Inconsistency-based Ranking of Knowledge Bases
Said Jabbour, Badran Raddaoui and Lakhdar Sais
CRIL - CNRS UMR 8188, University of Artois, Lens, France
Keywords:
Knowledge Representation, Measuring Inconsistency, Pareto Optimality.
Abstract:
Inconsistencies are a usually undesirable feature of many kinds of data and knowledge. Measuring incon-
sistency is potentially useful to determine which parts of the data or of the knowledge base are conflicting.
Several measures have been proposed to quantify such inconsistencies. However, one of the main problems
lies in the difficulty to compare their underlying quality. Indeed, a highly inconsistent knowledge base with
respect to a given inconsistency measure can be considered less inconsistent using another one. In this pa-
per, we propose a new framework allowing us to partition a set of knowledge bases as a sequence of subsets
according to a set of inconsistency measures, where the first element of the partition corresponds to the most
inconsistent one. Then we discuss how finer ranking between knowledge bases can be derived from an original
combination of existing measures. Finally, we extend our framework to provide some inconsistency measures
obtained by combining existing ones.
1 INTRODUCTION
There is considerable evidence that conflicts are often
inevitable in groups and organizations. Indeed, an-
alyzing inconsistency is central to many domains of
computer science and Artificial Intelligence (Bertossi
et al., 2005). It has been widely studied due to its sig-
nificant importance in many applications, including
network intrusion detection (McAreavey et al., 2011),
software specifications (Martinez et al., 2004), e-
commerce protocols (Chen et al., 2004), belief merg-
ing (Qi et al., 2005), news reports (Hunter, 2006), in-
tegrity constraints (Grant and Hunter, 2006), require-
ments engineering (Martinez et al., 2004), databases
(Martinez et al., 2007; Grant and Hunter, 2013), se-
mantic web (Zhou et al., 2009), and multi-agents sys-
tems (Hunter et al., 2014).
In recent years, several logic-based models have
been proposed to evaluate conflicts, including the
maximal η-consistency (Knight, 2002), measures
based on variables or via multi-valued models (Grant,
1978; Oller, 2004; Hunter, 2006; Ma et al., 2010;
Xiao et al., 2010; Ma et al., 2011), n-consistency
and n-probability (Doder et al., 2010), minimal incon-
sistent subsets based inconsistency measures (Hunter
and Konieczny, 2008; Mu et al., 2011; Mu et al.,
2012; Xiao and Ma, 2012), Shapley inconsistency
value (Hunter and Konieczny, 2010), minimal proof
based inconsistency measurement (Jabbour and Rad-
daoui, 2013), inconsistency degree based on parti-
tioning knowledge bases (Jabbour et al., 2014a), and
more recently inconsistency characterization using
prime implicates (Jabbour et al., 2014c; Jabbour et al.,
2014b).
There are different ways to categorize the pro-
posed approaches. One way is to distinguish between
syntax and semantics inconsistency measures. Se-
mantic based approaches aim to compute the propor-
tion of the language that is affected by the incon-
sistency, via for example paraconsistent semantics.
Whilst, syntax based approaches are concerned with
the minimal number of formulae that cause incon-
sistencies, often through minimal inconsistent sub-
sets. Additionally, some basic properties (Hunter and
Konieczny, 2010) such as Monotony (i.e., adding for-
mulae to a knowledge base must not decrease its in-
consistency value), are proposed to evaluate the qual-
ity of inconsistency measures.
However, using these different metrics, the incon-
sistency degree associated to a single formula or to
the whole knowledge base, is evaluated differently. In
other words, on the same set of inconsistent knowl-
edge bases, syntax and semantics based approaches
might lead to opposite conclusions. This observation
suggests that finding a general inconsistency measure
suitable for any knowledge base is clearly a hard and
challenging task. Each measure allows us to derive
some useful information about inconsistency. Usu-
414
Jabbour S., Raddaoui B. and Sais L..
Inconsistency-based Ranking of Knowledge Bases.
