Computing Inconsistency Using Logical Argumentation

Badran Raddaoui

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

Keywords:

Propositional Logic, Argumentation Theory, Measuring Inconsistency.

Abstract:

Measuring the degree of conﬂict of a knowledge base can help us to deal with inconsistencies. Several seman-

tic and syntax based approaches have been proposed separately. In this paper, we use logical argumentation as

a ﬁeld to compute the inconsistency measure for propositional formulae. We show using the complete argu-

mentation tree that our family of measures is able to express ﬁnely the inconsistency of a formula following

their context and allows us to distinguish between formulae. We extend our measure to quantify the degree

of inconsistency of set of formulae and give a general formulation of the inconsistency using some logical

properties.

1 INTRODUCTION

Inconsistencies arise naturally when working with

logic-based knowledge bases; they can come from on-

tology learning, merging of several knowledge bases,

decisions making, multi-agent system, or belief revi-

sion.

The need for handling inconsistencies in knowl-

edge bases has been well recognized in recent years.

Recently, the ﬁeld of inconsistency measurement has

gained some attention for knowledge representation

formalisms. Therefore, reasoning under inconsis-

tency is an important ﬁeld in Computer Science

(Bertossi et al., 2005) and in Artiﬁcial Intelligence in

particular and there are many logic-based proposals

for analysing inconsistent information. Then, inter-

est in quantifying inconsistency for knowledge bases

has grown rapidly in last years. This is because it

has been shown that measuring inconsistency is help-

ful to compare different knowledge bases and evalu-

ate their quality of information. For instance, if given

the opportunity to choose between different knowl-

edge bases, we may try to choose the one that is

least inconsistent. Already, measuring inconsistency

has been seen to be useful and attractive in diverse

applications including e-commerce protocols (Chen

et al., 2004), software speciﬁcations (Martinez et al.,

2004), belief merging (Qi et al., 2005), news reports

(Hunter, 2006), requirements engineering (Hunter

and Konieczny, 2006), integrity constraints (Grant

and Hunter, 2006), databases (Martinez et al., 2007),

ontologies (Zhou et al., 2009), semantic web (Zhou

et al., 2009), network intrusion detection (McAreavey

et al., 2011), and multi-agent systems (Hunter et al.,

2014).

To tackle this problem, a range of logic-based pro-

posals for analyzing and measuring the amount of in-

consistency of knowledge base have been presented

in literature, including the maximal n-consistency

(Knight, 2002), measures based on variables or via

multi-valued models (Grant, 1978; Hunter, 2002;

Oller, 2004; Hunter, 2006; Grant and Hunter, 2008;

Ma et al., 2010; Xiao et al., 2010; Ma et al.,

2011), n-consistency and n-probability (Doder et al.,

2010), measures based on minimal inconsistent sub-

sets (Hunter and Konieczny, 2008; Mu et al., 2011a;

Mu et al., 2012; Xiao and Ma, 2012), the Shapley

inconsistency value (Hunter and Konieczny, 2010),

inconsistency measurement based on minimal proofs

(Jabbour and Raddaoui, 2013), partitioning based in-

consistency measures (Jabbour et al., 2014a), and re-

cently inconsistency characterization using prime im-

plicates (Jabbour et al., 2014c; Jabbour et al., 2014b).

These proposals for measuring inconsistency can

be roughly divided into the two following fundamen-

tally categories. The complete comparison of them is

challenging. The ﬁrst one, called semantic measures,

aims to compute the proportion of language that is af-

fected by the inconsistencies. The measures belong-

ing to this ﬁrst class are often based on some paracon-

sistent semantics because we can still ﬁnd paracon-

sistent models for inconsistent knowledge bases. The

second approach, called syntactic measures, involves

counting the minimal number of formulae which are

164

Raddaoui B..

Computing Inconsistency Using Logical Argumentation.

DOI: 10.5220/0005221301640172

In Proceedings of the International Conference on Agents and Artiﬁcial Intelligence (ICAART-2015), pages 164-172

ISBN: 978-989-758-074-1

 2015 SCITEPRESS (Science and Technology Publications, Lda.)

responsible for the conﬂict. Viewing minimal incon-

sistent subsets as the purest form of inconsistency, it is

natural to derive syntax sensitive inconsistency mea-

sures for a knowledge base from the minimal incon-

sistent subsets of that base. The inconsistency mea-

sures considered in this work are deﬁned in terms of

minimal inconsistent subsets and belong to the second

class.

