Knowledge Base Compilation for Inconsistency Measures
Said Jabbour
1
, Badran Raddaoui
2
and Lakhdar Sais
1
1
CRIL - CNRS, UMR 8188, University of Artois, Lens, France
2
LIAS - ENSMA EA 6315, University of Poitiers, Poitiers, France
Keywords:
Knowledge Representation, Inconsistency Measure, Conflicting Variables, Prime Implicates, Compilation.
Abstract:
Measuring conflicts is recognized as an important issue for handling inconsistencies. Indeed, an inconsistency
measure can be employed to support the knowledge engineer in building a consistent knowledge base or
repairing an inconsistent one. Good measures are supposed to satisfy a set of rational properties. However,
defining sound properties is sometimes problematic. In (Jabbour et al., 2014c), the authors proposed a new
prime implicates based approach to identify the variables involved in the contradiction, and a refinement of the
notion of minimal inconsistent subsets (MUSes) in propositional knowledge bases. In this article, we establish
a bridge between the conflicting variables in knowledge bases and the three valued semantics by compiling
each formula of the base into its prime implicates. We then extend hitting sets for MUSes to hitting sets of
the set of deduced MUSes (DMUSes) based on prime implicates representation. This leads to an interesting
family of inconsistency metrics.
1 INTRODUCTION
Inconsistency is often encountered, especially in case
of multiple-source information. Common to most for-
malisms that have been studied to cope with inconsis-
tency is the idea that some pieces of information are
wrong, and thus responsible for the conflict. In this
view, consistency should be recovered by removing
incorrect pieces of information. This may be done, for
instance by analyzing and quantifying the amount of
contradictions of the set of contradictory information.
Measuring conflicts has gained a considerable atten-
tion in the field of Artificial Intelligence (Bertossi
et al., 2005). It is of particular importance for compar-
ing different knowledge bases by their inconsistency
levels (Grant, 1978). Indeed, it was proved useful
and attractive in diverse scenarios, including software
specifications (Martinez et al., 2004), e-commerce
protocols(Chen et al., 2004), belief merging(Qi et al.,
2005), news reports (Hunter, 2006), integrity con-
straints (Grant and Hunter, 2006), requirements engi-
neering (Martinez et al., 2004), databases (Martinez
et al., 2007; Grant and Hunter, 2013), semantic web
(Zhou et al., 2009), and network intrusion detection
(McAreavey et al., 2011), etc.
A number of logic-based inconsistency measures
have been studied and there are different ways to cat-
egorize them. One way is by their dependence to the
language or formula: the former aims to compute the
proportion of the language affected by inconsistency
(Grant, 1978; Hunter, 2002; Oller, 2004; Hunter,
2006; Grant and Hunter, 2008; Ma et al., 2010; Xiao
et al., 2010a; Ma et al., 2011; Xiao and Ma, 2012;
Jabbour and Raddaoui, 2013). Whilst, the latter is
concerned with the minimal number of formulas that
cause inconsistencies, often through minimal unsatis-
fiable subsets (Hunter and Konieczny, 2008; Mu et al.,
2011a; Mu et al., 2012; Grant and Hunter, 2013; Jab-
bour et al., 2014a). Some other measures are based
on both (Hunter and Konieczny, 2006; Hunter and
Konieczny, 2010). Different metrics can also be clas-
sified by being formula or knowledge base oriented.
For example, the inconsistency measures proposed in
(Hunter and Konieczny, 2006; Hunter and Konieczny,
2010) consist in quantifying the contribution of a for-
mula to the inconsistency of the whole knowledge
base containing it, while the other mentioned mea-
sures aim to quantify the inconsistency degree of the
whole knowledge base. Furthermore, some estab-
lished basic properties (Hunter and Konieczny, 2010)
such as consistency, monotony, free formula indepen-
dence, are proposed to evaluate the quality of incon-
sistency measures.
In this work, we focus on knowledge base oriented
inconsistency measures, from both the language and
the formula aspects. Our aim is to investigate novel
language and formula-based inconsistency measures
while answering the limitations of existing ones by
532
Jabbour, S., Raddaoui, B. and Sais, L.