DOI: 10.5220/0005210704140419
In Proceedings of the International Conference on Agents and Artificial Intelligence (ICAART-2015), pages 414-419
ISBN: 978-989-758-074-1
Copyright
c
 2015 SCITEPRESS (Science and Technology Publications, Lda.)

ally such information present some complementari-
ties that can be exploited to finely derive a better in-
consistency measure. As a summary, measuring in-
consistency is clearly a multi-criteria based evaluation
process.
Our main goal in the present paper is to exploit
the strength and the complementarities of some pro-
posed measures to better understand and quantify in-
consistency. Our contribution includes a Pareto-based
approach to decide which knowledge bases are domi-
nating given a set of knowledge bases with respect to
inconsistency. Then we discuss how finer ranking be-
tween knowledge bases can be obtained by combining
existing measures.
The remainder of this paper is organized as fol-
lows. Some preliminary definitions and notations
are given in the next section. In the section 3, we
present our approach for comparing different knowl-
edge bases using a well known social welfare mea-
sure, namely Pareto optimality. To our knowledge,
this is the first attempt to evaluate knowledge bases in
terms of its inconsistency degrees. Section 4 presents
a new family of inconsistency measures, before con-
cluding with some perspectives.
2 FORMAL DEFINITIONS
We assume, through this paper, a propositional lan-
guage L built from a finite set of atoms P using classi-
cal logical connectives {¬,∧,∨,→,↔}. We will use
letters such as p and q to denote propositional vari-
ables, and Greek letters like α and β to denote propo-
sitional formulae. The symbols > and ⊥ denote tau-
tology and contradiction, respectively.
A knowledge base K consists of a finite set of
propositional formulae over L. We denote by Var(K)
the set of variables occurring in K. For a set S, |S| de-
notes its cardinality. Moreover, a knowledge base K is
inconsistent if there is a formula α such that K ` α and
K ` ¬α, where ` is the deduction in classical propo-
sitional logic.
If the knowledge base K is inconsistent, then one
can define a minimal inconsistent subset of K as (1)
an inconsistent subset M of K, such that (2) all of its
proper subsets are satisfiable.
Definition 1 (MUS). Let K be a knowledge base and
M ⊆ K. M is a minimal unsatisfiable (inconsistent)
subset (MUS) of K iff:
1. M ` ⊥
2. ∀M
0
( M, M
0
0 ⊥
The set of all minimal unsatisfiable subsets of K is
denoted MUSes(K).
When a MUS is singleton, the single formula in it,
is called a self-contradictory formula.
Let us now, define a dual concept of MUS, called
maximal satisfiable subset (in short MSS).
Definition 2 (MSS). Let K be a knowledge base and
M ⊆ K. M is a maximal satisfiable (or consistent)
subset (MSS) of K iff:
1. M 0 ⊥
2. ∀α ∈ K \ M, M ∪ {α} ` ⊥
For a given inconsistent knowledge base K, and
M ⊂ K an MSS, the subset K\M is usually called a
minimal correction subset (in short MCS). Indeed, an
MCS gives us the minimal subset of the knowledge
base that when removed, we recover the consistency.
There exists a strong relationships between MUSes
and MCSs (Bailey and Stuckey, 2005; Liffiton and
Sakallah, 2008). This relationship can be stated sim-
ply: The set of MUSes of a knowledge base K and the
set of MCSes of K are ”hitting set duals” of one an-
other. Note that any MCS is the complement of some
MSS, and vice versa.
When a knowledge base K is inconsistent the clas-
sical inference relation is trivialized, i.e., any formula
and its negation can be inferred from K. To address
this problem, several techniques have been developed
to analyse inconsistency in terms of minimal incon-
sistent subsets and maximal consistent subsets. In-
deed, an inconsistency measure assigns a nonnegative
number to every knowledge base as its conflicting de-
gree. In the same vein, many logic-based frameworks
of inconsistency measures have been proposed. For
instance, I
MI
measure counts the number of minimal
inconsistent subsets of the knowledge base. Also,
I
C
value counts the sum of the number of maximal
consistent subsets together with the number of self-
contradictory formulae, but 1 must be subtracted to
make the measure equal to zero when the knowledge
base is consistent. Furthermore, the I
P
inconsistency
measure counts the number of formulae in minimal
inconsistent subsets of the knowledge base.