In this paper, we consider an argumentation-based

framework that uses classical logic as the underlying

formalism, which offers a more reasoned way to com-

pute the degree of inconsistency in knowledge bases.

Argumentation is an important cognitive process for

dealing with conﬂicting information by generating al-

ternative sets of arguments. It has been established as

an Artiﬁcial Intelligence keyword for the last ﬁfteen

years, especially for handling inconsistency in knowl-

edge bases. There are several approaches to formalize

argumentation. Among them is the so-called abstract

argumentation system by Dung (Dung, 1995), which

consists of a set of arguments and a binary relation

between them. A second approach is the deductive

or logic-based argumentation (Besnard and Hunter,

2001). We consider here the latter approach in which

an argument is a pair (support-conclusion) where the

support is a minimal consistent set of formulae that

entails the claim. Logical argumentation theory has

been exploited as a way to support the comparison

and selection of statements. Statements are repre-

sented as arguments and argumentation frameworks

support the reasoning on their acceptability.

The remainder of this paper is structured as fol-

lows: Section 2 introduces the argumentation ap-

proach for classical propositional logic, as proposed

by (Besnard and Hunter, 2001). Then, we recall sev-

eral inconsistency measures based on minimal incon-

sistent subsets and maximal consistent subsets. In

section 3, our new framework for measuring inconsis-

tency based on complete argumentation trees is pre-

sented. Next, we provide a generalization of our ap-

proach to evaluate the inconsistency of a subset of for-

mulae. Then, we study the logical properties of the

proposed measures. Finally, we conclude and give

some perspectives of this work.

2 FORMAL PRELIMINARIES

2.1 Propositional Logic

We assume a propositional language L built from a

ﬁnite set of propositional symbols P under the con-

ventional Boolean operators {¬,∧,∨,→,↔}, as well

as the truth constants ⊤ (for truth) and ⊥ (for falsity).

We will use lower case Roman letters a and b to de-

note propositional variables. We use Greek letters α

and β for propositional formulae and Φ and Ψ for sets

of formulae.

A knowledge base K is a ﬁnite set of propositional

formulae. We further assume a distinguished enumer-

ation for every subset of K, its canonical enumera-

tion. Importantly, it only serves to provide the total

order in which the formulae of any subset of K are to

be conjoined to yield a formula logically equivalent to

this subset. Therefore, no other constraint is imposed

on K, particularly K is not expected to be consistent.

It needs not even be the case that individual formulae

in K are consistent. We let ⊢ denote the classical con-

sequence relation. We write K ⊢ ⊥ to denote that K is

inconsistent.

If K is inconsistent, then one can deﬁne the notion

of minimal inconsistent subset as an unsatisﬁable set

of formulae M in K that is such that any of its subsets

is satisﬁable, i.e.:

Deﬁnition 1. Let K be a knowledge base and M ⊆

K. M is a minimal unsatisﬁable (inconsistent) subset

(MUS) of K iff:

1. M ⊢ ⊥

2. ∀M

′

( M, M

′

0 ⊥

Therefore, the set of all minimal inconsistent

subsets of K, denoted as MUSes(K), is deﬁned as

MUSes(K) = {M ⊆ K | M is a MUS of K}.

A formula α that is not involved in any MUS of

K is called free formula. The class of free formulae

of K is written free(K) = {α | ∄M ∈ MUSes(K), α ∈

M}. When a MUS is singleton, the single formula in

it is called a self-contradictory formula. We denote by

Sel fC(K) the set of self-contradictory formulae in K.

2.2 Logical Argumentation

In the present subsection, we focus on the logical ar-

gumentative model developed by Besnard and Hunter

(Besnard and Hunter, 2001). This framework adopts a

very common intuitive notion of an argument. Essen-

tially, an argument is a set of relevant formulae that

can be used to classically prove some point, together

with that point. Each point is represented by a for-

mula.

Deﬁnition 2. An argument A is a pair hΦ,αi s.t.:

1. Φ ⊆ K

2. Φ 6⊢ ⊥

3. Φ ⊢ α

4. ∀Φ

′

⊂ Φ, Φ

′

6⊢ α

ComputingInconsistencyUsingLogicalArgumentation

165

A is said to be an argument for α. The sets Φ

and α denote the support, i.e. Sup(A) = Φ, and the

conclusion of A, i.e. Conc(A) = α, respectively.

