs’ single-experiment reconstructibility has three additional forms in addition

to its current deﬁnition, and brieﬂy introduce the veriﬁcation algorithms we design for these new deﬁnitions.

These deﬁnitions and algorithms bring the following improvements to BCNs’ control theory. First, the solution

algorithms of single-experiment reconstruction are enriched to cope with more different scenarios. Second,

the veriﬁcation problem of single-experiment reconstructibility is simpliﬁed. Finally, the essential relationship

and difference between reconstruction and observation (which focuses on determining the initial state for a

BCN), is further clariﬁed.

1 INTRODUCTION

Inspired by the Boolean-type actions in genetic cir-

cuits (Jacob and Monod, 1961), Boolean networks

(BNs) were proposed by Kauffman as a popular and

well-established framework for modelling non-linear

and complex biological systems (Kauffman, 1968).

The main advantage of BNs is its simplicity and yet

can be used to capture the important dynamic proper-

ties of biological systems, thus facilitating the mod-

elling of large systems as a whole. In a BN, the nodes

are denoted as binary variables, being either ‘ex-

pressed’ or ‘not expressed’ and activation/inhibition

regulations between them are described by Boolean

functions.

Boolean control networks (BCNs) is a natural ex-

tension of BNs with external regulations and pertur-

bations (Ideker et al., 2001). A BCN has three differ-

ent types of nodes, namely, input-nodes, state-nodes,

and output-nodes. Their value vectors are called the

BCN’s input, state, and output, respectively. The rela-

tionship between these three sets of nodes is described

by the BCN’s updating rules. BCNs have been applied

to various problems and typical examples including

abduction-based drug target discovery (Biane and De-

laplace, 2017), and functional and structural analysis

a

https://orcid.org/0000-0002-5868-3049

b

https://orcid.org/0000-0002-4521-4112

of signalling and regulatory networks (Kaufman et al.,

1999; Klamt et al., 2006). The widespread use has

led to vast attention on the research of BCNs’ con-

trol theory ((Akutsu et al., 2007; Cheng and Qi, 2009;

Zhao et al., 2010; Cheng et al., 2011; Fornasini and

Valcher, 2013; Su et al., 2019; Mandon et al., 2019;

Zhang et al., 2015a)).

In this paper, we focus on BCN reconstruction

which plays an important role in state observer de-

sign and controller synthesis of BCNs (Fornasini

and Valcher, 2013). We propose that BCN’s single-

experiment reconstructibility, which was considered

to have only one form (Zhang et al., 2020a), has

three other different but equivalent deﬁnitions, and

design veriﬁcation algorithms for this new deﬁnitions

we proposed.

Before introducing BCN reconstruction, we dis-

cuss observation that is necessary for the quantita-

tive analysis of BCNs is of wide interest and remains

a topical issue ((Cheng and Qi, 2009; Zhao et al.,

2010; Cheng et al., 2011; Fornasini and Valcher,

2013; Zhang et al., 2020b; Zhu et al., 2021; Zhu et al.,

2022)). More speciﬁcally, observation is about how to

determine the initial state of a BCN by controlling its

inputs and observing its outputs (Zhang et al., 2020b).

Observation is further classiﬁed into three problems:

multiple-, single-, and arbitrary-experiment observa-

tion, to investigate how to determine the initial states

in three different situations, namely, (1) the BCN’s in-

Wu, G. and Pang, J.

Single-Experiment Reconstructibility of Boolean Control Networks Revisited.

DOI: 10.5220/0012161100003543

In Proceedings of the 20th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2023) - Volume 2, pages 85-93

ISBN: 978-989-758-670-5; ISSN: 2184-2809

Copyright © 2023 by SCITEPRESS – Science and Technology Publications, Lda. Under CC license (CC BY-NC-ND 4.0)

85

put can be controlled and initial state can be reset, (2)

input can be controlled and initial state cannot be re-

set, and (3) input cannot be controlled.

To study the solvability of these problems, ﬁve

types of observability Type-I, II, III, IV & V observ-

ability were proposed in the literature (Cheng and Qi,

2009; Zhao et al., 2010; Cheng et al., 2011; For-

nasini and Valcher, 2013; Wu et al., 2020). Type-I ob-

servability was proposed to formalise the solvability

of multiple-experiment observation problem, but was

later replaced by Type-II observability because Type-

II observability is easier to be satisﬁed (Zhang et al.,

2020b). Therefore, Type-II observability was named

as multiple-experiment observability. Type-I observ-

ability was then named as strong multiple-experiment

observability because it is stronger than Type-II ob-

servability. Type-III obser vability was considered as

single-experiment observability (Cheng et al., 2011;

Zhang et al., 2020b). More recently, a new notion

of Type-V observability, which is easier to be satis-

ﬁed than Type-III observability, was proposed to re-

place it in (Wu et al., 2020). Thus, Type-III observabil-

ity is now named as strong single-experiment observ-

ability. Type-IV observability was deﬁned to be the

single-experiment observability of BCNs (Fornasini

and Valcher, 2013).