Knowledge Base Compilation for Inconsistency Measures.
DOI: 10.5220/0005824305320539
In Proceedings of the 8th International Conference on Agents and Artificial Intelligence (ICAART 2016) - Volume 2, pages 532-539
ISBN: 978-989-758-172-4
Copyright
c
2016 by SCITEPRESS Science and Technology Publications, Lda. All rights reserved
satisfying the desired properties.
Inspired by the scenario given in (Xiao and Ma,
2012), suppose that there are n groups of people
polling on a set of policies {p
1
, ..., p
m
}. The poll re-
sult of each group is a set of propositional formulas.
For example, the set {p
1
¬p
2
, p
1
p
3
} expresses
that in this group there is one voter who votes p
1
but
votes against p
2
, and the other voter supports either p
1
or p
3
. Now consider the poll results of two groups:
γ
1
= {p
1
p
2
, ¬p
2
}, γ
2
= {p
1
, ¬p
1
p
2
, ¬p
2
, p
2
},
which are both inconsistent. Then, we can use var-
ious metrics to compare γ
1
and γ
2
. By ID
4
value
(Hunter, 2006), γ
1
contains one unit of inconsistency,
which seems reasonable because the conflict is mere
on p
2
within this group, but ID
4
treats γ
2
equiva-
lently even though there are indeed conflicts within
two subgroups. In contrast, ID
MUS
metric (Xiao and
Ma, 2012) considers that both poll results have two
units of inconsistency because p
1
and p
2
are all in-
volved in at least one subgroup with conflicts. In
short, ID
4
ignores some inconsistencies for γ
2
, while
ID
MUS
overestimates inconsistency in γ
1
, never men-
tioning ID
Q
(Hunter, 2002) that is always equal or
larger than ID
MUS
(Xiao and Ma, 2012). To improve
these language-basedmeasures, we revisit an interest-
ing notion, called conflicting variables, from which
we derive an inconsistency measure ID
c
MUS
that can
distinguish γ
1
and γ
2
. Compared with ID
4
and ID
Q
,
the MUS based measure I
MI
= |MUSes(K)| can dis-
tinguish γ
1
and γ
2
. However, as argued in (Mu et al.,
2011b), I
MI
does not satisfy the dominance property.
In this paper, we show how the compilation of a
given set of formulas into its prime implicates can
be used to characterize the conflicting variables in-
volved in conflicts (Jabbour et al., 2014c). We also
demonstrate that based on the same principle, we can
extend many syntactic based inconsistency measures.
This allows us to provide the user a way to explain
its knowledge since the final formula is compiled.
Furthermore, our approach allows to derive compu-
tational processes of such measures.
The paper is organized as follows: Section 2 pro-
vides basic notions and describes some inconsistency
measures relevant to the present work. In Section 3,
we recall the notion of conflicting variables proposed
in (Jabbour et al., 2014c). In Section 4, we show how
to use the compilation into prime implicates to iden-
tify the set of conflicting variables of a given knowl-
edge base. In Section 5, we demonstrate how the
DMUSes notion can be characterized based on prime
implicates compilation and then propose a hitting set
based approach for measuring inconsistencies.
2 PRELIMINARIES
Through this paper, We are given a propositional lan-
guage L built over a countably infinite set of propo-
sitional symbols P using classical logical connectives
, , , , ↔}. We will use letters such as p and q
to denote propositional variables, Greek letters like α
and β to denote propositional formulas. The symbols
and denote tautology and contradiction, respec-
tively. Sometimes, a propositional formula can be in
conjunctive normal form (CNF), i.e., a conjunction
of clauses, where a clause is a disjunction of literals
and a literal is either a propositional variable (p) or its
negation (¬p). For a set S, |S| denotes its cardinality.
We define a knowledge base K as a finite set
of consistent propositional formulas. We denote by
Var(K) the set of variables occurring in K. In ad-
dition, K is inconsistent if there is a formula α such
that K α and K ¬α, where is the deduction in
classical propositional logic. If K is inconsistent, the
Minimal Unsatisfiable Subset (MUS) of K is defined
as follows:
Definition 1 (MUS). Let K be a knowledge base and
M K. M is a minimal unsatisfiable (inconsistent)
subset (MUS) of K iff M and M
( M, M
0 .