An argument A is said to be an atomic argument

if its support is rooted by only one formula, i.e.,

Sup(A) = α.

Example 1. Let K = {a → b,¬b ∨ c,a,¬a ∨ ¬c,b ∧

d}. In view of K, some arguments are:

h{a,a → b, ¬b∨c},ci

h{a,¬a∨ ¬c},¬ci

h{b∧ d},c → di

h{a,a → b}, a∧ bi

h{¬b∨c},¬(b∧ ¬c)i.

Also, h{ b ∧ d},c → di, and h{¬b∨ c},¬(b∧ ¬c)i

are examples of atomic arguments.

Proposition 1. Let K be a knowledge base and Φ ⊆

K. hΦ,αi is an argument iff Φ ∪ {¬α} is a MUS of

K ∪ {¬α}.

Deﬁnition 3. Let K be a knowledge base. We say

that hΦ,αi is a free argument if and only if Φ ⊆ K \

S∈MUSes(K)

Proposition 2. Let K be a knowledge base. α ∈

free(K) if and only if there exists a free argument

hΦ,βi s.t. Φ ⊆ K and α ∈ Φ.

Arguments are not independent in the sense that

an argument can implicitly contain another. The fol-

lowing deﬁnition introduces a notion of subsumption

among arguments.

Deﬁnition 4. An argument hΦ,αi is more conserva-

tive than an argument hΨ,βi if and only if Φ ⊂ Ψ and

β ⊢ α.

That is if hΦ,αi is an atomic argument, then there

exists no argument hΨ,βi s.t. hΨ,βi is more conser-

vative than hΦ,αi, unless Ψ =

0 and β = ⊤.

Example 2. The argument h{a} , a ∨ bi is more con-

servative than the argument h{a,a → b},a ∧ bi.

It may happen that some arguments directly op-

pose the support of other arguments. This leads to

the notion of attacks, a major component of an ar-

gumentation system. In (Besnard and Hunter, 2001),

Besnard and Hunter capture a relation of attack be-

tween arguments as stated by the following deﬁnition.

Deﬁnition 5. An undercut of an argument hΦ,αi is an

argument hΨ,¬(β

∧ . .. ∧ β

)i s.t. {β

,. ..,β

} ⊆ Φ.

Example 3. Let K = {a,¬a∨ b,c, ¬c ∨ ¬a}. Then,

h{c,¬c ∨ ¬a},¬(a ∧ (¬a ∨ b))i is an undercut for

h{a,¬a ∨ b},bi. A less conservative undercut for

h{a,¬a∨ b},bi is h{c,¬c∨ ¬a},¬ai.

Deﬁnition 6. hΨ,βi is a maximal conservative un-

dercut of an argument hΦ,αi iff hΨ, βi is an under-

cut of hΦ,αi such that no other undercut for hΦ,αi is

strictly more conservative than hΨ,βi.

In other words, hΨ,βi is a maximal conservative

undercut of an argument hΦ, αi iff for all undercuts

hΨ

′

,β

′

i of hΦ,αi, if Ψ

′

⊆ Ψ and β ⊢ β

′

then Ψ ⊆ Ψ

′

and β

′

⊢ β.

The value of the next deﬁnition of canonical un-

dercut is that we only need to take the canonical un-

dercuts into account. These arguments are identiﬁed

by Besnard and Hunter as of relevant focus for gen-

erating counter-arguments. This means that we can

justiﬁably ignore the potentially very large number of

non-canonical undercuts.

Deﬁnition 7. hΨ,¬(β

∧ ··· ∧ β

)i is a canonical un-

dercut of hΦ,αi iff hΨ,¬(β

∧ . .. ∧ β

)i is a maximal

conservative undercut of hΦ,αi and hβ

,. ..,β

i is the

canonical enumeration of Φ.

Now, in order to obtain a structure gathering ar-

guments and counter-arguments for/against a speciﬁc

claim, Besnard and Hunter deﬁne the so-called ar-

gumentation trees that collate such arguments and

counter-arguments.

Deﬁnition 8. An argumentation tree for α is a tree

whose nodes are arguments such that:

1. The root is an argument for α

2. For every node hΨ,βi whose ancestor nodes are

hΨ

,β

i,...., hΨ

,β

i, there exists γ ∈ Ψ such that

for 1 ≤ i ≤ n, γ /∈ Ψ

3. Each child node is a canonical undercut of its par-

ent node.

