How to Decrease and Resolve Inconsistency of a Knowledge Base?
Dragan Doder
1
and Srdjan Vesic
2
1
Computer Science and Communication, University of Luxembourg, Luxembourg, Luxembourg
2
CRIL, CNRS – Univ. Artois, Lens, France
Keywords:
Decreasing Inconsistency, Measuring Inconsistency, Inconsistency Resolution.
Abstract:
This paper studies different techniques for measuring and decreasing inconsistency of a knowledge base. We
define an operation that allows to decrease inconsistency of a knowledge base while losing a minimal amount
of information. We also propose two different ways to compare knowledge bases. The first is a partial order
that we define on the set of knowledge bases. We study this relation and identify its link with a particular class
of inconsistency measures. We also study the links between the partial order we introduce and information
measures. The second way we propose to compare knowledge bases is to define a class of metrics that give
us a distance between knowledge bases. They are based on symmetric set difference of models of pairs of
formulae from the two sets in question.
1 INTRODUCTION
Reasoning under inconsistency is one of the key-
words of Artificial Intelligence. Some authors pro-
pose to get rid of inconsistency, some to reason with
it. Convincing arguments can be constructed even
to support the fact that inconsistency is not always
a bad thing (Gabbay and Hunter, 1991; Gabbay and
Hunter, 1993). The question how to measure incon-
sistency of a knowledge base has attracted attention
of researchers in recent years (Knight, 2002; Hunter
and Konieczny, 2005; Hunter and Konieczny, 2008;
Hunter and Konieczny, 2010; Mu et al., 2011; Grant
and Hunter, 2011b; Grant and Hunter, 2013). In some
papers, the measure of inconsistency depends on the
proportion of the language that is affected by the in-
consistency in a theory (Konieczny et al., 2003; Grant
and Hunter, 2006; Hunter and Konieczny, 2010). An-
other approach consists in considering the number of
formulae needed to produce a contradiction (Knight,
2002; Doder et al., 2010). According to Sorensen,
an inconsistent set S is better than an inconsistent
set S
0
, if the shortest derivation of a contradiction re-
quires more members of S than S
0
(Sorensen, 1988).
A related question is how to decrease its inconsis-
tency while keeping as much information as possi-
ble (Hunter and Konieczny, 2005; Grant and Hunter,
2011a; Grant and Hunter, 2011b). For example,
changing Σ = {x ∧z,¬x} to Σ
0
= {x ∧z,¬x ∨y} al-
lows us to obtain a consistent knowledge base. Grant
and Hunter propose three operations in order to de-
crease inconsistency of a knowledge base: deletion,
weakening and splitting (Grant and Hunter, 2011b;
Grant and Hunter, 2011a). Deleting a formula is the
most drastic operation, but it allows to easily get rid
of the conflicts. Weakening consists in changing a
formula by another formula logically implied by it.
Splitting a formula into its conjuncts may isolate the
really problematic conjuncts. It also allows at a later
step to delete just the portion of the conjunction in-
volved in the inconsistency.
In this paper, we aim at studying different ways to
compare knowledge bases in terms of inconsistency
and at proposing techniques for decreasing inconsis-
tency. We use just one operation to decrease the in-
consistency of the knowledge base: minimal weaken-
ing. More particularly, we are guided by the following
research objectives:
1. Define a partial order  on the set of knowledge
bases based on weakening. Identify the neces-
sary and sufficient conditions that an inconsis-
tency measure must satisfy so that Σ Σ
0
implies
that Σ
0
is less inconsistent than Σ. Study the link
between our partial order and measures of infor-
mation: does Σ  Σ
0
imply that Σ has more infor-
mation than Σ
0
?
2. Since  is a partial order, explore the possibilities
of defining a distance between knowledge bases Σ
and Σ
0
. Is it possible to do this by considering dis-
27
Doder D. and Vesic S..
How to Decrease and Resolve Inconsistency of a Knowledge Base?.