For semantic-based inconsistency measures, one
can cite the inconsistency degrees defined using para-
consistent semantics. The set of truth values for 4-
valued semantics (Arieli and Avron, 1998) contains
four elements: true, false, unknown and both, writ-
ten by t, f ,N,B, respectively. The truth value N al-
lows to express incompleteness of information. The
four truth values together with the ordering  de-
fined below form a lattice FOUR = ({t, f ,B, N},):
f  N  t, f  B  t,N 6 B,B 6 N. The 4-valued
semantics of connectives ∨,∧ are defined according
to the upper and lower bounds of two elements based
on the ordering , respectively, and the operator ¬ is
defined as ¬t = f ,¬ f = t, ¬B = B, and ¬N = N.
Inconsistency-basedRankingofKnowledgeBases
415

A 4-valued interpretation ρ is a 4-model of a
knowledge base K, denoted ρ |=
4
K, if for each for-
mula φ ∈ K, φ
ρ
∈ {t,B}. Let ρ be an interpretation un-
der 4-semantics. Then Conflict(K,ρ) = {p ∈ Var(K) |
p
ρ
= B} is called the conflicting set of ρ with respect
to K. Intuitively, in terms of size-wise minimality, the
larger the size of the conflicting set in 4-models of K,
the more inconsistent K is.
Definition 3 (ID
4
). The 4-semantics based inconsis-
tency degrees are defined as:
ID
4
(K) = min
ρ|=
4
K
|Con f lict(K,ρ)|
|Var(K)|
.
Example 1. Let K = {p,¬p ∨ q,¬q ∨ r,¬r,s ∨ u}.
Consider two 4-valued models ρ
1
, and ρ
2
of K de-
fined as: p
ρ
1
= t, q
ρ
1
= B, r
ρ
1
= f , s
ρ
1
= t, u
ρ
1
= N;
p
ρ
2
= B, q
ρ
2
= B, r
ρ
2
= B, s
ρ
2
= t, u
ρ
2
= N. So
ID
4
(K) = 1/5.
Recently, in (Xiao and Ma, 2012) the authors
prove that ID
4
is closed to the cardinality of min-
imal hitting sets H of the MUSes of K. Formally,
ID
4
(K) =
min
H
{|H| | ∀U ∈ MUSes(K), Var(U) ∩ H 6=
/
0}
|Var(K)|
3 TOWARDS
INCONSISTENCY-BASED
RANKING OF KNOWLEDGE
BASES
There exists many contradictory evaluation of the
inconsistency degree between existing inconsistency
measures (Grant and Hunter, 2011). Indeed, as an ex-
ample, by applying the two measures I
MI
and ID
4
on
the knowledge base K = {¬x
1
,x
1
∧x
2
,x
1
∧x
3
,. .., x
1
∧
x
n
}, we have I
MI
(K) = (n − 1) and ID
4
(K) =
1
n
. For
large values of n, K is considered strongly incon-
sistent according to I
MI
while with ID
4
this base is
viewed as marginally inconsistent.
This example illustrates the problem behind find-
ing a relevant measure to evaluate efficiently the in-
consistency of a knowledge base. The answer to this
issue appears challenging. One can find arguments in
favor of both contradictory conclusions. Indeed, it is
justified to attribute
1
n
as an inconsistency value since
all the conflicts involve a single variable. On the other
hand, there exists (n − 1) minimal conflicting subsets
in K which justify the (n − 1) value as an inconsis-
tency estimation.
Beyond these conflicting conclusions, both mea-
sures are complementary as they show different as-
pects of the inconsistency. More generally, this illus-
trating example also raises the problem of comparing
the inconsistency degree of several knowledge bases.