Next, we continue to introduce BCNs’ reconstruc-

tion. Reconstruction was divided into two problems:

single- and arbitrary-experiment reconstruction, to

investigate how to determine the current states of a

BCN in two different situations, namely, (1) its input

can be controlled and (2) input cannot be controlled.

It makes no sense to study multiple-experiment re-

construction, because in different experiments, the in-

puts being fed to the BCN may be different, and then

the corresponding current states may also be differ-

ent (Zhang et al., 2020a). As the tasks to be performed

in reconstruction and observation are similar, recon-

structibility has high similarity in deﬁnition with ob-

servability. To formalise BCN’s reconstructibility in

the above mentioned two different situations, Type-I

& II reconstructibility that are the counterparts corre-

sponding to Type-IV & III observability, were proposed

in (Fornasini and Valcher, 2013; Zhang et al., 2015a).

In this paper, ﬁrstly, we propose three new types of

reconstructibility for BCNs. Since single-experiment

observability has been redeﬁned as Type-V observabil-

ity in (Wu et al., 2020), we propose Type-III recon-

structibility as the counterpart corresponding to Type-

V observability. Moreover, we demonstrate that it

makes sense to study the counterparts (Type-IV & V

reconstructibility) corresponding to Type-I & II observ-

ability, even though these two types of observability

were proposed to study multiple-experiment obser-

Figure 1: The relationships between inputs, states, and out-

puts of BCNs.

vation. Secondly, we brieﬂy introduce the veriﬁca-

tion algorithms we design for these new types of re-

constructibility, and analyse the computational com-

plexity for them. We claim that with the proposal

of Type-V reconstructibility, the veriﬁcation problem

of the BCNs’ single-experiment reconstructibility can

be greatly simpliﬁed.Finally, we formally prove that

Type-II, III, IV& V reconstructibility are equivalent, even

if they are the counterparts corresponding to Type-III,

V, I& II observability which are not equivalent.

The remainder of this paper is organised as fol-

lows. We introduce necessary notations and the for-

mal deﬁnition of BCNs in Section 2. In Section 3,

we formally deﬁne Type-III, IV & V reconstructibility

and introduce the new solution algorithm designed

for the single-experiment reconstruction problem of

BCNs. We then present the veriﬁcation algorithms we

have designed for Type-III, IV & V reconstructibility in

Section 4. We summarise and compare the results on

reconstructibility and observability in Section 5. We

conclude the paper by discussing future research di-

rections in Section 6.

2 PRELIMINARIES

We ﬁrst introduce the following necessary notations:

• B: the set of Boolean values {0,1},

• T: the set of discrete time domain which is de-

noted by the set of natural numbers,

• v

i

2

x

: the x-dimensional Boolean vector whose dec-

imal value is equal to i,

• V

2

x

: the set {v

0

2

x

,. .. ,v

2

x

−1

2

x

} of Boolean vectors.

A Boolean control network (BCN) can be de-

scribed by the following state equation and output

equation (Ideker et al., 2001):

s(t +1) = f (i(t),s(t))

o(t) = h(s(t))

(1)

where t ∈ T; i(t) ∈ B

ℓ

, s(t) ∈ B

m

, and o(t) ∈ B

n

de-

note the vectors input, state, and output, respectively,

at time t; f : B

ℓ

× B

m

7→ B

m

and h : B

m

7→ B

n

are

logical functions. The relation between inputs, states,

and outputs of a BCN can be illustrated in Fig. 1,

ICINCO 2023 - 20th International Conference on Informatics in Control, Automation and Robotics

86

where 0, 1,. .. stand for time steps, i(0),i(1),.. . in-

puts, s(0),s(1),... states, o(0),o(1), ... outputs, and

arrows represent dependence. Moreover, as we repre-

sent an x-dimensional Boolean vector in the form v

i

2

x

,

the input set B

ℓ

, state set B

m

, and output set B

n

can

be replaced by V

L

, V

M

, and V

N

, respectively, where

L = 2

ℓ

, M = 2

m

, and N = 2

n

.

In order to discuss observability and recon-

structibility, we deﬁne the following classes of func-

tions to represent the relation between the input se-

quence, output sequence, and state sequence.

F

[t

0

,t]

: V

M

× (V

L

)

t−t

0

7→ (V

M

)

t−t

0

+1

,

F

[t

0

,t]

(s(t

0

),i(t

0

). .. i(t −1)) = s(t

0

). .. s(t),

(2)

H

[t

0

,t]

: V

M

× (V

L

)

t−t

0

7→ (V

N

)

t−t

0

+1

,

H

[t

0

,t]

(s(t

0

),i(t

0

). .. i(t −1)) = o(t

0

). .. o(t),

(3)

with t ≥ t

0

. For every state s(p) (t

0

< p ≤ t) in the

state sequence s(t

0

). .. s(t), s(p) = f (i(p − 1),s(p −

1)). For the output sequence o(t

0

). .. o(t), every o(p)

in it satisﬁes o(p) = h(s(p)).