DOI: 10.5220/0005176500270037
In Proceedings of the International Conference on Agents and Artificial Intelligence (ICAART-2015), pages 27-37
ISBN: 978-989-758-074-1
Copyright
c
 2015 SCITEPRESS (Science and Technology Publications, Lda.)

tances based on symmetric set difference of mod-
els of pairs of formulae (ϕ,ϕ
0
) where ϕ ∈ Σ and
ϕ
0
∈ Σ
0
? Can we prove that such a function is a
metric? Can we use this function to measure the
distance of a knowledge base to the closest con-
sistent knowledge base? What is the link between
such a function and the partial order ?
We study the above research objectives and pro-
vide answers to those questions. The rest of the paper
is organised as follows: the next section introduces
the setting, Section 3 studies and compares some in-
consistency measures from the literature. We con-
tinue by defining a partial order on the set of knowl-
edge bases and studying its properties in Section 4.
In Section 5, we apply our minimal weakening to
stepwise inconsistency resolution and analyse the ob-
tained result. Section 6 is devoted to defining a family
of metrics to measure distance between two knowl-
edge bases. The last section concludes.
2 FORMAL SETTING AND
NOTATION
We suppose a finite set Var of propositional letters is
used. We denote by L the set of well-formed clas-
sical propositional logic formulae made from Var,
` stands for classical entailment, and ≡ for logical
equivalence. For two sets S and S
0
, we write S ≡ S
0
if
and only if for every ϕ ∈S
0
, S `ϕ and for every ϕ ∈S,
S
0
` ϕ. We say that a set S is complete if and only if S
has exactly one model.
Our goal is to measure and decrease inconsistency
of a finite knowledge base. In doing so, we consider
changing its formulae. Our ambition in this paper is
to compare the inconsistency of knowledge bases ob-
tained in this process. Note that it may happen that
a knowledge base Σ = {x ∨y, x,¬x} is weakened to
Σ
0
= {x∨y,x∨y, ¬x}; observe that Σ
0
contains the for-
mula x ∨y twice. We do not want to lose this infor-
mation as we want to compare the “evolution” of Σ
towards a consistent knowledge base. This is why, in
this paper, we suppose that Σ, Σ
0
, Σ
00
... are multi-sets
rather than sets. So when we write {x ∨y, x ∨y,¬x}
we consider a multi-set where x ∨y appears two times
and ¬x appears once. To simplify the presentation, in
the rest of the paper, we sometimes use the word set
instead of multi-set when there is no danger of confu-
sion. For a multi-set S, we denote by set(S) the set of
formulae that appear at least once in S. We draw the
reader’s attention to the following detail. Through-
out the paper, we study different functions that are
defined on sets. If not stated otherwise, we suppose
that applying such a function to a multi-set yields (by
convention) a result that would be obtained by apply-
ing a given function on a set obtained from the given
multi-set by deleting duplicates. Formally, given a
function f defined on sets, if S is a multi-set, we de-
fine f (S) = f (set(S)).
We suppose that a knowledge base is a finite
multi-set of classical propositional logic formulae.
For a (multi-)set of formulae S, we denote by P (S)
the set of all propositional variables appearing in S.
For example, P ({x ∧y,x ∨¬(y → ¬z)}) = {x,y,z}.
We use the notation MinConf(Σ) for the set of
minimal (w.r.t. set inclusion) inconsistent subsets of
set(Σ). A formula ϕ is called a free formula of a
knowledge base Σ if and only if ϕ does not belong to
any minimal (w.r.t. set inclusion) inconsistent subset
of set(Σ).
3 INCONSISTENCY MEASURES
This section formally defines the notion of an incon-
sistency measure and presents related work. We start
by briefly recalling the definition of an inconsistency
measure (Grant and Hunter, 2011b).
Definition 1 (Inconsistency measure). An inconsis-
tency measure Inc is a function that for every finite
set of formulae returns a non-negative real number
and satisfies the following properties for all finite sets
Σ,Σ
0
⊆ L and all formulae ϕ, ψ ∈L:
• Inc(Σ) = 0 iff Σ is a consistent set (consistency)
• Inc(Σ ∪Σ
0
) ≥ Inc(Σ) (monotony)
• If ϕ is a free formula of Σ ∪{ϕ}, then
Inc(Σ ∪{ϕ}) = Inc(Σ) (free formula
independence)
An inconsistency measure gives a number indicat-
ing how conflicting a knowledge base is. For exam-
ple, the function Inc
MI
(Hunter and Konieczny, 2010)
is defined as the number of minimal inconsistent sub-
sets of Σ. (Recall that for a set X, |X | stands for its
cardinality.)