An argumentation tree aims at capturing the

way counter-arguments can take place as the dis-

pute develops. Condition 2 insists that each counter-

argument involves extra information thereby preclud-

ing cycles. These trees have noticeable properties. As

∆ is a ﬁnite set of formulae, it can be proved (Besnard

and Hunter, 2001) that there are only ﬁnitely several

argumentation trees for α and each of them is ﬁnite.

As several different argumentation trees for a

given formula α can co-exist, the following complete

argumentation tree concept aims to represent them in

a global manner by considering all possible attacks

and consequently all canonical undercuts.

Deﬁnition 9. A complete argumentation tree for α,

denoted as T (α), is an argumentation tree for α such

that the children nodes of a node A consist of all the

canonical undercuts of A that satisfy condition 2 of

Deﬁnition 8.

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Notation. To simplify the notation, from now on,

the conclusion of a canonical undercut is denoted ♦,

obviously, there is no ambiguity as to which formula

it stands for.

Example 4. Let K = {a ∧ b,¬b ∨ ¬c,¬a ∧

¬d, c,¬c,d,¬d ∨ c,e → f}.

The complete argumentation tree for the formula

b is visualised bellow.

h{a∧ b},bi

h{c,¬b∨ ¬c},♦ih{¬a∧¬d},♦i

h{d},♦i h{¬c},♦i

h{¬c,¬d ∨ c},♦ i h{d,¬d ∨ c},♦ i

h{d,¬d ∨ c,¬b∨ ¬c},♦i

Figure 1: Complete argumentation tree for b.

2.3 Inconsistency Measures

In this section, we describe some inconsistency mea-

sures deﬁned through minimal inconsistent subsets

and the properties usually used for their characteri-

zation. We limit our presentation to the most impor-

tant and related syntax based measures to the one pro-

posed in this paper.

Reasoning with minimal inconsistent sets is a

widely studied concept that gives rise to several mea-

sures of inconsistency of an entire knowledge base or

one of these formulae. These measures are mostly

based on two criteria which are the number of MUSes

and their size. In (Hunter, 2004), the authors deﬁne

the degree of inconsistency of a formula as the num-

ber of MUSes containing it. Extended to the entire

knowledge base, this measure has resulted in incon-

sistency measure which is deﬁned to be the number

of MUSes. Furthermore, in (Hunter and Konieczny,

2008) the authors introduced a family of inconsis-

tency measures called MinInc inconsistency values

MIV. For instance, the MIV

measure is a basic

one that assigns 1 if the formula belongs to a MUS

and 0, otherwise. When MIV

value is identical to

MIV

, and which associates a formula α the num-

ber of MUSes to which it belongs. Finally MIV

measure, deﬁned as MIV

(K,α) =

∑

|M|

such that

M ∈ MUSes(K) and α ∈ M, is a generalization of the

framework MIV

, since it takes into account the size

of each MUS containing α.

In contrast with the semantic measures, the ap-

proaches based on minimal inconsistent subsets have

some gaps. Indeed, such syntactic approaches do not

make a distinction in the degree of inconsistency be-

tween two different knowledge bases with exactly the

same size and the same number of minimal inconsis-

tent subsets, what motivated the new measure intro-

duced in (Mu et al., 2011b). This approach combines

both the minimal inconsistent subsets and the max-

imal consistent subsets in order to give an inconsis-

tency degree of the whole knowledge base. Another

approach that combines semantic and syntax based

approaches have been introduced in (Xiao and Ma,

2012). It is based on counting the variables of min-

imal inconsistent subsets and the minimal correction

subsets (Reiter, 1987).

However, while analyzing deeply the divergence

between the different approaches dealing with min-

imal inconsistent subsets, we notice that an impor-

tant point has not been taken into account in these

approaches. More precisely, the correlation between

MUSes have to be taken into account during the eval-

uation of the inconsistency degrees in the knowledge

bases.

In order to give the intuition behind the introduc-

tion of our new measures based on logical argumen-

tation, let us consider ﬁrstly the knowledge base K

such that K = {a, ¬a,a∨ b,¬b, b}. K is inconsistent.

For example, the inconsistency measure MIV

(resp.

MIV

) assigns the value 1 (resp. 1) to the formulae

a, a∨ b and b. However, we would like that a∨ b has

a better value, unlike to a and b, since the formulae

a, a ∨ b and b have the same properties except that

a∨ b belongs to a MUS more connected than a and b.