Intuitively, these two classes of functions rep-

resent the way to calculate the state sequence

s(t

0

). .. s(t) and output sequence o(t

0

). .. o(t) of a

BCN, respectively, in the time interval [t

0

,t], by its

state s(t

0

) and input sequence i(t

0

). .. i(t −1).

Then, we introduce the way to calculate the possi-

ble state set S

ps

(t) of a BCN, which contains all pos-

sible valuations of the BCN’s state s(t) that can be

deduced at time step t. Firstly, we deﬁne the function

ζ(S,i, o) to show how to calculate the state set S

ps

(t)

for a BCN by the state set S

ps

(t −1), input i(t −1), and

output o(t). Before deﬁning the function ζ(S,i,o), we

deﬁne the following function ξ(i,s).

ξ : (V

L

∪ {ε}) ×V

M

7→ V

M

,

ξ(i,s) =

f (i,s) i ̸= ε

s i = ε

.

(4)

The function ξ(i, s) is deﬁned to describe how the

BCN’s state s is affected by its input i. Compared

with the updating function s(t + 1) = f (i(t),s(t)), this

function could capture how the state s changes when

the input i = ε. Then we deﬁne function ζ(S,i,o).

ζ : 2

V

M

× (V

L

∪ {ε}) × (V

N

∪ {ε}) 7→ 2

V

M

ζ(S,i, o) =

{ξ(i,s) | s ∈ S,h(ξ(i,s)) = o} o ̸= ε

{ξ(i,s) | s ∈ S} o = ε

(5)

Then, we recursively deﬁne the following class of

functions G

[t]

(i(0). .. i(t − 1),o(0)...o(t)) to present

how to determine the set S

ps

(t) for a BCN by

analysing its input sequence i(0) ...i(t − 1) and out-

put sequence o(0). .. o(t).

G

[t]

: V

L

t

×V

N

t+1

7→ 2

V

M

(6)

These functions satisfy the following conditions.

• When t = 0, i(0) ...i(t − 1) = ε,

G

[t]

(i(0). .. i(t −1),o(0)... o(t)) = ζ(V

M

,ε, o(0)).

• When t > 0,

G

[t]

(i(0). .. i(t −1),o(0). .. o(t))

= ζ(S

ps

(t −1),i(t − 1),o(t))

where

S

ps

(t −1)

= G

[t−1]

(i(0). .. i(t −2),o(0). .. o(t −1)).

3 REVISITING

RECONSTRUCTIBILITY

In this section, we begin with two existing types of

reconstructibility and then introduce three new types

of reconstructibility that we propose in this work.

Deﬁnition 1 (Type-I reconstructibility (Fornasini and

Valcher, 2013)). A BCN satisﬁes Type-I recon-

structibility if there exists a ﬁnite number k ∈ T such

that for any input sequence I ∈ (V

L

)

p

where p ≥ k,

H

[0,p]

(s

′

,I) ̸= H

[0,p]

(s,I) holds for any two distinct

states s, s

′

∈ V

M

if their corresponding current states

s(p) and s

′

(p) are different.

The main steps to determine the current state for a

BCN with this property are as follows:

(1) Input to a BCN with an input sequence I ∈ (V

L

)

p

that has sufﬁcient length to distinguish all distinct

current states, and run the BCN to generate the

output sequence o(0). .. o(p);

(2) Return the current state s(p) which satisﬁes

s(p) ∈ G

[p]

(I,o(0) .. .o(p)).

Deﬁnition 2 (Type-II reconstructibility (Zhang et al.,

2015b)). A BCN satisﬁes Type-II reconstructibility

if there exists an input sequence I ∈ (V

L

)

p

for some

p ∈ T, such that for any two distinct states s, s

′

∈ V

M

,

H

[0,p]

(s

′

,I) ̸= H

[0,p]

(s,I) holds if their corresponding

current states s(p) and s

′

(p) are different.

Similarly, the reconstruction algorithm corre-

sponding to Type-II reconstructibility is shown as fol-

lows:

Single-Experiment Reconstructibility of Boolean Control Networks Revisited

87

(1) Input to a BCN with an input sequence I ∈ (V

L

)

p

which distinguishes all distinct current states, and

run the BCN to generate the output sequence

o(0). .. o(p);

(2) Return the current state s(p) which satisﬁes

s(p) ∈ G

[p]

(I,o(0) .. .o(p)).