The notion of minimal inconsistent subset is also
defined for CNF formula as stated in the following
definition.
Definition 2. A CNF formula α is minimally unsatis-
fiable (MUS) iff α and for any clause C α, we
have α\ {C} 0 .
Let MUSes(K) be the set of minimal inconsistent
subsets of K. Obviously, an inconsistent knowledge
base K can have multiple minimal inconsistent sub-
sets. A formula α that is not involved in any minimal
inconsistent set of K is called free formula. The set
of free formulas of K is written free(K) = {α |6 M
MUSes(K) s.t. α M}.
A hitting set of a collection of sets is a set inter-
secting every set of this collection. Formally,
Definition 3 (Hitting Set). H is a hitting set of a set
of sets if for all S , H S 6=
/
0. A hitting set H
of is irreducible if there is no other hitting set H
s.t. H
H. A hitting set H of is called a minimum
hitting set, denoted as HS
min
(), if H does not strictly
include any other hitting set.
That is, taking as set of MUSes involved in a
knowledge base K, the minimum hitting set of cap-
tures the minimum set of formulas that have to be re-
moved from K in order to resolve the inconsistency
(Reiter, 1987).
Knowledge Base Compilation for Inconsistency Measures
533
Prime implicates have been proposed as the mini-
mal elements w.r.t. in the set of all the clauses im-
plied by a formula α, formally defined as follows:
Definition 4 (Prime Implicate). A clause C is called
a prime implicate of a formula α if it satisfies the fol-
lowing conditions:
α C holds, and
for every clause C
, if α C
and π
C hold, then
C
C holds.
PI(α) denotes the set of prime implicates of α.
2.1 Paraconsistent Semantics
Different from classical two-valued (true, false) se-
mantics, multi-valued semantics (3-valued, 4-valued,
LPm, and Quasi Classical) are augmented with a third
truth value B to stand for the contradictory informa-
tion, hence able to evaluate inconsistency. Since 3-
valued, 4-valued, and LPm are the same in the context
of measuring inconsistency, but differ from the one
based on Quasi Classical (Xiao et al., 2010b), only 4-
valued and Quasi Classical semantics will be consid-
ered throughout the paper. In the sequel, we overview
some of these semantic based measures.
2.1.1 Four-valued Semantics (4-semantics)
The set of truth values for 4-valued semantics (Arieli
and Avron, 1998) contains four elements: true, false,
unknown (or undefined) and both (or overdefined,
contradictory). We use the symbols t, f, N, B, respec-
tively, for these truth values. The truth value N al-
lows to express incompleteness of information, i.e.
absence of any information about truth or falsity. 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.
Recall, that a 4-valued interpretation I is a 4-
model of a knowledge base K, denoted I |=
4
K, if for
each formula φ K, φ
I
{t, B}.
2.1.2 Quasi Classical Semantics (Q-semantics)
For the set of propositional variables A, let A
±
be
a set of objects defined as A
±
= {+p, p | p A},
where +p is a positive object, and p is a negative
object.
Definition 5 (Q-models (Besnard and Hunter, 1995)).
Suppose p A , C
1
, . . . , C
m
are clauses and l
1
, . . . , l
n
are literals. For I A
±
, the Q-satisfiability relation
|=
Q
is defined as follows:
I |=
Q
p iff +p I ;
I |=
Q
¬p iff p I ;
I |=
Q
l
1
. . . l
n
iff [I |=
Q
l
1
or . . . or I |=
Q
l
n
]
and [ for all i, I |=
Q
¬l
i
implies
I |=
Q
l
1
. . . l
i1
l
i+1
. . . l
n
];
I |=
Q
{C
1
, ..., C
m
} iff I |=
Q
C
i
(1 i m).
Q-semantics can also be regarded as assigning one
of the four truth values {B, t, f, N} to symbols in A in
the following way, which enables a uniform way to
define inconsistency degrees discussed below.
p
I
=
t iff +p I and p 6∈ I ;
f iff +p 6∈ I and p I ;
B iff +p I and p I ;
N iff +p 6∈ I and p 6∈ I .
