Knowledge Base Compilation for Inconsistency Measures

Said Jabbour

, Badran Raddaoui

and Lakhdar Sais

CRIL - CNRS, UMR 8188, University of Artois, Lens, France

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

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

, ..., p

}. The poll re-

sult of each group is a set of propositional formulas.

For example, the set {p

∧ ¬p

, p

∨ p

} expresses

that in this group there is one voter who votes p

but

votes against p

, and the other voter supports either p

or p

. Now consider the poll results of two groups:

= {p

∧ p

, ¬p

}, γ

= {p

, ¬p

∨ p

, ¬p

, p

which are both inconsistent. Then, we can use var-

ious metrics to compare γ

and γ

. By ID

value

(Hunter, 2006), γ

contains one unit of inconsistency,

which seems reasonable because the conﬂict is mere

on p

within this group, but ID

treats γ

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

and p

are all in-

volved in at least one subgroup with conﬂicts. In

short, ID

ignores some inconsistencies for γ

, while

MUS

overestimates inconsistency in γ

, never men-

tioning ID

(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

MUS

that can

distinguish γ

and γ

. Compared with ID

and ID

the MUS based measure I

= |MUSes(K)| can dis-

tinguish γ

and γ

. However, as argued in (Mu et al.,

2011b), I

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 Ω if for all S ∈ Ω, H ∩ S 6=

0. A hitting set H

of Ω is irreducible if there is no other hitting set H

′

s.t. H

′

⊂ H. A hitting set H of Ω is called a minimum

hitting set, denoted as HS

min

(Ω), if H does not strictly

include any other hitting set.

That is, taking Ω as set of MUSes involved in a

knowledge base K, the minimum hitting set of Ω 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 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 |=

K, if for

each formula φ ∈ K, φ

∈ {t, B}.

2.1.2 Quasi Classical Semantics (Q-semantics)

For the set of propositional variables A, let A

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

, . . . , C

are clauses and l

, . . . , l

are literals. For I ⊆ A

, the Q-satisﬁability relation

is deﬁned as follows:

I |=

p iff +p ∈ I ;

I |=

¬p iff −p ∈ I ;

I |=

∨ . . . ∨ l

iff [I |=

or . . . or I |=

]

and [ for all i, I |=

¬l

implies

I |=

∨ . . . ∨ l

i−1

∨ l

i+1

∨ . . . ∨ l

];

I |=

, ..., C

} iff 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.











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

value is deﬁned as I

(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

= 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

and ID

measures). Let K be a

knowledge base. The 4- and Q-semantics based in-

consistency degrees are deﬁned as:

(K) = min

I |=

|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

and I

of K deﬁned

as:

= t, q

= B, r

= f, s

= t, u

= N;

= B, q

= B, r

= B, s

= t, u

= N.

Then, we have ID

(K) =

. However, I

is not a Q-

model of K. In fact, I

is a Q-model of K with the least

amount of conﬂicts, thus we have ID

(K) =

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 = {p∧q, ¬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

(K), is deﬁned as follows: PM

(K) = {I | I |=

K, ∀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

(K). We de-

note ConfV

(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

(K) and ConfV

(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

(K) does not require that p ∈ Con flict(K, M )

for all M ∈ PM

(K). It is due to the fact that the

set ∩

M ∈PM

(K)

Conflict(K, M ) can be empty. For in-

stance, {p ∨ q, ¬p, ¬q} has two preferred 4-models

= B, q

= f and p

′

= f, q

′

= 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

(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

(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

′

) = ConfV

′

) = {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

= t, q

= B. Then,

we have ConfV

(K) = {q}. However by Deﬁnition 9,

ConfV(K) = {p, q} holds.

Note that in this example we also have

ConfV(K) = ConfV

(K) = {p, q}. Indeed, ConfV

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

, but it is not the case of ConfV

. Intuitively,

ConfV

gives a cautious estimation of the conﬂicting

variables. However, in the following example, we can

see that ConfV

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

(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

′

, 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

, . . . , α

} be a set of

propositional formulas. Then, we have ConfV(K) =

ConfV({PI(α

), . . . , PI(α

)}).

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

), . . . , PI(α

)}).

Using Proposition 1, the following result holds.

Proposition 2. Let K = {α

, . . . , α

} be a set of

propositional formulas. Then, we have ConfV(K) =

ConfV({PI(α

) ∧ . . . ∧ PI(α

)}).

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

, . . . , β

Let us consider β

′

= {β

′

, . . . , β

′

} such that β

′

⊆ β

(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

′

(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

, . . . , α

} be a set of

propositional formulas. Then, we have ConfV(K) =

Var(MUSes(PI(α

) ∧ . . . ∧ PI(α

))).

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

) ∧ . . . ∧ PI(α

), 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

, . . . , α

}. Then, ConfV(K) = Con fV

′

)

where K

′

= {PI(α

), . . . , PI(α

)}.

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

, . . . , φ

} ⊆ K and D = {α

, . . . , α

} 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

⊢ α

(2) ∀(1 ≤ i ≤ m), PI(α

) ⊆ PI(φ

(3) {α

, . . . , α

} is a MUS,

(4) ∀α ∈ {α

, . . . , α

} 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

, D

i, . . . , hS

, D

i} the set of its deduced

MUSes. Then, H is a hitting set of DMUSes(K) iff

H is a hitting set of {D

, . . . , D

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

) ∧ . . . ∧ (¬p ∨ q

), ¬p ∧ (p ∨

¬q

) ∧ . . . ∧ (p ∨ ¬q

)}. 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

∧ . . . ∧

, ¬p∧¬q

∧. . . ∧¬q

}. 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

, . . . , α

} be a knowledge

base. We have,

min

(DMUSes(K)) 6= HS

min

(DMUSes(α

∧. . . ∧α

))

Proof. Let us consider the two formulas

= p ∧ (¬p ∨ q) and α

= ¬p ∧ (p ∨ ¬q). We

have |HS

min

(DMUSes({α

, α

}))| = 2 while

|HS

min

(DMUSes(α

∧ α

))| = |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

, . . . , α

}) =

Var(MUSes({PI(α

) ∧ . . . ∧ PI(α

)). In similar way

we have the following proposition.

Proposition 5. Let K = {α

, . . . , α

} be a knowledge

base. H is a hitting set of DMUSes({α

, . . . , α

})) iff

H is a hitting set of PI(α

) ∧ . . . ∧ PI(α

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

) ∧ . . . ∧ PI(α

). 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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