Next, we introduce the new types of recon-

structibility. Firstly, as the BCNs’ single-experiment

observability was redeﬁned as Type-V obser vability

in (Wu et al., 2020), we deﬁne Type-III reconstructibil-

ity as the counterpart corresponding to Type-V observ-

ability. For this purpose, we propose the set of state

sets Set

S

(k) for BCNs which denotes the set that, for

every state set S ∈ Set

S

(k), the set S satisﬁes that only

k time steps are required to determine a BCN’s cur-

rent state s(t +k) by one experiment when its state set

S

ps

(t) = S. We recursively deﬁne the set Set

S

(k) of

BCNs as the following steps.

• When k = 0, then Set

S

(k) = {S ∈ 2

V

M

||S| = 1}.

• When k > 0, then

Set

S

(k) = {S ∈ (2

V

M

−

S

k−1

p=0

Set

S

(p))|∃i ∈ V

L

·

∀o ∈ V

N

· ∃p ≤ (k − 1) · ζ(S, i,o) ∈ Set

S

(p)}.

Intuitively, when |S| = |S

ps

(t)| = 1, i.e. the BCN’s

state s(t) is determined, we need 0 time step to deter-

mine its current state. When k > 0, we use the sets

Set

S

(0),.. ., Set

S

(k − 1) that have been deﬁned to de-

ﬁne the set Set

S

(k). Firstly, as the set Set

S

(k) should

not intersect with the sets Set

S

(0),.. ., Set

S

(k − 1), we

have for every state set S ∈ Set

S

(k), the condition

S ∈ (2

V

M

−

S

k−1

p=0

Set

S

(p)) should be met. Secondly,

as a BCN should require more time steps to determine

its current state at t than at t +1, those following con-

ditions also need to be satisﬁed.

Then, we can deﬁne the function Γ(S) to represent

the number of time steps needed to determine a BCN’s

state s(t) by one experiment, when its possible state

set S

ps

(t) = S.

Γ : (2

V

M

− {

/

0}) 7→ (T ∪ {∞})

(7)

satisﬁes the following conditions.

• If there exists a ﬁnite number k which satisﬁes that

S ∈ Set

S

(k), then Γ(S) = k.

• Otherwise, Γ(S) = ∞.

Now, Type-III reconstructibility can be deﬁend.

Deﬁnition 3 (Type-III Reconstructibility). A BCN sat-

isﬁes Type-III reconstructibility if for every o ∈ V

N

,

ζ(V

M

,ε, o) ̸=

/

0 implies Γ(ζ(V

M

,ε, o)) ̸= ∞.

We provide a reconstruction algorithm corre-

sponding to Type-III reconstructibility as well.

(1) Obtain the state set S

ps

(0) of this BCN by its ini-

tial output o(0), i.e. S

ps

(0) := ζ(V

M

,ε, o(0)), and

set the set variable S by S

ps

(0), i.e. S := S

ps

(0).

(2) Feed the BCN with an input i which satisﬁes

max

o

′

∈{o|ζ(S,i,o)̸=

/

0}

Γ(ζ(S,i, o

′

)) + 1 = Γ(S)

and run it to generate the new output o(t).

(3) Determine the new S

ps

(t) by the input i, output

o(t), and set variable S, i.e. S

ps

(t) = ζ(S,i, o(t)),

and update the set variable S by S

ps

(t), i.e. S :=

S

ps

(t).

(4) If the cardinal number |S| = 1, then return the

BCN’s current state s(t) which satisﬁes s(t) ∈ S.

Otherwise, update t, i.e. t = t + 1 and go to step 2.

Finally, we deﬁne Type-IV & V reconstructibility

as the counterpart corresponding to Type-I & II observ-

ability, respectively. We show that, for a BCN that sat-

isﬁes these two properties, the current state of it can

also be determined by performing one experiment.

Deﬁnition 4 (Type-IV reconstructibility). A BCN satis-

ﬁes Type-IV reconstructibility if for every state s ∈V

M

,

there exists an input sequence I ∈ (V

L

)

p

for some

p ∈ T such that for any state s

′

̸= s, H

[0,p]

(s

′

,I) ̸=

H

[0,p]

(s,I) holds if their corresponding current states

s(p) and s

′

(p) are different.

(1) Obtain the set S

ps

(0) of possible valuations of the

BCN’s initial state s(0) by its initial output o(0),

i.e. S

ps

(0) = ζ(V

M

,ε, o(0)), and set the set vari-

able S := S

ps

(0), the time variable T := 0, the

input sequence variable I := [] and the output se-

quence variable O := [], where [] denotes an empty

sequence.

(2) Assume one state s of the BCN under study as a

candidate initial state s(0) from set S.

(3) Feed the BCN with an input sequence I

′

∈ (V

L

)

p

that distinguishes the current state s(p) corre-

sponding to s(0) from other current states to gen-

erate new output sequence o(0)...o(p), and set

the time variable T := T + p, the input sequence

variable I := I + I

′

and the output sequence vari-

able O := O + o(0). .. o(p).