2.2 Inconsistency Measures for
Knowledge Bases
When a knowledge base is inconsistent the classical
inference relation is trivialized, since one can deduce
every formula of the language from the base. In par-
ticular, for knowledge bases that use classical logic
for knowledge representation, inconsistencies render
the whole knowledge base useless, due to the well-
known principle ex falso quodlibet. Moreover, nor-
mally when given two inconsistent sets of formulas,
they are not trivially equivalent. They do not con-
tain the same information and they do not contain the
same conflicts. To address this problem, numerous
works on analyzing and evaluating inconsistency have
been proposed, and different inconsistency measures
have been studied.
More formally, an inconsistency measure is a
function I that maps a knowledge base K to a non-
negative real number I(K) that quantifies the severity
or amount of the conflict in K. We go on by stating
the classical inconsistency metric that employs the set
of minimal inconsistent subsets in a simple manner
(Hunter and Konieczny, 2010). Minimal inconsistent
sets allow us to circumscribe the minimal sub-parts
of the knowledge base involved in the inconsistency.
Intuitively, this measure associates a greater value for
knowledge base containing more minimal inconsis-
tent sets. In other words, the more minimal inconsis-
tent subsets in the knowledge base K the greater the
inconsistency in K.
Definition 6. Let K be a knowledge base. Then, the
I
MI
value is defined as I
MI
(K) = |MUSes(K)|.
ICAART 2016 - 8th International Conference on Agents and Artificial Intelligence
534
Recently, there is an increasing interest in quan-
tifying inconsistency in inconsistent knowledge bases
through multi-valued semantics. This is because para-
consistent reasoning systems provide a natural start-
ing point for analyzing conflicts. Indeed, paracon-
sistent semantics lead to different proposals for mea-
sures of inconsistency. More formally, let I be an
interpretation under i-semantics (i = 4, Q). Then,
Conflict(K, I ) = {p Var(K) | p
I
= B} is called the
conflicting set of I with respect to the knowledge base
K. Intuitively, in terms of size-wise minimality, the
larger the size of the conflicting set in i-models of K,
the more severe the inconsistency in K.
Definition 7 (ID
4
and ID
Q
measures). Let K be a
knowledge base. The 4- and Q-semantics based in-
consistency degrees are defined as:
ID
i
(K) = min
I |=
i
K
|Conflict(K, I )|
|Var(K)|
where i {4, Q}.
Example 1. Let K = {p, ¬p q, ¬q r, ¬r, s u}.
Consider two 4-valued models I
1
and I
2
of K defined
as:
p
I
1
= t, q
I
1
= B, r
I
1
= f, s
I
1
= t, u
I
1
= N;
p
I
2
= B, q
I
2
= B, r
I
2
= B, s
I
2
= t, u
I
2
= N.
Then, we have ID
4
(K) =
1
5
. However, I
1
is not a Q-
model of K. In fact, I
2
is a Q-model of K with the least
amount of conflicts, thus we have ID
Q
(K) =
3
5
.
In the literature, several logical properties have
been studied to characterize inconsistency measures
(Hunter and Konieczny, 2010; Jabbour et al., 2014a;
Besnard, 2014). In this paper, we particularly focus
on the following four postulates.
Definition 8. For all finite sets K, K
, and all formulas
α, β, an inconsistency measure I should satisfy the
following properties:
(1) Monotony: I(K) I(K K
),
(2) Consistency: I(K) = 0 iff K is consistent,
(3) Free Formula Independence: if α is a free formula
in K {α}, then I(K {α}) = I(K),
(4) Dominance: If α β and α 0 , then I(K
{β}) I(K {α}).
Intuitively, we want I to be a function on knowl-
edge bases that is monotonically increasing with the
inconsistency in the knowledge base. If the knowl-
edge base K is consistent, I(K) shall be minimal. The
free formula independence property states that the set
of formulas not involved in any minimal inconsistent
subset does not influence the inconsistency measure.
Finally, the dominance property states that if a for-
mula is weakened, the knowledge base becomes less
inconsistent.