Definition 2 (MI inconsistency measure).
Inc
MI
(Σ) = | MinConf(Σ) |
Example 1. Let Σ = {ϕ,¬ϕ,ϕ → ψ,¬ψ,ω}. Then,
MinConf(Σ) = {C
1
,C
2
}, with C
1
= {ϕ, ¬ϕ} and C
2
=
{ϕ,ϕ → ψ,¬ψ}. Thus, Inc
MI
(Σ) = 2.
Another property that can be satisfied by an
inconsistency measure is dominance (Hunter and
Konieczny, 2010). The basic idea behind it is that if
a formula is weakened, the knowledge base becomes
less inconsistent.
ICAART2015-InternationalConferenceonAgentsandArtificialIntelligence
28

Definition 3 (Dominance). An inconsistency measure
Inc satisfies dominance if for all finite sets Σ ⊆L and
all formulae ϕ,ψ ∈ L such that ϕ ` ψ and ϕ 6` ⊥ we
have that Inc(Σ ∪{ϕ}) ≥ Inc(Σ ∪{ψ}).
Mu et al. claim that dominance is not appropri-
ate for characterizing variable-based measures for the
degree of inconsistency (Mu et al., 2011). They ar-
gue that {x,¬x} might be seen as more inconsis-
tent than {x ∧y, ¬x}, since the latter set contains a
non-contradictory information that y holds. However,
dominance implies the contrary. They also show that
Inc
MI
does not satisfy dominance.
An optional but not essential property is normali-
sation (Hunter and Konieczny, 2008). It says for every
knowledge base Σ we have 0 ≤Inc(Σ) ≤ 1.
4 DEFINING A PARTIAL ORDER
ON KNOWLEDGE BASES
Our goal is to decrease inconsistency of a knowledge
base Σ by changing it (possibly several times). To be
able to compare the knowledge bases Σ
0
, Σ
00
obtained
from Σ in this process, we define the relation  which
compares two multi-sets of the same cardinality.
Definition 4. Let Σ and Σ
0
be two knowledge bases.
We say that Σ Σ
0
if and only if Σ and Σ
0
can be writ-
ten as Σ = {ϕ
1
,...,ϕ
n
} and Σ
0
= {ϕ
0
1
,...,ϕ
0
n
} such
that for every i, it holds that `ϕ
i
→ ϕ
0
i
.
The reader can easily note that  is reflexive and
transitive. We also write Σ  Σ
0
, if Σ  Σ
0
and not
Σ
0
 Σ. Relation  corresponds to (one or several)
weakening(s).
For some results
1
, mostly in Sections 4 and 5, we
use the notion of complete conjunctive normal form
(Whitesitt, 1995). This allows us to identify minimal
changes that one can apply to a knowledge base, as
will be shown later. For a formula ϕ, we define its
complete conjunctive normal form (CCNF) denoted
CCNF(ϕ) as the formula ϕ
0
such that ϕ
0
≡ ϕ is in
conjunctive normal form (CNF) and it has the form
ϕ
0
= α
1
∧. .. ∧ α
n
where every α
i
is a disjunction
of literals and for every α
i
we have P ({α
i
}) = Var.
We call α
1
,...,α
n
the co-atoms of ϕ and we write
Coatoms(ϕ) = {α
1
,...,α
n
}
2
. For example if Var =
1
to be completely precise, for Definition 5, Propositions
1, 2, 3, 4, 5, 8, 9, 10 and 13, Lemma 2 and Corollary 1
2
The term “atom” comes from Boolean algebras where
it denotes an element whose intersection with any other el-
ement gives 0 or itself. In the Lindenbaum algebra of for-
mulae over Var, considered as a Boolean algebra, formulae
with a unique model are exactly the atoms, and the formu-
lae α
i
defined above are their complements, hereof called
{x,y,z} and ϕ = ¬x ∨y then CCNF(ϕ) = (¬x ∨y ∨z) ∧
(¬x ∨y ∨¬z) and Coatoms(ϕ) = {¬x ∨y ∨z,¬x ∨y ∨
¬z}.