Consequently, our goal is to use the interactions be-

tween MUSes as a main element in order to evaluate

the conﬂict brought by each formula of the knowledge

base.

One of the most known structure able to capture

the interactions between MUSes is undoubtedly the

argumentation tree. Argumentation tree is a well

known concept widely explored in the context of the

classical logic argumentation. This tree offers a faith-

ful image of the interactions of MUSes and allows

then to reason more ﬁnely about inconsistency, since

the attack relation deﬁnition ordinarily relates attacks

to inconsistency. Note that this is unlike the ori-

ented links between MUSes explored in (Benferhat

and Garcia, 1998) which says that the resolution of

one MUS allows automatically for the resolution of

the other. So, by looking inside MUSes and taking

into account the correlations between them, the pro-

posed analysis of MUSes structure lead to several in-

teresting measures different from the existing ones.

ComputingInconsistencyUsingLogicalArgumentation

167

3 ARGUMENTATION-BASED

INCONSISTENCY

MEASUREMENT

In this section, we discuss how to use logical argu-

mentation to address the problem of measuring the

degree of inconsistency in knowledge bases. Our ap-

proach offers considerable advantage as it actually

supports the use of diverse logics, not just propo-

sitional logic. In other words, our framework can

be naturally extended to other logics where argu-

ments are deﬁned like ﬁrst order logic (Besnard

and Hunter, 2005), conditional logic (Besnard et al.,

2013), modal logic (Raddaoui, 2013), description

logic (Black et al., 2009), resource logic (Besnard

et al., 2012), etc.

Before formalizing our inconsistency measure-

ment framework, we need further notations that will

be useful in the following section. Let T be a com-

plete argumentation tree, nodes(T ) denotes the set of

nodes of T . |T | is the number of nodes (i.e., argu-

ments) of T . Let n ∈ nodes(T ), we denote by H (n)

the height of n, i.e. the number of nodes from the root

to n. We will also use H (T ) to denote the height of

T . Given an argument hΦ, αi, Undercuts(Φ) is the

set of children of hΦ,αi in T . For instance, the set of

undercuts of an atomic argument hα,βi is denoted as

Undercuts(α).

We can now show that the inconsistency of the

knowledge base is rooted by the presence of conﬂict-

ual arguments.

Proposition 3. Let K be a knowledge base. If K is

inconsistent, then there exists at least one complete

argumentation tree T such that |T | > 1.

The following result states the relationship be-

tween minimal inconsistent subsets and attacks be-

tween arguments in the sense that if an argument at-

tacks another, then it must be that the support of the

former is inconsistent with the support of the latter.

Proposition 4. Let T be a complete argumentation

tree in K s.t. |T | ≥ 2. For each argument hΦ,αi ∈ T ,

there exists a MUS M ⊆ K such that M ⊆ Φ∪Ψ where

hΨ,βi ∈ Undercuts(Φ).

As shown by Proposition 4, the complete argu-

mentation tree can gather many minimal inconsistent

subsets of a given knowledge base in the same struc-

ture, and thus it takes the dependencies between these

MUSes into account.

Now, we characterize the notion of a free formula

in the light of the complete argumentation tree as fol-

lows.

Proposition 5. Let K be a knowledge base. Let α be

a formula in K. α is a free formula of K iff for each

complete argumentation tree T such that hα,βi is the

root of T , |T | = 1.

Now we can explore the advantages of consider-

ing argumentation to quantify the degree of conﬂict

in knowledge bases. In the following, we introduce

the degree of inconsistency measure of each formula

belonging to a given knowledge base K. More pre-

cisely, we give in the following a family of degree

of inconsistency measure, denoted as I

ARG

, in terms

of complete argumentation tree. These measures aim

specially to take into account not only the attacks be-

tween arguments and their number but also the quality

of each attack.

In logical argumentation, arguments may attack

each other, which is captured by logical conﬂict. This

requirement reﬂects a fundamental assumption in log-

ical argumentation, namely that the conﬂict among

arguments is related to attack between them. So, the

amount of the contradiction in the arguments can then

be viewed as the number of their attackers. More for-

mally, the following result holds.

Proposition 6. Let K be a knowledge base and hΦ,αi

be an atomic argument. Then, |Undercuts(Φ)| =

|{M | M ∈ MUSes(K), Φ∩ M 6=

0}|.

