1
Observability, reconstructibility and state observers
of Boolean Control Networks
Ettore Fornasini and Maria Elena Valcher
Index Terms—Boolean networks, Boolean control networks,
observability, reconstructibility, state observers.
Abstract—The aim of this paper is to introduce and charac-
terize observability and reconstructibility properties for Boolean
networks and Boolean control networks, described according to
the algebraic approach proposed by D. Cheng and co-authors in
the series of papers [3], [6], [7] and in the recent monography
[8]. A complete characterization of these properties, based both
on the Boolean matrices involved in the network description and
on the corresponding digraphs, is provided. Finally, the problem
of state observer design for reconstructible BNs and BCNs is
addressed, and two different solutions are proposed.
I. INTRODUCTION
Research interests in Boolean networks (BNs) have been
motivated by the large number of natural and artificial systems
whose describing variables display only two distinct configu-
rations, and hence take only two values. Originally introduced
to model simple neural networks, BNs have recently proved
to be suitable to describe and simulate the behavior of genetic
regulatory networks. Indeed, regulatory genes inside the cells
act just like switches, that may take either an “on” or an “off”
state (1 and 0, respectively), and this discovery led Kauffman
[14] to introduce random Boolean networks as models for
genetic networks (see also [23]). As a further application area,
BNs have also been used to describe the interactions among
agents and hence to investigate consensus problems [12], [21].
BNs are autonomous systems, since they evolve as automata,
whose dynamics is uniquely determined once the initial con-
ditions are assigned. On the other hand, when the network
behavior depends also on some (Boolean) control inputs, the
concept of BN naturally extends to that of Boolean control
network (BCN).
In the last decade, D. Cheng and co-workers have developed
an algebraic framework to deal with both BNs and BCNs [3],
[4], [5], [6], [7]. Their research efforts resulted in the recent
monography [8], where several theoretic problems, ranging
from stability and stabilizability to controllability, disturbance
decoupling and optimal control, have been investigated. Even
more, they stimulated further research in this area (see, for
instance [1], [9], [16], [18], [20]), aimed at deepening specific
control issues. The main idea underlying this approach is that
a Boolean network with n state variables exhibits 2
n
possible
configurations, and if any such configuration is represented
by means of a canonical vector of size 2
n
, all the logic maps
The Authors are with the Dip. di Ingegneria dell’Informazione, Univ. di
Padova, via Gradenigo 6/B, 35131 Padova, Italy, phone: +39-049-827-7795 -
fax: +39-049-827-7614, e-mail:fornasini,meme@dei.unipd.it.
that regulate the state-updating can be equivalently described
by means of 2
n
× 2
n
Boolean matrices. As a result, every
Boolean network can be described as a discrete-time linear
system. In a similar fashion, a Boolean control network can
be converted into a discrete-time bilinear system or, more
conveniently, it can be seen as a family of BNs, each of them
associated with a specific value of the input variables, and in
that sense it represents a switched system. As a consequence
of this algebraic set-up, logic-based problems can be converted
into algebraic problems and hence solved by resorting to the
standard mathematical tools available for linear state-space
models and, in particular, for positive state-space models [10],
[22], first of all graph theory.
In this paper, by following this stream of research, we first
address and characterize observability and reconstructibility
of Boolean networks. Then, we extend this analysis to the
class of BCNs. Finally, we address the problem of designing
a state observer for a BCN. In detail, the paper is organized
as follows: in section II we introduce and characterize ob-
servability, by first considering two elementary cases (BNs
consisting of a single cycle or of a single cycle and some
vertices accessing that cycle), and then moving to the general
case. Reconstructibility is the focus of section III, where it
is proved that this property is equivalent to the observability
of the reduced BN consisting of all the states of the BN
that belong to some cycle. Observability and reconstructibility
for BCNs are introduced and investigated in sections IV and
V, respectively. Finally, in section VI, the observer design
problem for BCNs (and hence, as a corollary, for BNs),
under the reconstructibility assumption, is analyzed, and two
different solutions are proposed. A preliminary version of the
first part of the paper has been accepted for presentation at
the next CDC 2012 conference [11].
Notation. Z
+
denotes the set of nonnegative integers. Given
two integers k, n ∈ Z
+
, with k ≤ n, by the symbol [k, n]
we denote the set of integers {k, k + 1, . . . , n}. We consider
Boolean vectors and matrices, taking values in B := {0, 1},
with the usual operations (sum +, product · and negation
¬
).