In other words, how one can identify a relevant or-
dering of different knowledge bases according to their
inconsistency degrees. The disagreement between I
MI
and ID
4
measures tends to suggest that such ordering
does not exist. However, all known measures agree on
the fact that a knowledge base with sparse MUSes hy-
pergraph induces a high inconsistency value. This ob-
servation allows us to claim that given a set of MUSes,
the maximal inconsistency value is obtained when the
MUSes are pairwise disjoint.
In the sequel, we face the problem of finding an
inconsistency ordering over a finite set S
K
of knowl-
edge bases according to a finite set of inconsistency
measures S
I
.
As discussed before, two measures (I
1
,I
2
) ∈ S
I
×
S
I
can lead to conflicting points of view about the
inconsistency of a given knowledge base. Conse-
quently, comparing the inconsistency degree of two
knowledge bases deserve to be further investigated.
This comparison is easier when the different knowl-
edge bases present some structure-based similarities,
for example, when all their associated MUSes are dis-
connected. However, for knowledge bases presenting
different structures, such comparison is clearly prob-
lematic.
Next, we define a proposal to characterize the
knowledge bases from S
K
that dominate others and
propose a combination to reach a better compromise
between them.
Definition 4 (Dominance). Let K
1
and K
2
be two
knowledge bases and I an inconsistency measure. We
say that K
1
dominates K
2
w.r.t. I (in short K
1

I
K
2
)
iff I(K
1
) ≥ I(K
2
).
Let us now introduce the notions of pareto domi-
nance and pareto optimality.
Definition 5 (Pareto Dominance). Let S
K
be a set of
knowledge bases and S
I
a set of inconsistency mea-
sures. A knowledge base K
1
∈ S
K
Pareto Dominates
a knowledge base K
2
∈ S
K
(in short K
1

S
I
K
2
) with
K
1
6= K
2
iff ∀ I ∈ S
I
, K
1

I
K
2
and ∃ J ∈ S
I
, K
1

J
K
2
.
A knowledge base is Pareto optimal if it is not
Pareto dominated by any other knowledge base w.r.t.
a given set of measures. Formally:
Definition 6 (Pareto Optimality). A knowledge base
K
1
∈ S
K
is Pareto optimal (or Pareto efficient) w.r.t.
S
K
and S
I
if there is no other knowledge base K
2
∈
S
K
s.t. K
1

S
I
K
2
.
The figure 1 illustrates the Pareto front, when S
I
=
{I
MI
,ID
4
}. Each dot represents a knowledge base.
ICAART2015-InternationalConferenceonAgentsandArtificialIntelligence
416

Figure 1: Pareto Optimality.
From the above definition, a set of knowledge
bases S
K
might contain several Pareto optimal knowl-
edge bases with respect to a set of measures S
I
.
One way to compute an ordering over the set of
knowledge bases S
K
, consists in iterating the pro-
cess of computing the pareto optimality to build a
sequence P = hS
1
K
,. .., S
n
K
i, where each S
i
K
corre-
sponds to the set of Pareto optimal knowledge bases
of S
K
\ (S
1
K
∪ . .. ∪ S
i−1
K
). This process allows us to
compute a partition P of the set of knowledge bases
S
K
. However, this approach does not allow us to de-
rive a total ordering over all the individual knowledge
base of S
K
, since the elements of S
i
K
are not compa-
rable. This can be seen as a form of stratification of
the set of knowledge bases.
Nevertheless, considering all the measures S
I
at
the same time, does not allow us to derive a total or-
dering of the individual knowledge bases of S
K
. To
allow a finer discrimination between all the individual
knowledge bases and particularly on those belonging
to the same element of the partition S
i
K
, we propose in
the second step to combine several measures to better
discriminate between them.
Indeed, imagine we have a set S
K
= {K
1
,. .., K
m
}
of knowledge bases and two inconsistency measures I
and I
0
. According to I and I
0
, suppose that S
K
can be
ordered as σ(K
1
)  ...  σ(K
m
) and σ
0
(K
1
)  ... 