3 CONFLICTING VARIABLES
In (Jabbour et al., 2014c; Jabbour et al., 2014b), the
authors introduced the notion of conflicting variable
in order to quantify the inconsistency of a knowledge
base by circumscribing the sub-parts of the knowl-
edge that create the contradictions. Indeed, conflict-
ing variables are defined to catch the elements of the
knowledge base that are really involved in conflicts.
This leads to a more fine-grained look on how incon-
sistencies are distributed over the pieces of informa-
tion in knowledge bases.
Definition 9 (Syntax-Based Conflicting Variable).
Let K be a knowledge base and p Var(K). p is a
conflicting variable in K if there exists S K, S
K,
and two sets of formulas D and D
such that the fol-
lowing conditions hold:
(1) |S | = |D|, α D, ! β S s.t. β α and
PI(α) PI(β),
(2) D 0 and D {p} ,
(3) |S
| = |D
|, α D
, ! β S
s.t. β α and
PI(α) PI(β),
(4) D
0 and D
p} ,
(5) D D
is a MUS.
We will denote the set of conflicting variables in-
volved in the knowledge base K by ConfV(K).
Intuitively, a conflicting variable p is a variable
such that both its associated literals are logically en-
tailed by sets of consistent logical consequences of K.
For example, for the knowledge base K = {pq, ¬p},
p is expected to be a conflicting variable, but not
q. A particular attention should be paid on the nec-
essary condition PI(α) PI(φ). To illustrate this,
let us consider the following knowledge base K =
{q r, q ¬r}. By taking S = {q r, q ¬r} and
D = {r ¬q, ¬r}, we have D {q} . Now, if
we consider S
= {q r} and D
= {q}, then we
have D
q} . Without considering the condi-
tion PI(α) PI(φ), q would be considered conflicting
while q does not contribute at all to the contradiction
in K. However, by considering the condition (1) of
Definition 9 (inclusion of prime implicates), it is clear
that D = {r ¬q, ¬r} can not be taken into account.
Consequently, q is not a conflicting variable.
The notion conflicting variables proposed in Def-
inition 9 is based on syntax. In the sequel, we recall
the semantic based version of this notion in order to
capture the set of problematic variables. Let us first
introduce the notion of preferred i-model.
Definition 10. For a knowledge base K and i
{4, Q}, the set of preferred i-models of K, written
PM
i
(K), is defined as follows: PM
i
(K) = {I | I |=
i
K, I
|=
i
K, Conflict(K, I ) Conflict(K, I
)}.
Knowledge Base Compilation for Inconsistency Measures
535
Definition 11 (Semantic-Based Conflicting Variable).
Let K be a knowledge base and p Var(K). p is
called a semantic-based conflicting variable in K if
p Conflict(K, M ) for some M PM
i
(K). We de-
note ConfV
i
(K) the set of conflicting variables under
semantics i, where i { 4, Q}.
The above general definition allows for a range of
possible conflicting variables to be defined in various
ways. Indeed, by considering different paraconsistent
semantics (4-semantics and Q-semantics), we get sev-
eral instantiations of semantics based conflicting vari-
ables (denoted ConfV
4
(K) and ConfV
Q
(K), respec-
tively). Note that the condition for calling a variable
problematic is based on the existence of a preferred
model in which the variable is valued B. That is, p
ConfV
i
(K) does not require that p Con flict(K, M )
for all M PM
i
(K). It is due to the fact that the
set
M PM
i
(K)
Conflict(K, M ) can be empty. For in-
stance, {p q, ¬p, ¬q} has two preferred 4-models
p
I
= B, q
I
= f and p
I
= f, q
I
= B whose conflict-
ing sets are disjoint.
Example 2. Let K = {p, ¬p, p r, ¬p ¬r} be an
inconsistent knowledge base. Then, K has only one
MUS, namely {p, ¬p}, so that ConfV(K) = {p}. But
ConfV
Q
(K) = {p, r}, since both p and r must be as-
signed to the value B to be a Q-model of K. In con-
trast, ConfV
4
(K) = {p} because there is no preferred
4-model of K assigning B to r. Now, if we consider
the knowledge base K
= {p q, ¬p, ¬q}, we have
ConfV(K
) = ConfV
Q
(K
) = ConfV
4
(K
) = {p, q}.