δ
i
k
will denote the ith canonical vector of size k, L
k
the
set of all k-dimensional canonical vectors, and L
k×n
⊂ B
k×n
the set of all k × n matrices whose columns are canonical
vectors of size k. Any matrix L ∈ L
k×n
can be repre-
sented as a row vector whose entries are canonical vectors in
L
k
, namely L = [ δ
i
1
k
δ
i
2
k
. . . δ
i
n
k
] , for suitable indices
i
1
, i
2
, . . . , i
n
∈ [1, k].
A permutation matrix Π is a nonsingular square matrix in
2
L
k×k
. In particular, a matrix
Π = C =
0 0 . . . 0 1
1 0 . . . 0 0
0 1
.
.
.
0 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0 0 . . . 1 0
= [ δ
2
k
δ
3
k
. . . δ
k
k
δ
1
k
]
(1)
is a k × k cyclic (permutation) matrix.
Given a matrix L ∈ B
k×k
(in particular, L ∈ L
k×k
), we
associate with it [2] a digraph D(L), with vertices 1, . . . , k.
There is an arc (j, `) from j to ` if and only if the (`, j)th
entry of L is unitary. A sequence j
1
→ j
2
→ . . . → j
r
→
j
r+1
in D(L) is a path of length r from j
1
to j
r+1
provided
that (j
1
, j
2
), . . . , (j
r
, j
r+1
) are arcs of D(L). A closed path
is called a cycle. In particular, a cycle γ with no repeated
vertices is called elementary, and its length |γ| coincides with
the number of (distinct) vertices appearing in it. Note that a k×
k cyclic matrix has a digraph that consists of one elementary
cycle with length k.
There is a bijective correspondence between Boolean vari-
ables X ∈ B and vectors x ∈ L
2
, defined by the relationship
x =
X
¬
X
.
We introduce the (left) semi-tensor product n between ma-
trices (and hence, in particular, vectors) as follows [8], [17],
[19]: given L
1
∈ R
r
1
×c
1
and L
2
∈ R
r
2
×c
2
(in particular,
L
1
∈ L
r
1
×c
1
and L
2
∈ L
r
2
×c
2
), we set
L
1
nL
2
:= (L
1
⊗I
T/c
1
)(L
2
⊗I
T/r
2
), T := l.c.m.{c
1
, r
2
},
where l.c.m. denotes the least common multiple. The semi-
tensor product represents an extension of the standard matrix
product, by this meaning that if c
1
= r
2
, then L
1
nL
2
= L
1
L
2
.
Note that if x
1
∈ L
r
1
and x
2
∈ L
r
2
, then x
1
nx
2
∈ L
r
1
r
2
. For
the various properties of the semi-tensor product we refer to
[8]. By resorting to the semi-tensor product, we can extend
the previous correspondence to a bijective correspondence
between B
n
and L
2
n
. This is possible in the following way:
given X = [ X
1
X
2
. . . X
n
]
>
∈ B
n
set
x :=
X
1
¬
X
1
n
X
2
¬
X
2
n . . . n
X
n
¬
X
n
.
This amounts to saying that
x =
2
6
6
6
6
4
X
1
X
2
. . . X
n−1
X
n
X
1
X
2
. . . X
n−1
¬
X
n
X
1
X
2
. . .
¬
X
n−1
X
n
.
.
.,
¬
X
1
¬
X
2
. . .
¬
X
n−1
¬
X
n
3
7
7
7
7
5
.
II. OBSERVABILITY OF BOOLEAN NETWORKS
A Boolean Network (BN) is described by the following
equations
X(t + 1) = f(X(t)),
Y (t) = h(X(t)), t ∈ Z
+
,
(2)
where X(t) and Y (t) denote the n-dimensional state variable
and the p-dimensional output variable at time t, taking values
in B
n
and B
p
, respectively. f and h are (logic) functions,
namely maps f : B
n
→ B
n
and h : B
n
→ B
p
. Upon
representing the state and the output vectors X(t) and Y (t)
by means of their equivalent x(t) and y(t) in L
N
and L
P
,
respectively, where N := 2
n
and P := 2
p
, the BN (2) can be
described [8] as
x(t + 1) = L n x(t) = Lx(t),
y(t) = H n x(t) = Hx(t), t ∈ Z
+
,
(3)
where L ∈ L
N×N
and H ∈ L
P ×N
are matrices whose
columns are canonical vectors of size N and P , respectively.
