BOOLEAN NORMAL FORMS, SHELLABILITY, AND RELIABILITY
COMPUTATIONS
∗
ENDRE BOROS
†
, YVES CRAMA
‡
, OYA EKIN
§
, PETER L. HAMMER
†
, TOSHIHIDE
IBARAKI
¶
, AND ALEXANDER KOGAN
†
SIAM J. D
ISCRETE MATH.
c
2000 Society for Industrial and Applied Mathematics
Vol. 13, No. 2, pp. 212–226
Abstract. Orthogonal forms of positive Boolean functions play an important role in reliability
theory, since the probability that they take value 1 can be easily computed. However, few classes
of disjunctive normal forms are known for which orthogonalization can be efficiently performed. An
interesting class with this property is the class of shellable disjunctive normal forms (DNFs). In
this paper, we present some new results about shellability. We establish that every positive Boolean
function can be represented by a shellable DNF, we propose a polynomial procedure to compute the
dual of a shellable DNF, and we prove that testing the so-called lexico-exchange (LE) property (a
strengthening of shellability) is NP-complete.
Key words. Boolean functions, orthogonal DNFs, dualization, shellability, reliability
AMS subject classifications. Primary, 90B25; Secondary, 05C65, 68R05
PII. S089548019732180X
1. Introduction. A classical problem of Boolean theory is to derive an orthog-
onal form, or disjoint products form, of a positive Boolean function given in DNF
(see section 2 for definitions). In particular, this problem has been studied exten-
sively in reliability theory, where it arises as follows. One of the fundamental issues in
reliability is to compute the probability that a positive Boolean function (describing
the state—operating or failed—of a complex system) take value 1 when each vari-
able (representing the state of individual components) takes value 0 or 1 randomly
and independently of the value of the other variables (see, for instance, [3, 25]). For
functions in orthogonal form, this probability is very easily computed by summing
the probabilities associated to all individual terms, since any two terms correspond
to pairwise incompatible events. This observation has prompted the development of
several reliability algorithms based on the computation of orthogonal forms (see, e.g.,
[18, 21]).
In general, however, orthogonal forms are difficult to compute and few classes of
DNFs seem to be known for which orthogonalization can be efficiently performed. An
interesting class with this property, namely, the class of shellable DNFs, has been in-
troduced and investigated by Ball and Provan [2, 22]. As discussed by these authors,
the DNFs describing several important classes of reliability problems (k-out-of-n sys-
tems, all-terminal connectedness, all-point reachability, etc.) are shellable. Moreover,
∗
Received by the editors May 21, 1997; accepted for publication October 14, 1999; published
electronically April 6, 2000. This research was partially supported by the National Science Founda-
tion (grants DMS 98-06389 and INT 9321811), NATO (grant CRG 931531), and the Office of Naval
Research (grant N00014-92-J1375).
http://www.siam.org/journals/sidma/13-2/32180.html
†
Rutgers Center for Operations Research, Rutgers University, 640 Bartholomew Road, Pis-
cataway, NJ 08854 (boros@rutcor.rutgers.edu, hammer@rutcor.rutgers.edu, kogan@rutcor. rut-
gers.edu).
‡
Ecole d’Administration des Affaires, Universit´edeLi`ege, 4000 Li`ege, Belgium (y.crama@ulg.
ac.be).
§
Department of Industrial Engineering, Bilkent University, Bilkent, Ankara 06533, Turkey
(karasan@bilkent.edu.tr).
¶
Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto Uni-
versity, Kyoto, Japan 606-8501 (ibaraki@kuamp.kyoto-u.ac.jp).
212
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BOOLEAN NORMAL FORMS, SHELLABILITY, AND RELIABILITY 213
besides its unifying role in reliability theory, shellability also provides a powerful the-
oretical and algorithmic tool in the study of simplicial polytopes, abstract simplicial
complexes, and matroids. (This is actually where the shellability concept originates
(see, e.g., [7, 8, 11, 16]); let us simply mention here, without further details, that
abstract simplicial complexes are in a natural one-to-one relationship with positive
Boolean functions.)