(4) Determine the new S

ps

(t) by the input sequence

I, output sequence O and reset variable S :=

S

ps

(t) = G

[T]

(I,O).

(5) If the cardinal number |S| = 1, then return the

BCN’s current state s(t) which satisﬁes s(t) ∈ S.

Otherwise, go to step 2.

Deﬁnition 5 (Type-V reconstructibility). A BCN sat-

isﬁes Type-V reconstructibility if for any two dis-

tinct states s, s

′

∈ V

M

, there exists an input sequence

ICINCO 2023 - 20th International Conference on Informatics in Control, Automation and Robotics

88

I ∈ (V

L

)

p

for some p ∈ T such that H

[0,p]

(s

′

,I) ̸=

H

[0,p]

(s,I) holds if their corresponding current states

s(p) and s

′

(p) are different.

(1) Obtain the set S

ps

(0) of possible valuations of the

BCN’s initial state s(0) by its initial output o(0),

i.e. S

ps

(0) = ζ(V

M

,ε, o(0)), and set the set vari-

able S = S

ps

(0), the time variable T := 0, the input

sequence variable I := [] and the output sequence

variable O := [].

(2) Assume two distinct states s and s

′

of the BCN

under study as a candidate initial state s(0) and

s

′

(0) from the state set S.

(3) Feed the BCN with an input sequence I

′

∈ (V

L

)

p

that distinguishes the current states s(p) corre-

sponding to s(0) from s

′

(0) to generate new out-

put sequence o(0)...o(p), and set the time vari-

able T := T + p, the input sequence variable I :=

I + I

′

and the output sequence variable O := O +

o(0). .. o(p).

(4) Determine the new S

ps

(t) by the input sequence I,

output sequence O and reset the state set variable

S := S

ps

(t) = G

[T]

(I,O).

(5) If the cardinal number |S| = 1, then return the

BCN’s current state s(t) which satisﬁes s(t) ∈ S.

Otherwise, go to step 2.

In the above two algorithms, we perform mul-

tiple consecutive experiments without resetting the

BCN’s initial state. These successive multiple ex-

periments can therefore be considered as single ex-

periment. For this reason, Type-IV reconstructibility

and Type-V reconstructibility can be utilised to study

the BCNs’ single-experiment reconstruction. This re-

ﬂects the difference between the reconstruction and

observation, that will further compared in Section 5.

Now we discuss how the above three algorithms

enrich the solution of BCN’s single-experiment re-

construction. Firstly, by using the reconstruction al-

gorithm corresponding to Type-III reconstructibility,

we can determine the current state for a BCN in

the shortest possible time steps, which reduces the

cost of experimentation. Secondly, as the compu-

tational cost for verifying Type-V reconstructibility is

lower than that for Type-II reconstructibility (will be

discussed Section 4), the reconstruction of more and

larger BCNs can be solved.

4 VERIFICATION OF

RECONSTRUCTIBILITY

In this section, we introduce the veriﬁcation algo-

rithms we design for Type- III, IV & V reconstructibility.

As the reconstructibility and observability of

BCNs have high similarity in their deﬁnitions, the ver-

iﬁcation algorithm for reconstructibility can be ob-

tained by modifying the corresponding observabil-

ity’s algorithm. As a consequence, the computational

complexity of their veriﬁcation algorithms remains

the same. For instance, the veriﬁcation algorithms

that are based on the deterministic ﬁnite automata

(DFA) for Type-I & II reconstructibility as introduced

in (Zhang et al., 2020a) were obtained by modifying

the veriﬁcation algorithms of Type-IV & III observabil-

ity provided in (Zhang et al., 2020b), and have the

same computational complexity as the latter two. Fol-

lowing this common practice, we design the veriﬁca-

tion algorithm for Type-III reconstructibility based on

the veriﬁcation algorithm of Type-V observability pro-

posed in (Wu et al., 2020), and the veriﬁcation algo-

rithms for Type-IV & V reconstructibility based on the

veriﬁcation algorithms of Type-I & II observability in-

troduced in (Zhang et al., 2020b). Then, the compu-

tational complexity of them can be easily obtained by

referring to the results of the papers (Wu et al., 2020;

Zhang et al., 2020b).

Given a BCN with ℓ input-nodes, m state-nodes,

and n output-nodes, the computational complexity of

the veriﬁcation algorithm for Type-III reconstructibility

is O(2

2

m

+ℓ−1

). For Type-IV reconstructibility, the com-

putational complexity of its veriﬁcation algorithm is

O(2

2

2m−1

ℓ

) (Zhang et al., 2020b).

For Type-V reconstructibility, we actually design

two algorithms for verifying it. The ﬁrst one

is based on DFAs, with computational complexity

O(2

4m+ℓ−2

). For every two distinct current states of

the BCN with Type-V reconstructibility, the input se-

quence that distinguishes them can be obtained from

the veriﬁcation result of this algorithm. The compu-

tational complexity of this algorithm is smaller than

the existing veriﬁcation algorithm (with the computa-

tional complexity of O(2

2

2m−1

l

) (Zhang et al., 2020a))

for Type-II reconstructibility that provides the input

sequence to distinguish all different current states.