Definition 1: Given a BN (3),
• two states x
1
= δ
i
N
and x
2
= δ
j
N
are said to be
indistinguishable, if the two output evolutions of the BN
starting at t = 0 from x (0) = x
1
and from x(0) = x
2
,
respectively
1
, coincide at every time instant t ∈ Z
+
;
otherwise they are distinguishable;
• the BN is said to be observable if every two distinct states
are distinguishable.
In order to analyze the observability problem, we introduce
a family of equivalence relations on the set L
N
of all states.
We say that x
1
and x
2
are indistinguishable in k steps (x
1
∼
k
x
2
) if the output evolutions, say y
1
(t) and y
2
(t), stemming
from x
1
(0) = x
1
and x
2
(0) = x
2
, respectively, coincide for
every t ∈ [0, k−1]. The equivalence relation ∼
k
partitions L
N
into disjoint classes. We let C
∼,k
be the set of such classes. It is
easily seen that if two states are indistinguishable in k+1 steps
then they are indistinguishable in k steps, while the converse
is not necessarily true. Therefore the cardinality of the set
C
∼,k+1
in general is greater than or equal to the cardinality of
C
∼,k
, and
|C
∼,1
| ≤ |C
∼,2
| ≤ |C
∼,3
| ≤ . . . . (4)
On the other hand, it can be shown [11] that if, for some
positive integer k, |C
∼,k
| = |C
∼,k+1
|, then C
∼,k
= C
∼,k+1
and
C
∼,k
= C
∼,k+`
for every ` ∈ Z
+
.
Note that condition |C
∼,1
| = 1 corresponds to the situation
when all the BN states produce the same output value, a
situation that surely prevents the BN from being observable.
So, when discussing observability, we will always assume that
ρ := |C
∼,1
|, which coincides with the number of nonzero rows
of H, is at least 2. Based on the above reasoning, we can prove
a preliminary lemma.
Lemma 1: Given a BN (3), with ρ ≥ 2, consider two
states x
1
and x
2
∈ L
N
, and let y
1
(t) and y
2
(t), t ∈ Z
+
,
be the output trajectories stemming from x
1
(0) = x
1
and
x
2
(0) = x
2
, respectively. Condition y
1
(t) = y
2
(t) for every
t ∈ [0, N − ρ] implies y
1
(t) = y
2
(t) for every t ∈ Z
+
.
Proof: We only need to show that the smallest k such
that |C
∼,k
| = |C
∼,k+1
| cannot be greater than N − ρ + 1. This
follows from the fact that |C
∼,1
| = ρ and, therefore, as far
1
In the following, we will denote the state and the output trajectories
stemming from x
i
by x
i
(t) and y
i
(t), t ∈ Z
+
, respectively. Accordingly,
we will use x
i
(0) = x
i
for the initial state.
3
as the sequence (4) is strictly increasing we have |C
∼,k
| ≥
k + ρ − 1. On the other hand, |C
∼,k
| ≤ N for every k. This
ensures that |C
∼,N−ρ+1
| = |C
∼,N−ρ+2
|. Therefore, two output
trajectories coincide if and only if they coincide on the time
interval [0, N − ρ].
As an immediate consequence of the previous lemma, a BN
(3) is observable if and only if, given the first N −ρ+1 samples
of any output trajectory of the BN, y(t), t ∈ [0, N − ρ], we
can determine the initial condition x(0) that has generated it.
In fact, (see Example 1, below), this bound is tight, by this
meaning that there exist BNs for which N − ρ + 1 output
values are required to determine the initial state. Lemma 1
immediately leads to the following condition, which reminds
of the analogous one obtained for linear systems.
Proposition 1: A BN (3) is observable if and only if the
observability matrix in N − 1 steps
O
N−1
:=
H
HL
HL
2
.
.
.
HL
N−2
(5)
has N distinct columns.
As a further step, we want to relate the observability of a
BN (3) to the structure of the associated digraph D(L). To this
goal, we first consider a BN whose digraph D(L) contains a
single cycle and all the other states access the cycle.