Shellability is the main topic of this paper. In section 2, we briefly review the
basic concepts and notations to be used in this paper. In section 3, we establish
that every positive Boolean function can be represented by a shellable DNF, and
we characterize those orthogonal forms that arise from shellable DNFs by a classical
orthogonalization procedure. In section 4, we prove that the dual (or, equivalently,
the inverse) of a shellable DNF can be computed in polynomial time. Finally, in
section 5, we define an important subclass of shellable DNFs, namely, the class of
DNFs which satisfy the so-called LE property, and we prove that testing membership
in this class is NP-complete.
2. Notations, definitions, and basic facts. Let B = {0, 1} and let n be a
natural number. For any subset S ⊆{1, 2,...,n}, 1
S
is the characteristic vector of
S, i.e., the vector of B
n
whose jth coordinate is 1 if and only if j ∈ S. Similarly,
0
S
∈ B
n
denotes the binary vector whose jth coordinate is 0 exactly when j ∈ S.
The lexicographic order ≺
L
on subsets of {1, 2,...,n} is defined as usual: for all
S, T ⊆{1, 2,...,n}, S ≺
L
T if and only if min{j ∈{1, 2,...,n}|j ∈ S \ T} <
min{j ∈{1, 2,...,n}|j ∈ T \ S}.
We assume that the reader is familiar with the basic concepts of Boolean algebra
and we introduce here only the notions that we explicitly use in the paper (see, e.g.,
[19, 20] for more information).
A Boolean function of n variables is a mapping f : B
n
−→ B. We denote by
x
1
,x
2
,...,x
n
the variables of a Boolean function and we let x =(x
1
,...,x
n
). The
complement of variable x
j
is x
j
=1− x
j
. A DNF is a Boolean expression of the form
Ψ(x
1
,...,x
n
)=
m
k=1
j∈I
k
x
j
j∈J
k
x
j
,(2.1)
where I
k
,J
k
⊆{1, 2,...,n} and I
k
∩ J
k
= ∅ for all 1 ≤ k ≤ m. The terms of Ψ are
the elementary conjunctions
T
k
(x
1
,...,x
n
)=T
I
k
,J
k
(x
1
,...,x
n
)=
j∈I
k
x
j
j∈J
k
x
j
(k =1, 2,...,m).
(By abuse of terminology, we sometimes call “terms” the pairs (I
k
,J
k
) themselves.)
It is customary to view any DNF Ψ (or, more generally, any Boolean expression)
as defining a Boolean function: for any assignment of 0 − 1 values to the variables
(x
1
,...,x
n
), the value of Ψ(x
1
,...,x
n
) is simply computed according to the usual
rules of Boolean algebra. With this in mind, we say that the DNF Ψ represents
the Boolean function f (and we simply write f =Ψ)iff(x)=Ψ(x) for all binary
vectors x ∈ B
n
. It is well known that every Boolean function admits (many) DNF
representations.
A Boolean function f is called positive if f(x) ≥ f(y) whenever x ≥ y, where the
latter inequality is meant componentwise. For a positive Boolean function f , there is
a unique minimal family of subsets of {1, 2,...,n}, denoted P
f
, such that f (1
S
)=1
if and only if S ⊇ P for some P ∈P
f
. A subset S for which f(1
S
) = 1 is called an
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214 BOROS, CRAMA, EKIN, HAMMER, IBARAKI, AND KOGAN
implicant set (or implicant, for short) of f, and if S ∈P
f
, then S is called a prime
implicant (set) of f.
Prime implicants of positive Boolean functions have a natural interpretation in
many applied contexts. For instance, in reliability theory, prime implicants of a co-
herent structure function are in one-to-one correspondence with the minimal pathsets
of the system under study, i.e., with those minimal subsets of elements which, when
working correctly, allow the whole system to work (see, e.g., [3, 25]).