σ
0
(K
m
) where σ and σ
0
are bijections from S
K
to S
K
.
As we discussed above, the different measures are
usually complementary as each one identifies some
aspects of the inconsistency. An issue to benefit from
such complementarities is to combine such measures
to obtain a more relevant and more discriminating
measure.
Indeed, a new combined measure can be defined
by aggregating the different measures of S
K
. One can
consider the average or their product as an aggregate
function. Such derived measure can be used to further
refine the partition. It can be seen as a way to tie-break
between the knowledge bases belonging to the same
element of the partition P .
4 COMBINING INCONSISTENCY
MEASURES
As shown before, while different inconsistency mea-
sures, based on semantics or syntax, provide an im-
portant way of evaluating inconsistency, they do have
some limitations. Indeed, several measures show dif-
ferent aspects of the inconsistency of an inconsistent
knowledge base.
By taking into account these different aspects, it
is possible to provide more fine grained criteria for
evaluating the conflict of knowledge bases.
Let us explore such possible combination of exist-
ing inconsistency degrees as follows.
Definition 7. Let K be a knowledge base. We define
I
comb1
(K) = |H
min
(K)| × |MUSes(K)|
Definition 7 define a new measure that aggregate
the ones considering either the minimal hitting set and
the one based on number of MUSes.
Proposition 1. I
comb1
is monotonic.
Proof. direct since a knowledge base augmented with
a new formula have a higher minimal hitting set and a
higher number of MuSes.
While fixing the number of MUSes, the knowl-
edge bases are compared to the size of their minimal
hitting set.
The second alternative measure can be defined by
substituting the multiplication operator in Definition
7 with the sum operator leading to Definition 8.
Definition 8. Let K be a knowledge base. We define
I
comb2
(K) =
(|H
min
(K)|+|MUSes(K)|)
2
I
comb2
consider the average between the minimal
hitting set and the number of MUSes.
The last alternative is based only on the hitting
sets. This is stated in Definition 9.
Definition 9. Let K be a knowledge base. We define
I
comb3
(K) =
(|H
min
(K)|+|H
max
(K)|)
2
Let us motivate this last measure. Inconsistency
are often linked to the cost of recovering consistency.
Hitting set of minimum size often represents this cost.
Clearly this vision is optimistic. In contract, the hit-
ting set of maximum size H
max
(K) is the worst case
allowing to recover consistency. I
comb3
allows us
Inconsistency-basedRankingofKnowledgeBases
417

to consider a compromise by summing up this two
bounds and dividing by 2 to consider their average.
Contrary to I
comb2
that combines H
min
(K) and
MUSes(K), I
comb3
considers that new MUSes that do
not involve the implication of new formulas in the in-
consistency have the same inconsistency.
Proposition 2. I
comb1
, I
comb2
, and I
comb3
are mono-
tonic
Proof. Adding new formula to a knowledge base in-
creases both its minimal hitting, its maximal one and
the number of MUSes.
The set of proposed measure can be ranked as fol-
lows:
I
comb3
≤ I
comb2
≤ I
comb1
Note that for all the considered measures and
when the number of MUSes is fixed, knowledges
bases with disjoint MUSes are associated with higher
inconsistency value.
Example 2. Let us reconsider again the knowledge
base K = {¬x
1
,x
1
∧x
2
,x
1
∧x
3
,. .., x
1
∧x
n
}. We have:
I
comb1
(K) = n, I
comb2
(K) =
n+1
2
, and I
comb3
=
n
2
Now in order to have a fair comparison, let
us consider the case of the knowledge base K =
{x
1
,¬x
1
,x
2
,¬x
2
,. .., x
n
,¬x
n
}. We have : I
comb1
(K) =
n
2
, I
comb2
(K) = n, and I
comb3
= n. Indeed, in this par-
ticular case, the size of the hitting set and the number
of MUSes are the same. Suppose now we augmented
K with the following set K
0
= {¬x
1
∨ ¬x
2
,¬x
2
∨
¬x
3
,. .., ¬x
n−1
∨ ¬x
n
}. We have now 2n − 1 MUSes
and the minimal hitting set remains the same i.e.,
breaking the MUSes of K allows in the same time to
break the ones of K
0
.