As shown by the previous example, syntax-based
conflicting variables always coincide with the con-
flicting variables under the 4-semantics. However, it
is not true in the general case, as shown by the follow-
ing example.
Example 3. Given K = {p, ¬p q, ¬q, q}, the sin-
gle preferred 4-model I of K is p
I
= t, q
I
= B. Then,
we have ConfV
4
(K) = {q}. However by Definition 9,
ConfV(K) = {p, q} holds.
Note that in this example we also have
ConfV(K) = ConfV
Q
(K) = {p, q}. Indeed, ConfV
Q
and ConfV coincide due to the fact that there are rea-
sonable deductions of both p (p K) and ¬p (p
q, ¬q} ¬p) from the knowledge base K. While the
reduction of p is trivial, we claim that the reduction
of ¬p should not be neglected as under ConfV and
ConfV
Q
, but it is not the case of ConfV
4
. Intuitively,
ConfV
4
gives a cautious estimation of the conflicting
variables. However, in the following example, we can
see that ConfV
Q
estimates more variables as conflict-
ing than ConfV.
Example 4. Consider K = {p, ¬p, p r, ¬p ¬r}.
We have ConfV(K) = {p}, since K is in CNF
and it possesses a single MUS {p, ¬p}. However,
ConfV
Q
(K) = {p, r} since the deduction of ¬r from
{p, ¬p ¬r} and that of r from p, p r} are both
considered. But, the definition of conflicting variables
restricts deductions of literals merely to those formu-
las forming a MUS (see condition (5) of Definition 9).
The previous examples ensure that Definition 9
and Definition 11 characterize differently the notion
of conflicting variable.
For what follows, we will examine more relations
between syntax and semantic based conflicting vari-
ables in the context of measuring inconsistency.
4 KNOWLEDGE BASE
COMPILATION FOR
SEMANTIC CONFLICTING
VARIABLES
In the section, we provide a deeper analysis of the
semantic based characterization of the notion of con-
flicting variables. In particular, in semantic based
applications, it could be useful to show how multi-
valued semantics can be used to capture problematic
variables of an inconsistent knowledge base. As men-
tioned earlier, the conflicting variables using multi-
valued semantics are different from the ones based on
syntax (see Definition 9). Therefore, in the sequel, we
push further our reasoning in order to highlight the re-
lationship between them through the prime implicates
representation.
Firstly, let us recall that problematic variables in
mutli-valued semantics are captured by minimal pre-
ferred models. Here, we present an approach that in-
volves compiling each formula of the knowledge base
to its set of prime implicates, and then generating con-
flicting variables from the new canonical representa-
tion.
Let us start by showing that multi-valued seman-
tics is not prime implicate tolerant. That is, given
two knowledge bases K and K
where K
is composed
by the prime implicates of formulas in K, then K
may have a multi-valued inconsistency measure dif-
ferent from K
. To illustrate this, let us consider the
knowledge base K = {p (¬p q), ¬p (p ¬q)}.
Here, K possesses a unique minimal model such that
Conflict(K, I ) = {p}. Now, by replacing each for-
mula in K by the set of its prime implicates, we obtain
a new equivalent knowledge base K
= {p q, ¬p
¬q}. Clearly, K
contains always a unique model
where Conf lict(K, I
) = {p, q}. Moreover, the set
of conflicting variables is the same for both K and
K
, i.e., ConfV(K) = ConfV(K
) = {p, q}. So, this
ICAART 2016 - 8th International Conference on Agents and Artificial Intelligence
536
example gives an intuition on how to get the link
between multi-valued semantics and conflicting vari-
ables.
Let us recall that for a knowledge base K repre-
sented as a set of clauses, the set of conflicting vari-
ables ConfV corresponds exactly to the variables in-
volved in the set of minimal inconsistent subsets of K,
which is equal to the union of the conflicts of the min-
imal preferred models of K. However, as illustrated in
the previous example, such correspondence ceases to
be true in general, i.e., when K is not in CNF.
In the sequel, we show how to retrieve this rela-
tionship for non clausal knowledge bases.