Every positive Boolean function can be represented by at least one positive DNF,
i.e., by a DNF of the form
Φ(x
1
,...,x
n
)=
m
k=1
j∈I
k
x
j
.(2.2)
Clearly, if I
k
⊆ I
l
for some k = l, then the Boolean function represented by (2.2)
does not change when we drop the term corresponding to I
l
. Hence, Φ represents f
if and only if the (containment wise) minimal subsets of I = {I
1
,...,I
m
} are exactly
the prime implicants of f.
Besides its representations by positive DNFs, every positive Boolean function can
also be represented by a variety of nonpositive DNFs. Let us record the following fact
for further reference.
Lemma 2.1. If the DNF Ψ given by (2.1) represents a positive Boolean function
f, then f =
m
k=1
j∈I
k
x
j
(and f ≡ 1 if I
k
= ∅ for some k ∈{1, 2,...,m}).
Proof. If Ψ represents f, then f (x)=Ψ(x) ≤
m
k=1
j∈I
k
x
j
for all x ∈ B
n
(since
the inequality holds termwise).
To prove the reverse inequality, assume that
m
k=1
j∈I
k
x
∗
j
= 1 for some x
∗
∈ B
n
.
Then there is an index k,1≤ k ≤ m, such that
j∈I
k
x
∗
j
= 1, or, equivalently,
1
I
k
≤ x
∗
.Now,f(1
I
k
)=Ψ(1
I
k
) = 1 and hence, since f is positive, f(x
∗
)=1.
As explained in the Introduction, this paper pays special attention to orthogonal
DNFs: the DNF (2.1) is said to be orthogonal (or is an ODNF, for short) if, for every
pair of terms T
k
,T
l
(k, l ∈{1, 2,...,m},k = l) and for every x ∈ B
n
, T
k
(x)T
l
(x)=0.
Equivalently, (2.1) is orthogonal if and only if (I
k
∩ J
l
) ∪ (I
l
∩ J
k
) = ∅ for all k = l.
In subsequent sections, we use the following basic properties of ODNFs.
Lemma 2.2. Let us assume that (2.1) is an ODNF of a positive Boolean function,
let k ∈{1, 2,...,m}, and let A
k
= {I
l
| l ∈{1, 2,...,m}, and I
l
∩J
k
= ∅}. Then J
k
is
a minimal transversal of A
k
, and S ∩ I
k
= ∅ holds for all other minimal transversals
S = J
k
of A
k
.
Proof. Let us assume that S is a transversal of A
k
for which S ∩ I
k
= ∅. Then
0
S
≥ 1
I
k
, and hence Ψ(0
S
) ≥ Ψ(1
I
k
) = 1. Furthermore, for every term T
l
(x) of
Ψ, l = k, we have T
l
(0
S
) = 0, since either I
k
∩ J
l
= ∅ or I
l
∩ J
k
= ∅, i.e., I
l
∈A
k
and hence I
l
∩ S = ∅, and in both cases the literals of T
l
corresponding to these
intersections have value 0 at the vector 0
S
.ThusT
k
(0
S
) = 1 must hold, and hence
J
k
⊆ S is implied.
On the other hand, J
k
itself is a transversal of A
k
(by definition of A
k
), which
proves that J
k
is the only minimal transversal of A
k
which is disjoint from I
k
.
Lemma 2.3. Let us assume that (2.1) is an ODNF of a positive Boolean function,
let k ∈{1, 2,...,m}, and let Ψ
k
denote the disjunction of all terms of Ψ but term
T
k
. Then Ψ
k
represents a positive Boolean function if and only if J
k
∩ I
l
= ∅ for all
l ∈{1, 2,...,m}\k.
Proof. Assume first that Ψ
k
represents a positive Boolean function and let l ∈
{1, 2,...,m},l= k. Then, by Lemma 2.1, Ψ
k
(1
I
l
) = 1. On the other hand, T
k
(0
J
k
)=
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BOOLEAN NORMAL FORMS, SHELLABILITY, AND RELIABILITY 215
1 and, since Ψ is an ODNF, all terms other than T
k
vanish at 0
J
k
, so that Ψ
k
(0
J
k
)=0.