Proof. We see from Proposition 4, for each undercut

hΨ,βi for hΦ,αi, there exists a MUS M s.t. M ⊆ Ψ∪

Φ. Then, it easy to see that the number of undercuts

of hΦ,αi is equal to the number of MUSes containing

Φ.

Proposition 6 suggests that the existence of under-

cuts of a given argument depends on the set of MUSes

involving its support. So, the evaluation of the contra-

diction of a formula (i.e. support of the argument)

is linked to the set of counter-arguments of the ar-

gument containing this formula. According to this

observation, one can notice that the MIV

measure is

simply the number of canonical undercuts that defeat

the argument hα,βi, i.e., MIV

(α) = |Undercuts(α)|.

Then, we can see that each time a counter-argument

exists, we increase the degree of inconsistency by 1.

However,according to the argumentation tree, the ini-

tial argument can be challenged, as well as counter-

arguments to the initial argument can themselves be

challenged, and so on, recursively. This means that

the amount of conﬂict is splitting among the whole

argumentation tree, telling that the intuition behind

that the conﬂict lie in the counter-arguments, counter

counter-arguments, etc.

Our goal is then to claim that the degree of incon-

sistency of the formula supported the initial argument

must decrease when the counter-arguments of this ar-

gument are themselves be attacked, and so on. This is

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168

the direct formulation of the idea that the more argu-

ments needed to produce the argumentation tree, the

less inconsistency there is in the root. To this end,

it should be natural to take all attacks among argu-

ments in order to evaluate more ﬁnely the inconsis-

tency of the formulae. Starting from this, it is obvious

to see that MIV

measure provides a local evaluation

of the inconsistency since it considers only the neigh-

borhood of α (e.g., only MUSes containing α), and

consequently only the counter-arguments of the ini-

tial argument are taken into account. For instance, the

MIV

measure assigns the same blame to each for-

mula belonging to the same number of interconnected

MUSes. This result shows that the MIV

measure is

not sufﬁciently discriminating for our purposes since

just one level of inconsistency is considered. Note

that the same reasoning can be obtained if we con-

sider the MIV

or the MIV

measures.

In the sequel, we will present different measures

able to consider such aspects by tacking the whole

structure of the argumentation tree into account to

evaluate the responsibility/contribution of each for-

mula in the inconsistency of the knowledge base.

To address this need, let us now introduce a uni-

form deﬁnition of an inconsistency degree under the

complete argumentation tree in order to make a dis-

tinction among the formulae of the knowledge base

according to their participation in the inconsistency.

Deﬁnition 10. Let K be a knowledge base s.t. α ∈ K.

Let hα,βi be an atomic argument. Let T be a com-

plete argumentation tree s.t. hα,βi is the root of T .

The inconsistency degree of α, denoted I

ARG

(α,K), is

deﬁned as:

ARG

(α,K) =







0 if |T | = 1

|Undercuts(α)|

|T |−1

otherwise

ARG

measure assigns as an inconsistency degree

the ratio between the number of counter-arguments

of hα,βi and the size of the argumentation tree, but

1 must be subtracted to not count the root of the

tree. Hence, more no challenged counter-arguments

for the initial argument means higher degree of in-

consistency. This allows us to draw a more precise

picture of the inconsistencies of the formulae in the

knowledge base. Note that assigning the maximum

value of 0 as a the degree of conﬂict of a free for-

mula seems to be very natural, since a free formula

has nothing to do with the conﬂicts of the knowledge

base.

Next, we show that the inconsistency measure de-

ﬁned above satisﬁes the consistency, and free formula

independence properties.

Proposition 7. Let K be a knowledge base and α ∈ K.

The inconsistency measure I

ARG

satisﬁes the follow-

ing properties:

• I

ARG

(α,K) = 0 if K is consistent (consistency)

• I

ARG

(α,K) = 0 iff α ∈ free(K) (free formula in-

dependence)

Note that according to the I

ARG

measure, the in-

consistency can decrease when new formulae are

added in the knowledge base. This is explained by

the fact that the number of MUSes can increase as

well as the number of counter-arguments in the argu-

mentation tree. Hence, the I

ARG

measure is not mono-

tonic.

Example 5. Let us consider the knowledge base of

Example 4. Then, the I

ARG

value gives as result:

ARG

({a ∧ b},K) =

, I

ARG

({¬a ∧ ¬d}, K) =

ARG

({¬b∨¬c},K) =

, I

ARG

({c},K) =

ARG

({d},K) =

, I

ARG

({¬c},K) =

, I

ARG

({e →

f},K) = 0.