Proposition 2: Consider a BN (3), with
L =
W 0
T C
∈ L
N×N
,
where C is a k × k cyclic matrix and W is nilpotent. The BN
is observable if and only if
i) [distinguishability of states before state merging] if i 6= j,
condition Lδ
i
N
= Lδ
j
N
implies Hδ
i
N
6= Hδ
j
N
;
ii) [distinguishability of states belonging to the cycle] every
state belonging to the cycle, δ
i
N
, i ∈ [N − k + 1, N ],
generates a periodic output trajectory with minimal period
k.
Proof: [Necessity] If condition i) was not satisfied,
the two initial states x
1
(0) = δ
i
N
and x
2
(0) = δ
j
N
would
produce the same state trajectory, starting at t = 1, and the
corresponding output trajectories, y
1
(t) and y
2
(t), t ∈ Z
+
,
would coincide for every t ∈ Z
+
. Hence the two states would
be indistinguishable. On the other hand, if condition ii) is
not satisfied, there would be a state belonging to the cycle,
say x(0) = δ
i
N
, i ∈ [N − k + 1, N], generating a periodic
output trajectory with minimal period
¯
k, a proper divisor of
k, and hence it would be y(t +
¯
k) = y(t) for every t ∈ Z
+
.
Consequently, the two states x(0) 6= x(
¯
k), belonging to the
cycle, would be indistinguishable.
[Sufficiency] We want to prove that if conditions i) and ii)
hold, then the BN is observable. Suppose it is not. Then two
distinct states x
1
(0) = x
1
and x
2
(0) = x
2
can be found that
produce the same output trajectories, i.e. y
1
(t) = y
2
(t), ∀ t ∈
Z
+
. If the two state trajectories eventually coincide, then there
exists a minimum t
0
∈ Z
+
such that x
1
(t
0
+ 1) = x
2
(t
0
+ 1).
But then assumption i) is contradicted for the two distinct
states x
1
(t
0
) = δ
i
N
and x
2
(t
0
) = δ
j
N
. So, we now assume that,
at every time t ∈ Z
+
, x
1
(t) 6= x
2
(t). Consider the sequence of
pairs (x
1
(t), x
2
(t)), t ∈ Z
+
. Since all such pairs take values
in the finite set L
N
×L
N
, there exist t
m
, t
M
∈ Z
+
, with t
m
<
t
M
, such that (x
1
(t
m
), x
2
(t
m
)) = (x
1
(t
M
), x
2
(t
M
)). As both
the trajectories x
1
(t) and x
2
(t) are periodic starting (at least)
from t = t
m
, this means that x
1
(t
m
) = δ
i
N
and x
2
(t
m
) =
δ
j
N
, i 6= j, are indistinguishable states corresponding to the
cyclic part C. So, both these states, belonging to the cycle,
generate a periodic output trajectory whose minimal period is
smaller than k.
Remark 1: Condition ii) in Proposition 2 can be
simply restated by saying that the ordered k-tuple
(y(0), y(1), . . . , y(k − 1)) corresponding to any periodic
state trajectory is irreducible, namely it cannot be obtained
by repeating a shorter ordered sequence.
Example 1: Consider the BN (3) with L = C an N × N
cyclic matrix and
H = [ δ
1
2
δ
1
2
. . . δ
1
2
δ
2
2
] ∈ L
2×N
.
We may describe the BN by means of a suitable digraph,
obtained by adding to D(L) the information regarding the
static output map H. This can be achieved by associating with
each node of D(L) a dashed arrow, labeled by the value of
the corresponding output.
1"
y"="δ
2
1"
2"
y"="δ
2
1"
3"
y"="δ
2
1"
4"
y"="δ
2
1"
N
"
y"="δ
2
2"
N*1"
y"="δ
2
1"
N*2"
y"="δ
2
1"
5"
y"="δ
2
1"
FIG. 1. Digraph corresponding to the BN of Example 1.
By Proposition 2 , the BN is observable. In order to deter-
mine x(0), consider the sequence y(0), y(1), . . . , y(N − 2).