Thus we conclude that 1
I
l
≤ 0
J
k
, or, equivalently, J
k
∩ I
l
= ∅.
Let us assume next that Ψ
k
does not represent a positive Boolean function. In
particular, Ψ
k
does not represent
m
l=1
l=k
j∈I
l
x
j
. Hence there exists l = k and there
exists a set S containing I
l
such that Ψ
k
(1
S
) = 0. On the other hand, since Ψ defines
a positive Boolean function, Ψ(1
S
) = 1 must hold (by Lemma 2.1). This implies that
T
k
(1
S
) = 1, i.e., J
k
∩ S = ∅. Therefore, J
k
∩ I
l
= ∅ follows.
3. Shellable DNFs. As mentioned earlier, any positive Boolean function can be
represented by a variety of DNFs. We now introduce one particular way of generating
such a DNF representation.
In what follows, the symbol I always denotes an arbitrary family of subsets of
{1, 2,...,n}, and π denotes a permutation of the sets in I. Let us denote by π(I) the
rank of the set I ∈I(i.e., its placement order) in the order of π.
Definition 3.1. For every family I of subsets of {1, 2,...,n}, every permutation
π of the sets in I, and every set I ∈I, the (I,π)-shadow J
I,π
(I) of I is the set
J
I,π
(I)={j ∈{1, 2,...,n}|∃I
∈I,π(I
) <π(I),I
\ I = {j}}.(3.1)
Lemma 3.1. For every permutation π of the sets of I, the positive Boolean
function f =
I∈I
j∈I
x
j
is represented by the DNF
Ψ
I,π
=
I∈I
j∈I
x
j
j∈J
I,π
(I)
x
j
.(3.2)
Proof. Clearly, f(x) ≥ Ψ
I,π
(x) for every Boolean vector x. In order to prove the
reverse inequality, let us consider any Boolean vector x
∗
such that f(x
∗
) = 1. Denote
by I ∈Ithe first set (according to the permutation π) for which
j∈I
x
∗
j
=1. We
claim that x
∗
j
= 0 for all j ∈ J
I,π
(I), from which there follows
j∈I
x
∗
j
j∈J
I,π
(I)
x
∗
j
=
1 and Ψ
I,π
(x
∗
) = 1, as required. To establish the claim, notice that, for every
j ∈ J
I,π
(I), there is a set I
∈Isuch that I
⊆ I ∪{j} and π(I
) <π(I). By choice
of I,
k∈I
x
∗
k
= 0, and thus x
∗
j
=0.
Example 3.1. Let us consider the family I = {I
1
= {1, 2},I
2
= {2, 3},I
3
= {3, 4}}
and the permutation π =(I
1
,I
3
,I
2
). Then J
I,π
(I
1
)=J
I,π
(I
3
)=∅, J
I,π
(I
2
)={1, 4},
and thus the positive Boolean function f = x
1
x
2
∨ x
2
x
3
∨ x
3
x
4
is also represented by
the DNF
f =Ψ
I,π
= x
1
x
2
∨ x
3
x
4
∨ x
1
x
2
x
3
x
4
.
The notion of “shadow” has been put to systematic use by Ball and Provan [2] in
their discussion of shellability and upper bounding procedures for reliability problems,
and by Boros [9] in his work on “aligned” Boolean functions (a special class of shellable
functions). Let us now recall one of the definitions of shellable DNFs.
Definition 3.2. A positive DNF Ψ=
I∈I
j∈I
x
j
is called shellable if there
exists a permutation π of I (called shelling order of I,orofΨ) with the following
property: for every pair of sets I
1
,I
2
∈Iwith π(I
1
) <π(I
2
), there exists j ∈ I
1
∩
J
I,π
(I
2
) (or equivalently: there exists j ∈ I
1
and I
3
∈I such that π(I
3
) <π(I
2
) and
I
3
\ I
2
= {j}).