The following result suggests that free arguments

are not challenged by any other arguments. This

means that these arguments have nothing to do with

the conﬂicts of the knowledge base.

Proposition 8. Let K be a knowledge base and α ∈ K.

If hα,βi is a free argument, then I

ARG

(α,K) = 0.

It is interesting to note that the argumentation tree

associated to the formula α does not usually involve

all MUSes of the knowledge base but only those in-

terconnected with the MUSes containing α. What

happens is that while such interconnected MUSes do

not contain cycles (e.g. each argument is not dupli-

cated in many branches of the tree), the number of

nodes remains the same for each formula belonging

to these interconnected MUSes. In this case, the in-

consistency measure is only sensitive to the number

of the canonical undercuts of the initial argument. As

consequently, the size and the dependencies between

the set of MUSes the formula α belongs to, can have

an impact on the evaluation of the inconsistency. To

illustrate this, let us consider the following example.

Example 6. Let us consider the knowledge bases K

and K

such that K

= {a,¬a∧b,¬a∧¬b}, and K

{¬a,a,¬a ∨ b,b,¬b}. From this, we obtain the fol-

lowing complete argumentation trees for a∨b. Then,

we have I

ARG

(a,K

) =

while I

ARG

(a,K

) =

The evaluation of the inconsistency value of each

formula given by I

ARG

is still very rough. In partic-

ular, the I

ARG

measure is not yet able to distinguish

ﬁnely between the formulae of the knowledge base.

Indeed, if we consider again the knowledge base K

Example 6, the values of inconsistency of ¬a, ¬a∨b

ComputingInconsistencyUsingLogicalArgumentation

169

h{a},a∨bi

h{¬a∧¬b},♦ih{¬a∧b},♦i

h{¬a∧¬b},♦i h{ ¬a ∧ b},♦i

Figure 2: Complete argumentation tree for a∨ b.

h{a},a∨bi

h{¬a∨b,¬b},♦ih{¬a},♦i

h{b},♦i

Figure 3: Complete argumentation tree for a∨ b.

and ¬b are equal. So it could prove better not to sim-

ply take the number of counter-arguments of the root

of an argument tree, but to take into account the height

of the argumentation tree as well as the distance be-

tween nodes.

To address this need, we see in the following that

by taking a more reﬁned inconsistency measure on

knowledge bases, we get a better assignment to for-

mulae.

Deﬁnition 11. Let K be a knowledge base such that

α ∈ K. Let hα,βi be an atomic argument. Let T be a

complete argumentation tree s.t. hα,βi is the root of

T . The inconsistency degree of α is deﬁned as:

∗

ARG

(α,K) = |Undercuts(α)| × f(T )

where f is a function that takes as input the complete

argumentation tree T .

The above deﬁnition is a general deﬁnition that al-

lows for a range of possible measures to be proposed.

Note that instances of I

∗

ARG

depend on the choice of

function f.

Next we will introduce three types of functions as

follows:

(α) =

H (T )

(α) =

∑

n∈nodes(T )

H (n)

(α) =

∑

n∈nodes(T )

H (n)

The differently deﬁned functions lead to differ-

ent inconsistency measures. Let us explain the re-

sulting measures as follows: Firstly, I

ARG

(α,K) =

|Undercuts(α)| × f

(α) takes into account the height

of the complete argumentation tree associated to α.

secondly, I

ARG

(α,K) = |Undercuts(α)| × f

(α) con-

siders the ratio between the counter-arguments of the

initial argument and the sum of all other arguments

of the complete argumentation tree where each one

is represented by its distance to the root node of the

tree. Finally I

ARG

(α,K) = |Undercuts(α)| × f

(α)

computes the weighted sum of each node of the tree,

where the weight is the inverse of the hight of the cor-

responding node.

Although, the three above inconsistency measures

are quite different, they allow to analyse more deeply

the structure of the complete argumentation tree by

taking into account the dependencies between MUSes

in the knowledge base. This allows us to deﬁne a

much more precise view of the inconsistency, as il-

lustrated in the following example.