We have: I
comb1
(K) = (2n − 1) × n, I
comb2
(K) =
3n
2
, and I
comb3
= n
As we can see, I
comb1
, and I
comb2
are more discrim-
inative compared to I
comb3
.
Let us finally consider a case of combined seman-
tic and syntactic measures. To show the intuition be-
hind its introduction, let us consider the following
knowledge base K
1
= {x
1
∨ . .. ∨ x
n
,¬x
1
∧ y
,
.. .¬x
n
∧
y
1
,. .., ¬x
1
∧ y
n
,. .., ¬x
n
∧ y
n
} and K
2
= {x
1
∨ . .. ∨
x
n
,¬x
1
∧ y
,
.. .¬x
n
∧ y
1
,. .., ¬x
1
∧ y
n
,. .., ¬x
n
∧ y
n
}.
K
1
contains n
2
+1 formulas leading to n
n
MUSes.
While n takes higher values, reasoning on MUSes be-
comes harder. Let us notice that this formula contains
(2n + 1) variables. So, clearly from semantic point of
view, conflicts concern the (2n + 1) variables. Then,
combining semantic measures should be done with
syntactic measures not leading to exponential value.
Definition 10. Let K be a knowledge base. We define
I
comb4
(K) =
ID
4
(K) × |Var(K)|
max{|M|, M ∈ MSSes(K)}
I
comb4
(K) considers a high inconsistent knowledge
base as the one maximizing the number of variables
linking the MUSes while minimizing the maximum
MSS. A similar version ofI
comb4
(K) can be rewritten
as
ID
4
(K) × |Var(K)| × |H
max
(K)|.
5 CONCLUSION
Several measures have been proposed to evaluate the
inconsistency degree of a given knowledge base. In
this paper, we identified the difficulty to compare their
relevance or quality. A highly inconsistent knowledge
base with respect to a given measure can be consid-
ered less inconsistent using another one. This issue
induces even more difficulty to compare the conflict
of different knowledge bases. To deal with such prob-
lems, we first proposed a new framework allowing us
to partition a set of knowledge bases as a sequence of
subsets according to a set of inconsistency measures,
where the first element of the partition corresponds
to the most inconsistent ones. We also pointed out
that such partition sequence can be refined thanks to
a combination of existing measures. As future works,
we plan to study what is the adequate aggregation op-
erator and what is the most relevant set S
I
that can be
considered.
REFERENCES
Arieli, O. and Avron, A. (1998). The value of the four val-
ues. Artificial Intelligence, 102:97–141.
Bailey, J. and Stuckey, P. J. (2005). Discovery of mini-
mal unsatisfiable subsets of constraints using hitting
set dualization. In PADL, pages 174–186.
Bertossi, L. E., Hunter, A., and Schaub, T. (2005). Intro-
duction to inconsistency tolerance. In Inconsistency
Tolerance, pages 1–14.
Chen, Q., Zhang, C., and Zhang, S. (2004). A verification
model for electronic transaction protocols. In APWeb,
pages 824–833.
Doder, D., Raskovic, M., Markovic, Z., and Ognjanovic, Z.
(2010). Measures of inconsistency and defaults. Int.
J. Approx. Reasoning, 51(7):832–845.
Grant, J. (1978). Classifications for inconsistent theories.
Notre Dame Journal of Formal Logic, 19(3):435–444.
Grant, J. and Hunter, A. (2006). Measuring inconsistency in
knowledgebases. J. Intell. Inf. Syst., 27(2):159–184.
Grant, J. and Hunter, A. (2011). Measuring consistency
gain and information loss in stepwise inconsistency
resolution. In ECSQARU, pages 362–373.
Grant, J. and Hunter, A. (2013). Distance-based measures
of inconsistency. In ECSQARU, pages 230–241.