For a set S, we define Coatoms(S) =
S
ϕ∈S
Coatoms(ϕ), i.e. Coatoms(S) is the union
of all co-atoms of all formulae in S. Note that both
functions CCNF and Coatoms depend on Var, but we
do not write CCNF
Var
nor Coatoms
Var
since Var is
fixed, thus there is no danger of confusion.
By convention, we define CCNF(ϕ) = > if and
only if ϕ is a tautology. We do not distinguish be-
tween logically equivalent formulae. For example, we
consider (x ∨y) ∧(x ∨¬y) and (x ∨¬y) ∧(y ∨x) to be
the same formula.
The idea of using CCNF is to make all proposi-
tional variables explicit in order to allow for some
basic operations on formulae we define later to be
executed more easily. Of course, most of the time
users might see using CCNF too restrictive. Note,
however, that this question is not in the focus of our
paper. Every formula can be represented in CCNF,
thus we can allow users to insert formulae in any form
in the knowledge base and suppose that they will be
automatically translated into CCNF. There are also
concerns about computational complexity regarding
translating formulae into CCNF and back (for exam-
ple for the purpose of shortening them before display-
ing them to a user). While we are aware that this ques-
tion is of great interest, it is out of the scope of the
present paper.
Note that  is a partial order if the formulae are in
CCNF.
Proposition 1. Relation  is a partial order, i.e. a
reflexive, antisymmetric and transitive relation if all
the formulae are in CCNF.
Proof. It is easy to see that  is reflexive and transi-
tive. Let us show that it is antisymmetric. Suppose
that Σ  Σ
0
and Σ
0
 Σ. Then Σ and Σ
0
can be written
as Σ = {ϕ
1
,...,ϕ
n
} and Σ
0
= {ϕ
0
1
,...,ϕ
0
n
} such that
` ϕ
i
→ ϕ
0
i
, for every i ∈ {1,2,. .., n} (1)
and there exist a permutation π of the set {1,2,. ..,n}
such that
` ϕ
0
i
→ ϕ
π(i)
, for every i ∈ {1,2,. .., n}. (2)
Let i ∈{1,2,. ..,n}. From (1), we conclude `ϕ
π(i)
→
ϕ
0
π(i)
. From (2) we obtain ` ϕ
0
π(i)
→ ϕ
π
2
(i)
. By tran-
sitivity of implication, we have that ` ϕ
0
i
→ ϕ
π
2
(i)
.
Repeating the same argument, we can prove ` ϕ
0
i
→
ϕ
π
k
(i)
, for every positive integer k. Note that there is `
such that π
`
is identity on {1,2,. .., n}, so
` ϕ
0
i
→ ϕ
i
, for every i ∈ {1,2,. .., n}. (3)
co-atoms.
HowtoDecreaseandResolveInconsistencyofaKnowledgeBase?
29

From (1) and (3) we have ` ϕ
0
i
↔ ϕ
i
for each i. If
the formulas are in CCNF, then ϕ = ϕ
0
for each i, and,
consequently, Σ = Σ
0
, so the relation is antisymmet-
ric.
We now present a useful characterization of con-
sistency for knowledge bases in CCNF. Namely, a
knowledge base Σ is consistent if and only if there
exists a co-atom that is absent from all formulae in Σ.
Proposition 2. Let Σ be a knowledge base in CCNF.
Σ is consistent if and only if there exists a co-atom α
i
(i.e. a disjunction of literals such that P (α
i
) = Var)
such that there exists no ϕ ∈Σ with α
i
∈Coatoms(ϕ).
Proof. We first show that if every co-atom appears
in at least one formula then Σ ` ⊥. Suppose that
for every co-atom α, there exists ϕ ∈ Σ such that
α ∈ Coatoms(ϕ). Since every co-atom of Σ is in at
least one formula of Σ, then Σ ` Coatoms(Σ). Ob-
serve that for every valuation, at least one co-atom α
j
is false. Thus, Coatoms(Σ) ` ⊥. Consequently, Σ is
inconsistent.