The following proposition states that the set of
conflicting variables remains unchanged when using
the prime implicates based compilation approach.
Proposition 1. Let K = {α
1
, . . . , α
n
} be a set of
propositional formulas. Then, we have ConfV(K) =
ConfV({PI(α
1
), . . . , PI(α
n
)}).
Proof. Let D and D
be two sets of formulas satis-
fying the requirements of Definition 9. Hence, D
and D
form a MUS and the prime implicates of
each formula in D and D
are subsets of prime im-
plicates of a formula in K. Consequently, by com-
piling each formula in D and D
, we deduce that
ConfV(K) = Con fV({PI(α
1
), . . . , PI(α
n
)}).
Using Proposition 1, the following result holds.
Proposition 2. Let K = {α
1
, . . . , α
n
} be a set of
propositional formulas. Then, we have ConfV(K) =
ConfV({PI(α
1
) . . . PI(α
n
)}).
Proof. As shown by Proposition 1, if D and D
are
two sets satisfying the requirements of Definition 9,
then each formula in D and D
can be considered as
conjunction of a subset of the prime implicates of a
formula in K. We assume that D D
= {β
1
, . . . , β
m
}.
Let us consider β
= {β
1
, . . . , β
m
} such that β
i
β
i
(for all 1 i m) and β
. As β
is inconsistent,
then there exists two literals p and ¬p implied by β
.
Now, assuming that any formula β
′′
obtained by re-
moving any clause (prime implicate) from β
i
(for all
1 i m) leads to consistency, i.e., β
′′
6⊢ , we also
deduce that β
is a MUS.
According to the previous result, we have the fol-
lowing interesting proposition.
Proposition 3. Let K = {α
1
, . . . , α
n
} be a set of
propositional formulas. Then, we have ConfV(K) =
Var(MUSes(PI(α
1
) . . . PI(α
n
))).
Proposition 3 provides a characterization of con-
flicting variables through the compilation of each ini-
tial formula into its set of prime implicates. Such
compilation based approach allows us to consider the
conjunction of the resulting prime implicates formu-
las as a single canonical formula from which con-
flicting variables can be computed. Indeed, using
PI(α
1
) . . . PI(α
n
), we have just to enumerate the
set of minimal inconsistent sets in order to obtain the
set of conflicting variables or to enumerate minimal
preferred models and then consider the union of the
underlying conflicts.
The following theorem states the characterization
of conflicting variables through the compilation of
knowledge bases.
Theorem 1. Let K be a knowledge base such that
K = {α
1
, . . . , α
n
}. Then, ConfV(K) = Con fV
4
(K
)
where K
= {PI(α
1
), . . . , PI(α
n
)}.
Proof. Direct consequence of Proposition 3.
5 HITTING SETS BASED
INCONSISTENCY METRICS
Different inconsistency metrics have been defined in
the light of minimal inconsistent subsets. In this sec-
tion, we explore the notion of hitting set using de-
duced MUSes. Indeed, the notion of deduced MUS
is introduced in (Jabbour et al., 2014c; Jabbour et al.,
2014b) as an original characterization that allows us
to summarize the conflict that arise in knowledge
bases. More precisely, we show how inconsistency
measures based on hitting set of the minimal inconsis-
tent sets (e.g., (Mu, 2015)) can be extended to hitting
sets of deduced MUSes.
For that we need some further notation.
Definition 12 (deduced MUS (Jabbour et al., 2014c)).
Let K be a knowledge base and M = hS, Di such
that S = {φ
1
, . . . , φ
m
} K and D = {α
1
, . . . , α
m
} a
set of formulas (S is called the support of D). M is
a MUS modulo logical deduction of K, denoted as
DMUS(K), if:
(1) (1 i m), φ
i
α
i
,
(2) (1 i m), PI(α
i
) PI(φ
i
),
(3) {α
1
, . . . , α
m
} is a MUS,
(4) α {α
1
, . . . , α
m
} there is no α
such that:
(a) α
is weaker that α (α α
but α
0 α),
(b) (D \ {α}) {α
} is a MUS.
The notion of hitting sets can be extended to the
set of deduced MUSes as follows.
