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Journal ArticleDOI

Boolean Normal Forms, Shellability, and Reliability Computations

01 Apr 2000-SIAM Journal on Discrete Mathematics (Society for Industrial and Applied Mathematics)-Vol. 13, Iss: 2, pp 212-226

TL;DR: It is established that every positive Boolean function can be represented by a shellable DNF, a polynomial procedure to compute the dual of a shellability DNF is proposed, and it is proved that testing the so-called lexico-exchange (LE) property (a strengthening of shellability) is NP-complete.
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.

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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
jI
k
x
j
jJ
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
)=
jI
k
x
j
jJ
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 )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
jI
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
jI
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
jI
k
x
j
for all x B
n
(since
the inequality holds termwise).
To prove the reverse inequality, assume that
m
k=1
jI
k
x
j
= 1 for some x
B
n
.
Then there is an index k,1 k m, such that
jI
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
jI
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)-shadow J
I
(I) of I is the set
J
I
(I)={j ∈{1, 2,...,n}|∃I
∈I(I
) (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
jI
x
j
is represented by the DNF
Ψ
I
=
I∈I
jI
x
j
jJ
I
(I)
x
j
.(3.2)
Proof. Clearly, f(x) Ψ
I
(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
jI
x
j
=1. We
claim that x
j
= 0 for all j J
I
(I), from which there follows
jI
x
j
jJ
I
(I)
x
j
=
1 and Ψ
I
(x
) = 1, as required. To establish the claim, notice that, for every
j J
I
(I), there is a set I
∈Isuch that I
I ∪{j} and π(I
) (I). By choice
of I,
kI
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
(I
1
)=J
I
(I
3
)=, J
I
(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
I
= 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
jI
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
) (I
2
), there exists j I
1
J
I
(I
2
) (or equivalently: there exists j I
1
and I
3
∈I such that π(I
3
) (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
defined by (3.2) is orthogonal.
Proof. Consider any two terms, e.g., T
1
=
jI
1
x
j
jJ
I
(I
1
)
x
j
and T
2
=
jI
2
x
j
jJ
I
(I
2
)
x
j
of Ψ
I
, for which π(I
1
) (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
(I
2
). This shows that Ψ
I
is orthogonal.
Conversely, assume that Ψ
I
is orthogonal. If I
1
J
I
(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
(I
2
)=,
and assume further that I J
I
(I
2
) = for all I ∈Isuch that π(I) (I
1
) (if this
is not the case, simply replace I
1
by I in the proof). Since Ψ
I
is orthogonal, there
must be some index j in I
2
J
I
(I
1
). By Definition 3.1, there exists a set I
3
∈I
with π(I
3
) (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
(I
2
) may not be empty (since
π(I
3
) (I
1
)); on the other hand, I
3
J
I
(I
2
) must be empty, since j ∈ J
I
(I
2
) and
I
1
J
I
(I
2
)=.
Observe that the DNF Ψ
I
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
) (I
2
),
the “positive part”
jI
1
x
j
of the first term is orthogonal to the “negative part”
jJ
I
(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
jI
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
jI
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
(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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Citations
More filters

Journal ArticleDOI
Emanuele Borgonovo1Institutions (1)
TL;DR: This work builds a unified framework for the utilization of J and D in both coherent and non-coherent systems and proposes an algorithm that enables the numerical estimation of DT by varying one probability at a time, making it suitable in the analysis of complex systems.
Abstract: In the management of complex systems, knowledge of how components contribute to system performance is essential to the correct allocation of resources. Recent works have renewed interest in the properties of the joint (J) and differential (D) reliability importance measures. However, a common background for these importance measures has not been developed yet. In this work, we build a unified framework for the utilization of J and D in both coherent and non-coherent systems. We show that the reliability function of any system is multilinear and its Taylor expansion is exact at an order T. We then introduce a total order importance measure ( D T ) that coincides with the exact portion of the change in system reliability associated with any (finite or infinitesimal) change in component reliabilities. We show that D T synthesizes the Birnbaum, joint and differential importance of all orders in one unique indicator. We propose an algorithm that enables the numerical estimation of D T by varying one probability at a time, making it suitable in the analysis of complex systems. Findings demonstrate that the simultaneous utilization of D T and J provides reliability analysts with a complete dissection of system performance.

60 citations


Cites background from "Boolean Normal Forms, Shellability,..."

  • ...By a fundamental result of Boolean logics [Boros et al (2000)]: = M_ i=1 Oi (24) Since is a Bernoulli variable, then E[ ] = P ( = 1) = E[O1 _O2 _ ::: _OM ] = P ([Mi=1Oi = 1) (25) There follows that P ( = 1) = MX s=0 P (Os = 1) X i s P ((Oi = 1) \ (Os = 1)) + :::+ ( 1)MP (\Mi=1(Oi = 1)) (26) Then,…...

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  • ...In these instances, multilinear functions play a crucial role and many reliability results are a consequence of the properties of Boolean functions [Birnbaum (1969), Agrawal and Barlow (1984), Ball and Provan (1988), Fishman (1989), Boros et al (2000), Khachiyan et al (2007).]...

    [...]

  • ...Core of the theory is the representation of a system as a set of N components, each of which can be in two possible states [Birnbaum (1969), Ball and Provan (1988), Boros et al (2000), Giglio and Wynn (2004), Khachiyan et al (2007).]...

    [...]