ICAART2015-InternationalConferenceonAgentsandArtificialIntelligence
418

Hunter, A. (2006). How to act on inconsistent news: Ignore,
resolve, or reject. Data Knowl. Eng., 57(3):221–239.
Hunter, A. and Konieczny, S. (2008). Measuring inconsis-
tency through minimal inconsistent sets. In KR, pages
358–366.
Hunter, A. and Konieczny, S. (2010). On the measure of
conflicts: Shapley inconsistency values. Artif. Intell.,
174(14):1007–1026.
Hunter, A., Parsons, S., and Wooldridge, M. (2014). Mea-
suring inconsistency in multi-agent systems. Un-
stliche Intelligenz.
Jabbour, S., Ma, Y., and Raddaoui, B. (2014a). Inconsis-
tency measurement thanks to mus decomposition. In
AAMAS, pages 877–884.
Jabbour, S., Ma, Y., Raddaoui, B., and Sa
¨
ıs, L. (2014b). On
the characterization of inconsistency: A prime impli-
cates based framework. In ICTAI, pages 146–153.
Jabbour, S., Ma, Y., Raddaoui, B., and Sa
¨
ıs, L. (2014c).
Prime implicates based inconsistency characteriza-
tion. In ECAI, pages 1037 – 1038.
Jabbour, S. and Raddaoui, B. (2013). Measuring inconsis-
tency through minimal proofs. In ECSQARU, pages
290–301.
Knight, K. (2002). Measuring inconsistency. J. Philosoph-
ical Logic, 31(1):77–98.
Liffiton, M. H. and Sakallah, K. A. (2008). Algorithms
for computing minimal unsatisfiable subsets of con-
straints. J. Autom. Reasoning, 40(1):1–33.
Ma, Y., Qi, G., and Hitzler, P. (2011). Computing inconsis-
tency measure based on paraconsistent semantics. J.
Log. Comput., 21(6):1257–1281.
Ma, Y., Qi, G., Xiao, G., Hitzler, P., and Lin, Z. (2010).
Computational complexity and anytime algorithm for
inconsistency measurement. Int. J. Software and In-
formatics, 4(1):3–21.
Martinez, A. B. B., Arias, J. J. P., and and, A. F. V.
(2004). On measuring levels of inconsistency in multi-
perspective requirements specifications. In PRISE’04,
pages 21–30.
Martinez, M. V., Pugliese, A., Simari, G. I., Subrahmanian,
V. S., and Prade, H. (2007). How dirty is your re-
lational database? an axiomatic approach. In EC-
SQARU, pages 103–114.
McAreavey, K., Liu, W., Miller, P., and Mu, K. (2011).
Measuring inconsistency in a network intrusion detec-
tion rule set based on snort. Int. J. Semantic Comput-
ing, 5(3).
Mu, K., Liu, W., and Jin, Z. (2011). A general framework
for measuring inconsistency through minimal incon-
sistent sets. Knowl. Inf. Syst., 27(1):85–114.
Mu, K., Liu, W., and Jin, Z. (2012). Measuring the blame
of each formula for inconsistent prioritized knowledge
bases. J. Log. Comput., 22(3):481–516.
Oller, C. A. (2004). Measuring coherence using lp-models.
J. Applied Logic, 2(4):451–455.
Qi, G., Liu, W., and Bell, D. A. (2005). Measuring conflict
and agreement between two prioritized belief bases.
In IJCAI, pages 552–557.
Xiao, G., Lin, Z., Ma, Y., and Qi, G. (2010). Computing in-
consistency measurements under multi-valued seman-
tics by partial max-sat solvers. In KR.
Xiao, G. and Ma, Y. (2012). Inconsistency measurement
based on variables in minimal unsatisfiable subsets.
In ECAI, pages 864–869.
Zhou, L., Huang, H., Qi, G., Ma, Y., Huang, Z., and Qu, Y.
(2009). Measuring inconsistency in dl-lite ontologies.
In Web Intelligence, pages 349–356.
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