Definition 13. Let K be a knowledge base and
{hS
1
, D
1
i, . . . , hS
m
, D
m
i} the set of its deduced
MUSes. Then, H is a hitting set of DMUSes(K) iff
H is a hitting set of {D
1
, . . . , D
n
}.
Knowledge Base Compilation for Inconsistency Measures
537
The following example shows that the size of the
minimum hitting set obtained according to DMUSes
(Definition 13) can differ from the one obtained with
MUSes.
Example 5. Let us consider the knowledge
base K = {p (¬p q), ¬p (p ¬q)}.
We have |HS
min
(MUSes(K))| = 1 while
|HS
min
(DMUSes(K))| = 2. Indeed, removing a
formula from K allows to make it consistent. How-
ever using the DMUSes of K, we have to remove two
formulas from K as it involves two independent sets.
Generally, and similarly to the difference between
MUSes and DMUSes, the minimum hitting set of
DMUSes can be more larger than the one of MUSes.
To illustrate such case, let us consider the knowledge
base K = {p (¬p q
1
) . . . (¬p q
n
), ¬p (p
¬q
1
) . . . (p ¬q
n
)}. Obviously, the size of the
minimum hitting set of MUSes of K is equal to 1 for
any value of n while for DMUSes this value is equal
to n + 1. Since compiled, K will be {p q
1
. . .
q
n
, ¬p¬q
1
. . . ¬q
n
}. These differences, question
us about the representation model that will be used to
express agents knowledge.
As prime implicates are elementary knowledge
representing each formula, the notion of DMUSes
makes distinction between the sub-parts of each for-
mula concerned by the conflict.
Proposition 4. Let K = {α
1
, . . . , α
n
} be a knowledge
base. We have,
HS
min
(DMUSes(K)) 6= HS
min
(DMUSes(α
1
. . . α
n
))
Proof. Let us consider the two formulas
α
1
= p (¬p q) and α
2
= ¬p (p ¬q). We
have |HS
min
(DMUSes({α
1
, α
2
}))| = 2 while
|HS
min
(DMUSes(α
1
α
2
))| = |HS
min
(DMUSes(p
(¬p q) ¬p (p ¬q))| = 1.
Let us recall an interesting result proved be-
low for conflicting variables Con fV({α
1
, . . . , α
n
}) =
Var(MUSes({PI(α
1
) . . . PI(α
n
)). In similar way
we have the following proposition.
Proposition 5. Let K = {α
1
, . . . , α
n
} be a knowledge
base. H is a hitting set of DMUSes({α
1
, . . . , α
n
})) iff
H is a hitting set of PI(α
1
) . . . PI(α
n
).
The result of Proposition 5 allows to have a mean
to compute the set of hitting sets of DMUSes of a K.
Indeed, by compiling each formula of the knowledge
base into its set of prime implicates, Proposition 5 al-
lows us to reduce the computation of the hitting sets
of DMUSes of K to the computation of the hitting sets
of the underlying prime implicates CNF formula, i.e.,
PI(α
1
) . . . PI(α
n
). This result opens interesting
research avenue for practical computation using well-
known available tools such as maximum satisfiability
(MaxSAT) solvers.
6 CONCLUSION
In this paper, we have proposed a new framework for
characterizing inconsistency. To this end, we have
presented a compilation of a knowledge base that uses
the prime implicates of the formulas of that base for
characterizing conflicting variables. Secondly, based
on the notion of conflicting variables, we have pre-
sented a family of measures aimed at helping the
modeler in evaluating inconsistency in propositional
knowledge bases. Lastly, we have shown how incon-
sistency measures based on hitting set of the minimal
inconsistent sets can be extended to hitting sets of de-
duced MUSes. This last result opens an interesting
direction for practical computation.
This work opens several other new research per-
spectives. First, we plan to analyze the behavior of
existing inconsistency metrics based on MUSes in the
light of deduced MUS. Secondly, based on the pro-
posed inconsistency metrics, we will also work on
developing a direct stepwise resolution of conflict so
that inconsistency can be decreased whilst minimiz-
ing information loss. Finally, a comparative empirical
evaluation of our inconsistency measures is clearly an
interesting perspective.
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