The second one we design is based on graph the-

ory, and its computational complexity is O(2

2m+ℓ−1

)

which is equivalent to the computational complexity

of the optimised veriﬁcation algorithm for Type-II re-

constructibility as introduced in (Zhang et al., 2020a).

However, this algorithm and the optimised veriﬁca-

tion algorithm for verifying Type-II reconstructibil-

ity (Zhang et al., 2020a) can only provide an an-

swer whether the BCN to be veriﬁed satisﬁes single-

experiment reconstructibility or not. In other words,

if and only if the veriﬁcation result of the second algo-

rithm shows that the BCN under study satisﬁes Type-

V reconstructibility, we need the ﬁrst one to obtain the

Single-Experiment Reconstructibility of Boolean Control Networks Revisited

89

input sequences that distinguish the pairs of different

current states. This also explains why the ﬁrst one is

more complex.

In the following, we mainly focus on the ﬁrst

veriﬁcation algorithm we design for Type-V recon-

structibility and give its details.

1

Our main idea is

to construct the weighted pair graph for a BCN ﬁrst,

then construct the DFA for the pairs of different states

that produce the same output from the weighted pair

graph, and ﬁnally obtain the input sequences that

distinguish the pairs of different current states from

the DFA. We need to ﬁrst introduce the notion of

weighted directed graph for BCNs.

Deﬁnition 6 (Weighted Pair Graph (Zhang et al.,

2020a)). Given a BCN, a weighted directed

graph G = (V ,E,W ), where V denotes the

vertex set, E ⊂ V × V denotes the edge set,

and W denotes the weight function, is called

the weighted pair graph of the BCN if V =

{{s,s

′

}|s,s

′

∈ V

M

,s ̸= s

′

,h(s) = h(s

′

)}, for all

[{s

1

,s

′

1

},{s

2

,s

′

2

}] ∈ V × V , [{s

1

,s

′

1

},{s

2

,s

′

2

}] ∈ E iff

there exists i ∈ V

L

such that f (i,s

1

) = f (i,s

2

) and

f (i,s

′

1

) = f (i,s

′

2

) or f (i,s

1

) = f (i,s

′

2

) and f (i,s

′

1

) =

f (i,s

2

); for all edges e = [{s

1

,s

′

1

},{s

2

,s

′

2

}] ∈ E,

W (e) = {i ∈ V

L

| f (i,s

1

) = f (i,s

2

) and f (i,s

′

1

) =

f (i,s

′

2

) or f (i,s

1

) = f (i,s

′

2

) and f (i,s

′

1

) = f (i,s

2

)}.

After introducing the notions of weighted pair

graph, we use the BCN shown in Table 1 as an ex-

ample to illustrate how to utilise them to obtain the

input sequences that distinguish the pairs of different

current states for a BCN which satisﬁes Type-V recon-

structibility. For this BCN, its weighted pair graph is

shown the top part of Fig. 2, and all the DFA con-

structed from the graph are shown in the lower part of

this ﬁgure. For every vertex {s

i

,s

′

i

} (e.g. {v

1

8

,v

3

8

}) in

the weighted pair graph, we construct a DFA A

i

(the

second DFA shown in the lower part of Fig. 2) for it.

In this DFA, the vertex {s

i

,s

′

i

} (e.g. {v

1

8

,v

3

8

}) is the

start state q

0

. The other states of the DFA A

i

are com-

posed of the vertices ({v

1

8

,v

2

8

} and {v

4

8

,v

5

8

}) reach-

able by {s

i

,s

′

i

} ({v

1

8

,v

3

8

}), in the weighted pair graph.

For every two state q (e.g. {v

1

8

,v

2

8

}) and q

′

({v

4

8

,v

5

8

})

in this DFA, q

′

= σ(q,a) holds if their correspond-

ing vertices are connected in the weighted pair graph.

Moreover, all states (e.g. {v

1

8

,v

2

8

} and {v

4

8

,v

5

8

}) in the

DFA A

i

are accepting states. Then, for the pair of dif-

ferent BCN’s states {s

i

,s

′

i

} ({v

1

8

,v

2

8

}), every word w ∈

L(A

i

) (in this case, L(A

i

) = (v

1

2

,v

0

2

)

∗

(ε∪ v

1

2

)) accepted

by this DFA, is the input sequence that could not dis-

tinguish the corresponding current BCN’s states of the

1

Due to the page limit, we will not show the details of

other veriﬁcation algorithms we brieﬂy mentioned in this

paper.