Example 7. Let us consider the knowledge base K =

{¬a,a,¬a ∨ b, b,¬b}. Then, we have:

• I

ARG

(a) = 1, I

ARG

(¬a∨ b) =

ARG

(¬a) =

• I

ARG

(a) =

ARG

(¬a∨ b) =

ARG

(¬a) =

• I

ARG

(a) = 5, I

ARG

(¬a∨ b) = 2,I

ARG

(¬a) =

Interestingly, we note that by using the family I

∗

ARG

of inconsistency measures we have the following re-

lation: I

∗

ARG

(¬a) = I

∗

ARG

(b) < I

∗

ARG

(¬a∨ b) < I

∗

ARG

(a) =

∗

ARG

(¬b). We can notice that now we can make a dis-

tinction between the formulae ¬a, ¬a∨ b, and ¬b.

3.1 Logical analysis

As seen earlier, the I

ARG

measure combines the car-

dinality of the set of canonical undercuts, and the

inverse of the height of the complete argumentation

tree in order to quantify the participation of each for-

mula in the inconsistencies. Note that adding new

formulae to a knowledge base may grow the height

of the argumentation tree (by adding new counter-

arguments in the tree), and consequently the I

ARG

value decreases. This means that I

ARG

is not mono-

tonic. To illustrate, let us consider the knowledgebase

K = {a,¬a∧b,¬b∧c,¬c∧d, ¬d∧e}. Then, we have

ARG

(a) =

. Now if we add the new formula ¬e to

K, then the degree of inconsistency of a in K ∪ {¬e}

becomes

In contrast, I

ARG

value counts the inverse of all

distances from the root node to each argument in

the tree. Moreover, adding new formulae cannot de-

crease the number of counter-arguments, and cannot

increase the distance of existing MUSes from the root

node, thus the inverse of the distances will be non-

decreasing. Consequently, the I

ARG

measure is mono-

ICAART2015-InternationalConferenceonAgentsandArtificialIntelligence

170

tonic. By the same reasoning, I

ARG

is a monotonic

measure.

3.2 Quantifying the Conﬂict of a Set of

Formulae

In this section, we consider another inconsistency

measure which aims to evaluate the amount of con-

ﬂict of a set of formulae. To do this, the I

ARG

mea-

sure can be naturally extended to a consistent subset

of formulae, by just taking this subset as a support of

the root argument in the complete argumentation tree.

Now, the inconsistency measure I

ARG

can be de-

ﬁned as follows:

Deﬁnition 12. Let K be a knowledge base and S a

consistent subset of K. Let T be a complete argumen-

tation tree s.t. hS ,αi is the root of T . The degree of

inconsistency of S is deﬁned as:

ARG

(S , K) =







0 if |T | = 1

|Undercuts(S )|

|T |−1

otherwise

This deﬁnition allows us to deﬁne to what extent a

subset of formulae inside a formula is concerned with

the inconsistencies of the knowledge base.

Note that instances of I

∗

ARG

measure can be obvi-

ously extended to evaluate the inconsistency of a set

of formulae by just considering a consistent subset of

the knowledge base as a support of the root argument

of the complete argumentation tree.

Example 8. Let the knowledge base K = {a ∧

b,c,a → ¬b ∨ c, a → ¬b,d,¬d, ¬c,d → a}. Then

ARG

({a∧ b,a → ¬b∨c},K) =

Now, we can show that for a particular case of in-

terconnected MUSes, the following result holds.

Proposition 9. Let K be a knowledge base and S ⊆ K

s.t. S 0 ⊥. If there exists no chain of interconnected

MUSes of a length greater then 2, then the maxi-

mum value reached by I

ARG

(S , K) is equal to 1 and

ARG

(S , K) is equal to |MUSes(K)| − |sel fC(K)|.

4 CONCLUSION

In this paper, we have presented a new framework for

deﬁning inconsistency values that allow to associate

each formula with its degree of contribution for the

conﬂict of the whole base. Our approach is based

on logical argumentation which is shown to be a use-

ful approach to take the interaction between MUSes

into account. We have also shown that such a frame-

work can be extend to quantify the degree of conﬂict

of a consistent subset of formulae. We also proposed

some logical properties to characterize our inconsis-

tency measures.

In future work, we plan to investigate the com-

putational complexity of I

ARG

family of inconsistency

measures, and develop algorithms and implementa-

tions, possibly based on techniques of the computa-

tion of arguments (Besnard et al., 2010). Addition-

ally, we will study how our inconsistency measures

could be used to direct step-wise resolution of incon-

sistency. Finally, we plan to undertake case studies

of applications of our framework of inconsistency de-

grees.

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