If all such vectors are equal to δ
1
2
, then x(0) = δ
1
N
, otherwise
if y(t) = δ
2
2
for some t ∈ [0, N − 2], then x(t) = δ
N
N
and
hence x(0) = δ
N−t
N
. Note that, in general, it would not be
possible to determine the initial state by stopping the output
observation before time t = N − 2. Finally, the observability
matrix in N − 1 steps is
4
O
N−1
=
H
HL
HL
2
.
.
.
HL
N−2
=
δ
1
2
δ
1
2
. . . δ
1
2
δ
1
2
δ
2
2
δ
1
2
δ
1
2
. . . δ
1
2
δ
2
2
δ
1
2
δ
1
2
δ
1
2
. . .
δ
2
2
δ
1
2
δ
1
2
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
δ
1
2
δ
2
2
. . . δ
1
2
δ
1
2
δ
1
2
.
Proposition 3: Consider a BN (3), with
L = blockdiag{D
1
, D
2
, . . . , D
r
} ∈ L
N×N
,(6)
and D
ν
=
W
ν
0
T
ν
C
ν
∈ L
n
ν
×n
ν
, (7)
where W
ν
is a (n
ν
− k
ν
) × (n
ν
− k
ν
) nilpotent matrix, and
C
ν
is a k
ν
× k
ν
cyclic matrix. The BN is observable if and
only if
i) [distinguishability of states before state merging] if i 6= j,
condition Lδ
i
N
= Lδ
j
N
implies Hδ
i
N
6= Hδ
j
N
;
ii) [distinguishability of states belonging to a cycle] for
every ` ∈ [1, r], the ordered k
`
-tuple (Hδ
i+1
N
, Hδ
i+2
N
,
. . . , Hδ
i+k
`
N
), with
2
i := (n
1
+n
2
+. . .+n
`−1
)+(n
`
−k
`
),
is irreducible;
iii) [distinguishability of states belonging to different cycles]
if `, d ∈ [1, r], ` 6= d, and k
`
= k
d
=: k, the two ordered
k-tuples
(Hδ
i+1
N
, Hδ
i+2
N
, . . . , Hδ
i+k
N
), (Hδ
j+1
N
, Hδ
j+2
N
, . . . , Hδ
j+k
N
),
with i = (n
1
+ n
2
+ . . . + n
`−1
) + (n
`
− k
`
) and j =
(n
1
+ n
2
+ . . . + n
d−1
) + (n
d
− k
d
), neither coincide nor
can be obtained one from the other by means of cyclic
permutations.
Proof: [Necessity] The necessity of conditions i) and ii)
follows immediately from Proposition 2. On the other hand, if
condition iii) was not satisfied, there would be two initial states
corresponding to two distinct cycles and generating the same
periodic output trajectory, thus contradicting observability.
[Sufficiency] We want to prove that if conditions i), ii) and
iii) hold, then the BN is observable. Suppose it is not. Then
two distinct states x
1
(0) = x
1
and x
2
(0) = x
2
could be
found that produce the same output trajectory. If x
1
and x
2
correspond to the same block D
ν
then, by the same reasoning
adopted in the proof of Proposition 2, either i) or ii) would
be contradicted. So, assume that such indistinguishable states
correspond to different blocks, say D
`
and D
d
, ` 6= d.
Since both state trajectories eventually become periodic of
periods k
`
and k
d
, respectively, the corresponding output
trajectories become periodic, too, and since condition ii) holds,
the minimal periods of the outputs coincide with the minimal
periods of the state trajectories. Since the output trajectories
coincide, by the indistinguishability of the states, it follows
that k
`
= k
d
. Set k := k
`
= k
d
and let t
m
∈ Z
+
be the
smallest time t such that x
1
(t) = δ
i+1
N
for i = (n
1
+n
2
+. . .+
n
`−1
) + (n
`
− k
`
), and assume that x
2
(t
m
) = δ
h+1
N
for some
2
We assume n
0
:= 0.
h ∈ [(n
1
+n
2
+. . .+n
d−1
)+(n
d
−k
d
), n
1
+n
2
+. . .+n
d
−1].
Then the two sequences
(Hδ
i+1
N
, HLδ
i+1
N
, . . . , HL
k−1
δ
i+1
N
), (Hδ
h+1
N
, HLδ
h+1
N
, . . . , HL
k−1
δ
h+1
N
)
coincide, thus contradicting iii).