Table 1: A BCN used to illustrate the veriﬁcation algorithm.

s(t) v

0

8

v

1

8

v

2

8

v

3

8

v

4

8

v

5

8

v

6

8

v

7

8

i(t)

v

2

8

v

3

8

v

3

8

v

6

8

v

1

8

v

3

8

v

4

8

v

3

8

v

0

2

s(t +1)

v

7

8

v

5

8

v

1

8

v

4

8

v

2

8

v

6

8

v

0

8

v

0

8

v

1

2

o(t) v

0

4

v

1

4

v

1

4

v

1

4

v

2

4

v

2

4

v

3

4

v

3

4

·

Figure 2: The weighted directed graph and DFAs con-

structed for the BCN shown in Table 1.

states (i.e. v

1

8

and v

2

8

), because the accepting state

σ

∗

(q

0

,w) of this DFA denotes the pair of different

BCN’s states ({v

1

8

,v

2

8

} or {v

4

8

,v

5

8

}) that produce the

same output. Thus, the language L = Σ

∗

− L(A

i

) con-

tains all input sequences that can distinguish the cor-

responding current states of s

i

and s

′

i

(e.g. v

1

8

and

v

3

8

). If the language L = Σ

∗

−L(A

i

) =

/

0, which means

L(A

i

) = Σ

∗

, then there is no input sequence can be

used to distinguish the corresponding current states

of s

i

and s

′

i

, this BCN thus does not satisfy Type-V re-

constructibility. However, if the BCN to be veriﬁed

does not satisfy Type-V reconstructibility, this algo-

rithm would not be used since the second algorithm

we design (mentioned in the previous paragraphs)

would show the veriﬁcation result.

Finally, we analyse the computational complexity

of this algorithm. For a BCN, for every two different

initial states s

i

and s

′

i

, the size of the DFA A

i

is no

greater than that of the weighted pair graph. Since

at most 2

m

(2

m

− 1)/2 DFA need to be checked and

the size of the graph is O(2

2m+l−1

), the computational

complexity of this algorithm is O(2

4m+l−2

).

5 COMPARING

RECONSTRUCTIBILITY AND

OBSERVABILITY

Before comparing the reconstructibility and observ-

ability of BCNs, we need to present all ﬁve existing

types of observability. The initial state determining

algorithms corresponding to all ﬁve types of observ-

ICINCO 2023 - 20th International Conference on Informatics in Control, Automation and Robotics

90

ability will be omitted due to the page limit. Type-I &

II observability were proposed to study the solvability

of the BCNs’ multiple-experiment observation.

Deﬁnition 7 (Type-I observability (Cheng and Qi,

2009)). A BCN satisﬁes Type-I observability if for

every states s ∈ V

M

there exists an input sequence

I ∈ (V

L

)

p

for some p ∈ T, such that for any state s

′

̸= s,

H

[0,p]

(s

′

,I) ̸= H

[0,p]

(s,I).

Type-I observability was later replaced by Type-II

observability (Zhang et al., 2020b).

Deﬁnition 8 (Type-II observability (Zhao et al.,

2010)). A BCN satisﬁes Type-II observability if for

any two distinct states s, s

′

∈ V

M

there exists an in-

put sequence I ∈ (V

L

)

p

for some p ∈ T, such that

H

[0,p]

(s

′

,I) ̸= H

[0,p]

(s,I).

Type-III & V observability were proposed to re-

search single-experiment observation.

Deﬁnition 9 (Type-III observability (Cheng et al.,

2011)). A BCN satisﬁes Type-III observability if there

exists an input sequence I ∈ (V

L

)

p

for some p ∈

T, such that for any two distinct states s, s

′

∈ V

M

,

H

[0,p]

(s

′

,I) ̸= H

[0,p]

(s,I).

Type-V observability was proposed to redeﬁne

single-experiment observability. As the state s(t) can

be guaranteed to be determined in a ﬁnite time steps

of k, by performing one experiment if and only if

• |S

ps

(t +k)| = 1, i.e. s(t + k) is determined.

• For every t

0

: t + 1 ≤ t

0

≤ t + k, for every s ∈

S

ps

(t

0

), there exists only one s

′

∈ S

ps

(t

0

− 1) that

satisﬁes s = f (i(t

0

−1),s

′

), such that s(t

0

−1) can

be uniquely determined by s(t

0

) and i(t

0

−1). The

state s(t) then can be uniquely determined step by

step by s(t +k) and i(t). .. i(t +k − 1).

The authors of (Wu et al., 2020) proposed the

set of state sets

˙

Set

S

(k) for BCNs to denote the set

that, for every state set S ∈

˙

Set

S

(k), the set S satis-

ﬁes that only k time steps are required to determine a

BCN’s state s(t) by one experiment when its state set

S

ps

(t) = S.

˙

Set

S

(k) is recursively determined:

• When k = 0, then

˙

Set

S

(k) = {S ∈ 2

V

M

||S| = 1}.