Definition 3.2 is due to Ball and Provan [2], who observe that it is essentially
equivalent (up to complementation of all sets in I) to the “classical” definition of
shellability used, for instance, in [8, 11, 16]. The connection between Lemma 3.1 and
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216 BOROS, CRAMA, EKIN, HAMMER, IBARAKI, AND KOGAN
the notion of shellability is clarified in the next lemma (this result is implicit in [2],
where alternative characterizations of shellability can also be found).
Lemma 3.2. Permutation π is a shelling order of I if and only if the DNF Ψ
I,π
defined by (3.2) is orthogonal.
Proof. Consider any two terms, e.g., T
1
=
j∈I
1
x
j
j∈J
I,π
(I
1
)
x
j
and T
2
=
j∈I
2
x
j
j∈J
I,π
(I
2
)
x
j
of Ψ
I,π
, for which π(I
1
) <π(I
2
).
Assume first that π is a shelling order of I. By Definition 3.2, there is an index
j in I
1
∩ J
I,π
(I
2
). This shows that Ψ
I,π
is orthogonal.
Conversely, assume that Ψ
I,π
is orthogonal. If I
1
∩ J
I,π
(I
2
) is nonempty, then I
1
and I
2
satisfy the condition in Definition 3.2. So, assume now that I
1
∩ J
I,π
(I
2
)=∅,
and assume further that I ∩ J
I,π
(I
2
) = ∅ for all I ∈Isuch that π(I) <π(I
1
) (if this
is not the case, simply replace I
1
by I in the proof). Since Ψ
I,π
is orthogonal, there
must be some index j in I
2
∩ J
I,π
(I
1
). By Definition 3.1, there exists a set I
3
∈I
with π(I
3
) <π(I
1
) such that I
3
\I
1
= {j}. Now, we derive the following contradiction:
on the one hand, by our choice of I
1
, I
3
∩ J
I,π
(I
2
) may not be empty (since
π(I
3
) <π(I
1
)); on the other hand, I
3
∩J
I,π
(I
2
) must be empty, since j ∈ J
I,π
(I
2
) and
I
1
∩ J
I,π
(I
2
)=∅.
Observe that the DNF Ψ
I,π
associated to a shelling order π of I is orthogonal in
a rather special way: namely, for any two terms T
1
and T
2
such that π(I
1
) <π(I
2
),
the “positive part”
j∈I
1
x
j
of the first term is orthogonal to the “negative part”
j∈J
I,π
(I
2
)
x
j
of the second term (this follows directly from Definition 3.2).
As one may expect, not every positive DNF is shellable: a minimal counterexam-
ple is provided by the DNF
Φ(x
1
,...,x
4
)=x
1
x
2
∨ x
3
x
4
.
On the other hand, it can be shown that every positive Boolean function can be
represented by shellable DNFs (see also [9, Theorem 1]).
Theorem 3.3. Every positive Boolean function f can be represented by a shellable
DNF.
Proof. As a first proof, let us consider the DNF
Φ=
I∈I
j∈I
x
j
,
where I denotes the family of all implicants of the function f , and let π be a permu-
tation ordering these implicants in a nonincreasing order by their cardinality. Then
Φ represents f, and it is easy to see by Definition 3.2 that π is a shelling order of Φ.
Since the above DNF can, in general, be very large compared to the number of
prime implicants of f, let us show below another construction, using only a smaller
subset of the implicants.
Call a leftmost implicant of f any implicant I of f for which I \{h(I)} is not an
implicant of f, where h(I) denotes the highest-index element of the subset I. Let L
denote the family of leftmost implicants of f. Clearly, all prime implicants of f are in
L; therefore f is represented by the DNF Ψ
L
=
I∈L
j∈I
x
j
. Let us now consider
the permutation π of L induced by the lexicographic order of these implicants. We
claim that π is a shelling order of L.
To prove the claim, let I
1
and I
2
be two leftmost implicants of f with I
1
≺
L
I
2
,
and let j = min{i|i ∈ I
1
\ I
2
}.Ifj = h(I
1
), then j ∈ I
1
∩ J
I,π
(I
2
) (take I
3
= I
1
in
Definition 3.2), and we are done. So, assume next that j<h(I
1
). Let T = I
2
∪{j} and
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