Remark 2: Conditions i), ii) and iii) of Proposition 3 can
be expressed in a rather compact form if we block-partition
the matrix H, according to the block partition of L. Indeed,
if we assume that
H = [ H
1n
H
1c
| H
2n
H
2c
| . . . | H
rn
H
rc
] ,
where each block [ H
νn
H
νc
] has size P × n
ν
, while H
νc
has size P × k
ν
, then the BN (3) is observable if and only if
i) each block
W
ν
0
T
ν
C
ν
H
νn
H
νc
has all distinct columns;
ii) each block H
νc
cannot be seen as the juxtaposition of two
or more copies of the same block (say Φ
ν
), i.e. H
νc
6=
[ Φ
ν
Φ
ν
. . . Φ
ν
];
iii) if ` 6= d, the blocks H
`c
and H
dc
are distinct and
cannot be obtained one from the other by means of cyclic
permutations of the columns, i.e. 6 ∃ a cyclic matrix C and
h ∈ Z
+
such that H
dc
= H
`c
C
h
.
Proposition 3 provides a general characterization of observ-
ability for Boolean networks. This is due to the fact that the
matrix L of every BN can be reduced to the block diagonal
form (6)-(7), by means of a suitable permutation matrix.
Indeed, every state trajectory of a BN takes only a finite
number of distinct values and hence it eventually becomes
periodic (possibly constant). The fact that all columns of L are
canonical vectors implies that the set of all states of the BN can
be partitioned into say r (disjoint) domains of attraction, each
of them consisting of an elementary cycle (called equilibrium
point, in case it consists of a single state) and a number of
states that eventually converge to it.
Proposition 4: Given a BN (3), there exists r ∈ N and a
permutation matrix Π such that
Π
>
LΠ = blockdiag{D
1
, D
2
, . . . , D
r
} ∈ L
N×N
,(8)
with D
ν
=
W
ν
0
T
ν
C
ν
∈ L
n
ν
×n
ν
, (9)
where W
ν
is a (n
ν
− k
ν
) × (n
ν
− k
ν
) nilpotent matrix, and
C
ν
is a k
ν
× k
ν
cyclic matrix.
Proof: Let Z
1
, Z
2
, . . . , Z
r
be the distinct elementary
cycles (possibly equilibrium points) of the system, and let
D
1
, D
2
, . . . , D
r
be the corresponding disjoint domains of
attraction. Clearly, a permutation matrix Π can be found such
that Π
>
LΠ is block diagonal as in (8). Even more, we can
order the states of each domain D
ν
so that those belonging
to Z
ν
are the last ones. So, it entails no loss of generality
assuming that each diagonal block is as in (9), where C
ν
is
the cyclic matrix associated with Z
ν
. All the states of D
ν
\Z
ν
5
produce trajectories that belong to Z
ν
after a finite number of
steps, which means that, for a sufficiently high k,
D
k
ν
=
W
k
ν
0
∗ C
k
ν
=
0 0
∗ C
k
ν
,
and hence W
ν
is nilpotent.
Propositions 3 and 4 have mainly a theoretical value, as
they connect the topological structure of the digraph D(L)
to the observability property. However, the evaluation of the
permutation matrix Π is computationally demanding, as it
requires to determine the limit cycles and the corresponding
domains of attraction. As a matter of fact, observability of
a BN can be characterized in a much simpler way, whose
practical feasibility will be addressed in Remark 4. To this
goal it is sufficient to notice that every elementary cycle of
length k corresponds to k distinct periodic state trajectories,
depending on the specific choice of the initial state. On the
other hand, each of these k periodic trajectories is uniquely
associated with the ordered k-tuple (x(0), x(1), . . . , x(k−1)).
By taking this perspective, we can merge the distinguishability
conditions ii) and iii) into a single one.
Theorem 1: A BN (3) is observable if and only if
i) [distinguishability of states before state merging] for
every x
1
, x
2
∈ L
N
, with x
1
6= x
2
, condition Lx
1
= Lx
2
implies Hx
1
6= Hx
2
;
ii) [distinguishability of states belonging to (the same or
different) cycles] for every pair of distinct periodic state
trajectories of the same minimal period k, described by
the two ordered k-tuples
(x
1
, x
2
, . . . , x
k
) 6= (
¯
x
1
,
¯
x
2
, . . . ,
¯
x
k
),
the corresponding output trajectories are periodic with
(minimal) period k and described by two different ordered
k-tuples, i.e.