• When k > 0, then

˙

Set

S

(k) = {S ∈ (2

V

M

−

S

k−1

p=0

˙

Set

S

(p))|∃i ∈ V

L

·

(|ζ(S,i, ε)| = |S|)&(∀o ∈ V

N

· ∃p ≤ (k − 1) ·

ζ(S,i, o) ∈

˙

Set

S

(p))}.

Intuitively, when |S| = |S

ps

(t)| = 1, i.e. s(t) is de-

termined, we need 0 time step to determine it. Thus,

we set

˙

Set

S

(k) = {S ∈ 2

V

M

||S| = 1} when k = 0. When

k > 0, we use the sets

˙

Set

S

(0),.. .,

˙

Set

S

(k − 1) that

have been deﬁned to deﬁne the set

˙

Set

S

(k). Firstly,

as the set

˙

Set

S

(k) should not intersect with the sets

˙

Set

S

(0),.. .,

˙

Set

S

(k − 1), we have for every state set

S ∈

˙

Set

S

(k), the condition S ∈ (2

V

M

−

S

k−1

p=0

˙

Set

S

(p))

should be satisﬁed. Secondly, as the BCN’s state s(t)

should be determined by its state s(t + 1) and input

i(t), those following conditions also need to be satis-

ﬁed as we discussed in the previous paragraph.

Now, the function

˙

Γ(S) to represent the number of

time steps needed to determine a BCN’s state s(t) by

one experiment, when its state set S

ps

(t) = S, can be

deﬁned as follows.

˙

Γ : (2

V

M

− {

/

0}) 7→ (T ∪ {∞})

(8)

satisﬁes the following conditions.

• If there exists a ﬁnite number k which satisﬁes that

S ∈

˙

Set

S

(k), then

˙

Γ(S) = k.

• Otherwise,

˙

Γ(S) = ∞.

Thus, the BCNs’ single-experiment observability

can be deﬁned as the following Type-V observability.

Deﬁnition 10 (Type-V observability (Wu et al., 2020)).

A BCN satisﬁes Type-V observability if for every pos-

sible S

ps

(0) of this BCN,

˙

Γ(S

ps

(0)) ̸= ∞.

Finally, we introduce Type-IV observability.

Deﬁnition 11 (Type-IV observability (Fornasini and

Valcher, 2013)). A BCN satisﬁes Type-IV observabil-

ity if there exist a ﬁnite number k ∈ T such that for any

input sequence I ∈ (V

L

)

p

where p ≥ k, H

[0,p]

(s

′

,I) ̸=

H

[0,p]

(s,I) holds for any two distinct states s, s

′

∈ V

M

.

Now we can compare reconstructibility and ob-

servability for BCNs. Firstly, for every type of observ-

ability and the reconstructibility which as the coun-

terpart of this type of observability, this type of ob-

servability implies the counterpart of it. The reason

is that once the initial state of a BCN can be deter-

mined, its current can also be determined by the ini-

tial state and inputs of the BCN. Secondly, all types of

observability are not equivalent. The relationship be-

tween them is shown in Fig. 3, where arrows means

“implies”. Thirdly, the four types of reconstructibility

namely Type-II, III, IV & V reconstructibility are equiv-

alent, which is formuated by the following theorem.

Theorem 1. Type-II, III, IV & V reconstructibility are

equivalent.

Due to space limitation, we omit the detailed

proof of this main theorem. Intuitively, it can be

proved by showing that Type-V reconstructibility im-

plies Type-II reconstructibility. This proposition can

be easily proved by constructing an input sequence

that distinguishes all current states for a BCN with

Single-Experiment Reconstructibility of Boolean Control Networks Revisited

91

Figure 3: The relationships between all types of the BCNs’

observability and reconstructibility.

Type-V reconstructibility. The propositions that the re-

constructibilities Type-II implies Type-III, Type-III im-

plies Type-IV, and Type-IV implies Type-V are obvi-

ous. Therefore, all these properties are equivalent,

and the relation between all types of observability

and reconstructibility can be illustrated in Fig. 3. It

is worth noting that, successive multiple experiments

can therefore be considered as single experiment in

the reconstruction of a BCN, is the main reason for

the difference between these two problems.

6 CONCLUSION AND FUTURE

WORK

It still requires a signiﬁcant amount of computa-

tional overhead to verify the single-experiment recon-

structibility for large scale BCNs, due to the com-

putational complexity we have discussed in the pa-

per. Thus, in future, we plan to improve the scala-

bility of the veriﬁcation algorithm for BCNs’ single-

experiment reconstructibility. We plan to research

whether it is possible to determine the current state

for a BCN, without any information given in advance,

about which input should be fed to the BCN at every

time step. If the answer to this question is positive, the

veriﬁcation of the single-experiment reconstructibil-

ity of BCN will be further simpliﬁed, and the com-

putational complexity of the corresponding algorithm

will also be reduced.

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