We now show that if there exists a co-atom α
i
that is absent from all the formulae of Σ, we can con-
struct a model of Σ. Define a valuation v
α
i
: Var −→
{true,false} in the following manner: for every
x
i
∈Var, let v
α
i
(x
i
) = true if and only if ¬x
i
appears
in α
i
. (We have thus that for every x
i
, v
α
i
(x
i
) = false
if and only if x
i
appears in α
i
.) Let us show that all
the formulae of Σ are true in this valuation. Let ϕ ∈ Σ
with ϕ = α
i
1
∧... ∧α
i
m
. Let us show that every co-
atom of ϕ is true in valuation v
α
i
. Consider α
i
j
from
ϕ. We have α
i
6= α
i
j
; thus there exists a variable x
k
such that either (x
k
appears in α
i
j
and ¬x
k
appears in
α
i
) or (¬x
k
appears in α
i
j
and x
k
appears in α
i
). In
both cases, α
i
j
is true. Since this holds for all co-
atoms of all formulae, Σ is consistent.
Using the notion of co-atom allows to define the
minimal operations for decreasing inconsistency of a
formula or a (multi)-set of formulae. The first opera-
tion is deleting a co-atom from a formula, the second
one consists in deleting a co-atom from all formulae
of a set, and the third specifies the notation for delet-
ing a co-atom from a formula of a set without chang-
ing other formulae in that set.
Definition 5. Let ϕ be a formula in CCNF. We define
ϕ
del(α
i
)
=
^
α
k
∈Coatoms(ϕ)\{α
i
}
α
k
.
For a (multi-)set S of formulae in CCNF, we define
S
del(α
i
)
= {ϕ
del(α
i
)
| ϕ ∈ S}.
S(ϕ,del(α
i
)) = S \{ϕ}∪{ϕ
del(α
i
)
}.
S can be a set or a multi-set; in both cases, S
del(α
i
)
is obtained from S by deleting α
i
from all the formulae
of S. In case when S is a set, S(ϕ,del(α
i
)) is also a
set. In case when S is a multi-set, S(ϕ,del(α
i
)) is also
a multi-set (where α
i
was deleted from all occurrences
of formula ϕ).
Recall that, in CCNF, we identify the empty for-
mula with the tautology.
Example 2. Let ϕ
1
= (¬x ∨y ∨z) ∧(¬x ∨y ∨¬z) and
ϕ
2
= (¬x ∨y ∨z) ∧(x ∨y ∨z). Denote S = {ϕ
1
,ϕ
2
}
and α
1
= (¬x ∨y ∨z). Then, ϕ
del(α
1
)
= ¬x ∨y ∨¬z,
S
del(α
1
)
= {¬x∨y∨¬z,x∨y∨z}and S(ϕ
1
,del(α
1
)) =
{¬x ∨y ∨¬z,(¬x ∨y ∨z) ∧(x ∨y ∨z)}.
An important advantage of representing a knowl-
edge base in CCNF is that removing a co-atom from
a formula is a minimal weakening in the semantical
sense, since it enlarges the set of models of the for-
mula for exactly one valuation.
In what follows, Mod(ϕ) denotes the set of mod-
els of ϕ.
Proposition 3. Let ϕ be a formula in CCNF and α
i
be
a co-atom of ϕ. Let v
α
i
: Var −→ {true, false} be
the valuation such that for every x
i
∈ Var, v
α
i
(x
i
) =
true if and only if ¬x
i
appears in α
i
. Then
Mod(ϕ
del(α
i
)
) = Mod(ϕ) ∪{v
α
i
}.
Proof. The proof follows directly from the fact that ϕ
is in CCNF.
We now show that the basic operation we propose
in Definition 5 is a minimal change, in the sense that
for every two knowledge bases such that Σ Σ
0
, there
exists a minimal change of Σ resulting in a knowledge
base between Σ and Σ
0
.
Proposition 4. For every two knowledge bases Σ and
Σ
0
in CCNF, if Σ  Σ
0
then there exists ϕ ∈Σ and α
i
∈
Coatoms(ϕ) such that Σ  Σ(ϕ, del(α
i
))  Σ
0
.