(Hx
1
, Hx
2
, . . . , Hx
k
) 6= (H
¯
x
1
, H
¯
x
2
, . . . , H
¯
x
k
).
III. RECONSTRUCTIBILITY OF BOOLEAN NETWORKS
In the previous section we have seen that, as for linear state-
space models, observability corresponds to the possibility of
uniquely determining the system initial condition x(0) from
the observation of the corresponding output evolution in some
interval [0, T ]. Reconstructibility property may be introduced
along the same perspective, as the possibility of determining
the system final state x(T ) from the corresponding output
evolution in [0, T ].
Definition 2: A BN (3) is said to be reconstructible if there
exists T ∈ Z
+
such that the knowledge of the output trajectory
y(t), t ∈ [0, T ], allows to uniquely determine x(T ) (and hence
x(t) for every t ≥ T ). If this is the case, the smallest such T
will be denoted by T
min
.
It is clear that observability implies reconstructibility. On
the other hand, it is also obvious that if all the states of a BN
belong to some cycle (or, in particular, are equilibrium points),
then once the state x(T ) has been uniquely identified, the state
x(0) can be determined by moving backward. This means
that, for a BN whose digraph D(L) is the union of cycles,
observability and reconstructibility are equivalent properties.
In the general case, it turns out that reconstructibility is
equivalent to the fact that all states that belong to the cycles
are distinguishable one from the other.
Theorem 2: Given a BN (3), the following facts are equiv-
alent:
i) the BN is reconstructible;
ii) the reduced BN, obtained from (3) by considering only
the states that belong to some cycle
3
, is observable;
iii) for every pair of distinct periodic state trajectories of the
same minimal period k, described by the two ordered
k-tuples
(x
1
, x
2
, . . . , x
k
) 6= (
¯
x
1
,
¯
x
2
, . . . ,
¯
x
k
),
the corresponding output trajectories are periodic of (min-
imal) period k and described by two different k-tuples,
i.e.
(Hx
1
, Hx
2
, . . . , Hx
k
) 6= (H
¯
x
1
, H
¯
x
2
, . . . , H
¯
x
k
).
Moreover, when the BN is reconstructible, T
min
≤ T
r
+
¯
N −1,
where T
r
is the minimum number of steps after which the state
of the BN surely belongs to a cycle, while
¯
N is the number
of states of the BN that belong to a cycle, i.e.
T
r
:= min{t ∈ Z
+
: L
t
x ∈ ∪
r
i=1
Z
i
, ∀ x ∈ L
N
}|,
¯
N := |{x ∈ L
N
: x ∈ Z
i
, ∃ i ∈ [1, r]}|,
Z
i
, i ∈ [1, r], being the distinct cycles of the BN.
Proof: i) ⇒ iii) If iii) were not satisfied, there would be
two distinct initial states, x
1
and
¯
x
1
, that produce two distinct
periodic state trajectories of the same minimal period and the
same output trajectory. Hence at every time t, we would not
be able to distinguish x(t) from
¯
x(t), the states reached at
time t starting from x
1
and from
¯
x
1
, respectively. So, for
every choice of T , from the output in [0, T ], we could not
determine the state at time T .
iii) ⇒ ii) Follows from Theorem 1.
ii) ⇒ i) Set T := T
r
+
¯
N − 1. Every state trajectory x(t) is
surely periodic for t ≥ T
r
, which amounts to saying that x(T
r
)
belongs to some cycle Z
i
, i ∈ [1, r]. If
¯
N = 1, namely the
reduced BN is one-dimensional, then clearly reconstructibility
holds for T = T
r
. So, assume now that
¯
N > 1 (and hence
ρ, the number of nonzero rows of H, is not smaller than 2).
By the observability of the reduced BN and Lemma 1, we can
claim that upon observing y(t), t ∈ [T
r
, T
r
+
¯
N − 2], we can
identify the state x(T
r
). So, in particular, we can determine
x(T
r
+
¯
N − 1). This ensures that the BN is reconstructible
and that T
min
≤ T
r
+
¯
N − 1.
3
Note that the definition is well posed, since cycles are invariant sets, and
indeed each state belonging to a cycle can only access one and only one state
belonging to the same cycle.