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01 Jan 2017-
TL;DR: This chapter deals with the paradigm of handling system reliabilityAnalysis in the Boolean domain as a supplement to (rather than a replacement to) analysis in the probability domain, and explains some important properties of the concept of Boolean quotient.
Abstract: This chapter deals with the paradigm of handling system reliability analysis in the Boolean domain as a supplement to (rather than a replacement to) analysis in the probability domain. This paradigm is well-established within the academic circles of reliability theory and engineering, albeit virtually unknown outside these circles. The chapter lists and explains arguments in favor of this paradigm for systems described by verbal statements, fault trees, block diagrams, and network graphs. This is followed by a detailed exposition of the pertinent concept of the Real or Probability Transform of a switching (two-valued Boolean) function, and that of a Probability-Ready Expression (PRE). Some of the important rules used in generating a PRE are presented, occasionally along with succinct proofs. These include rules to achieve disjointness (orthogonality) of ORed formulas, and to preserve statistical independence, as much as possible, among ANDed formulas. Recursive relations in the Boolean domain are also discussed, with an application to the four versions of the AR algorithm for evaluating the reliability and unreliability of the k-out-of-n:G and the k-out-of-n:F systems. These four versions of the algorithm are explained in terms of signal flow graphs that are compact, regular, and acyclic, in addition to being isomorphic to the Reduced Ordered Binary Decision Diagram (ROBDD). An appendix explains some important properties of the concept of Boolean quotient, whose expectation in Boolean-based probability is the counterpart of conditional probability in event-based probability.

28 citations


Journal ArticleDOI
TL;DR: The paper treats the problem of survival reliability which is the probability of successful migration of a specific species from a critical habitat patch to destination habitat patches via heterogeneous imperfect corridors and contributes methods for computing a new measure of reliability that arises when paths to destination habitats patches share common corridors.
Abstract: This paper attempts to set the stage for a prospective interplay between ecology and reliability theory concerning the common issues of network connectivity and redundancy. The paper treats the problem of survival reliability which is the probability of successful migration of a specific species from a critical habitat patch to destination habitat patches via heterogeneous imperfect corridors. The paper surveys techniques of system reliability in an ecological setting and contributes methods for computing a new measure of reliability that arises when paths to destination habitat patches share common corridors. Care is taken to ensure that the reliability expressions obtained are as compact as possible and to check them for correctness.

21 citations


Journal ArticleDOI
TL;DR: An efficient method based on the inclusion-exclusion principle to compute the reliability of systems in the presence of epistemic uncertainty is presented and it is implied that the bounding interval of the system's reliability can be obtained with two simple calculations using methods similar to those of classical probabilistic approaches.
Abstract: We present an efficient method based on the inclusion-exclusion principle to compute the reliability of systems in the presence of epistemic uncertainty. A known drawback of belief functions and other imprecise probabilistic theories is that their manipulation is computationally demanding. Therefore, we investigate some conditions under which the measures of belief function theory are additive. If this property is met, the application of belief functions is more computationally efficient. It is shown that these conditions hold for minimal cuts and paths in reliability theory. A direct implication of this result is that the credal state (state of beliefs) about the failing (working) behavior of components does not affect the credal state about the working (failing) behavior of the system. This result is proven using a reliability analysis approach based on belief function theory. This result implies that the bounding interval of the system's reliability can be obtained with two simple calculations using methods similar to those of classical probabilistic approaches. A discussion about the applicability of the discussed theorems for non-coherent systems is also proposed.

14 citations


Cites background from "Boolean Normal Forms, Shellability,..."

  • ...Corollary 1 cannot be extended to non-coherent systems because the structure function of some non-coherent systems can be expressed by more than one structure function [64]....

    [...]


References
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Book
01 Jan 1979-
Abstract: This is the second edition of a quarterly column the purpose of which is to provide a continuing update to the list of problems (NP-complete and harder) presented by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NP-Completeness,’’ W. H. Freeman & Co., San Francisco, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed. Readers having results they would like mentioned (NP-hardness, PSPACE-hardness, polynomial-time-solvability, etc.), or open problems they would like publicized, should send them to David S. Johnson, Room 2C355, Bell Laboratories, Murray Hill, NJ 07974, including details, or at least sketches, of any new proofs (full papers are preferred). In the case of unpublished results, please state explicitly that you would like the results mentioned in the column. Comments and corrections are also welcome. For more details on the nature of the column and the form of desired submissions, see the December 1981 issue of this journal.

39,992 citations




Book
01 Jun 1981-
TL;DR: A number of new classes of life distributions arising naturally in reliability models are treated systematically and each provides a realistic probabilistic description of a physical property occurring in the reliability context, thus permitting more realistic modeling of commonly occurring reliability situations.
Abstract: : This is the first of two books on the statistical theory of reliability and life testing. The present book concentrates on probabilistic aspects of reliability theory, while the forthcoming book will focus on inferential aspects of reliability and life testing, applying the probabilistic tools developed in this volume. This book emphasizes the newer, research aspects of reliability theory. The concept of a coherent system serves as a unifying theme for much of the book. A number of new classes of life distributions arising naturally in reliability models are treated systematically: the increasing failure rate average, new better than used, decreasing mean residual life, and other classes of distributions. As the names would seem to indicate, each such class of life distributions provides a realistic probabilistic description of a physical property occurring in the reliability context. Also various types of positive dependence among random variables are considered, thus permitting more realistic modeling of commonly occurring reliability situations.

3,824 citations


"Boolean Normal Forms, Shellability,..." refers background in this paper

  • ...In the context of reliability theory, the prime implicants of f represent the minimal cutsets of the system under study, namely, the minimal subsets of elements whose failure causes the whole system to fail (see [3, 25])....

    [...]

  • ...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 variable (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])....

    [...]


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