Proof. If Σ  Σ
0
, then Σ  Σ
0
, so Σ = {ϕ
1
,...,ϕ
n
}
and Σ
0
= {ϕ
0
1
,...,ϕ
0
n
} where ` ϕ
i
→ ϕ
0
i
for every i.
From Σ 6= Σ
0
we obtain that there is j such that ϕ
j
6=
ϕ
0
j
. Since ϕ
j
and ϕ
0
j
are in CCNF, we conclude that
Coatoms(ϕ
0
j
) ⊂ Coatoms(ϕ
j
). Let us choose a co-
atom α
k
∈ Coatoms(ϕ
j
) \Coatoms(ϕ
0
j
). Obviously,
`ϕ
j
→ϕ
j
del(α
k
)
and `ϕ
j
del(α
k
)
→ϕ
0
j
. Consequently,
Σ  Σ(ϕ
j
,del(α
k
))  Σ
0
.
Also, whenever Σ  Σ
0
then Σ
0
can be obtained
from Σ by minimal changes.
Proposition 5. For every two knowledge bases Σ and
Σ
0
in CCNF, if Σ Σ
0
then there exist n ∈N, ϕ
1
,...,ϕ
n
and α
k
1
∈Coatoms(ϕ
1
),. ..,α
k
n
∈Coatoms(ϕ
n
) such
that Σ = Σ
0
 Σ
1
 ···  Σ
n
= Σ
0
, where Σ
i
=
Σ
i−1
(ϕ
i
,del(α
k
i
)) for i ∈ {1,.. .,n}.
ICAART2015-InternationalConferenceonAgentsandArtificialIntelligence
30

Proof. First, note that, by Proposition 4, if Σ Σ
0
then
there exist ϕ
1
∈ Σ and α
k
1
∈ Coatoms(ϕ
1
) such that
Σ  Σ(ϕ
1
,del(α
k
1
))  Σ
0
. If Σ(ϕ
1
,del(α
k
1
)) = Σ
0
,
we are done. Otherwise, we can apply Proposition
4 to Σ
1
= Σ(ϕ
1
,del(α
k
1
)) and Σ
0
. More generally,
if Σ  Σ
1
 ···  Σ
i
 Σ
0
, by Proposition 4 there
are ϕ
i+1
∈ Σ
i
and α
k
i+1
∈ Coatoms(ϕ
i+1
) such that
Σ Σ
1
···Σ
i
Σ
i+1
Σ
0
. Since Var is finite, and
all the formulae are in CCNF, there are only finitely
many different knowledge bases of given cardinality.
Thus, there is n ∈ N such that Σ
n
= Σ
0
.
According to the previous result, every weaken-
ing of a knowledge base in CCNF may be obtained as
a sequence of consecutive minimal weakenings. We
call this procedure (leading to a consistent knowledge
base) stepwise inconsistency resolution based on min-
imal weakening. The next section shows that those
changes lead to the most informative theories.
We also prove for a number of inconsistency mea-
sures Inc that the partial order  we introduced on
the knowledge bases is strongly related to the incon-
sistency of a knowledge base. We characterize the
case when Σ  Σ
0
implies that Σ is more inconsistent
than Σ
0
. Namely, the previous statement holds if and
only if the corresponding inconsistency measure en-
joys strong dominance.
Definition 6 (Strong dominance). An inconsistency
measure Inc satisfies strong dominance if and only if
for every knowledge base Σ and for every two formu-
lae ϕ, ψ
ϕ ` ψ implies Inc(Σ ∪{ϕ}) ≥ Inc(Σ ∪{ψ}).
Note that the only difference between dominance
and strong dominance is that the former also stipu-
lates that the inconsistency of a knowledge base can-
not increase if a contradiction from Σ is exchanged
with another formula. This condition is naturally
adopted in a semantic-based approach, like the one
based on CCNF, where two logically equivalent for-
mulae are considered equal. Namely, in CCNF, con-
tradiction contains all the possible co-atoms. Thus,
deleting one co-atom cannot increase inconsistency.
Obviously, strong dominance implies dominance.
Also, we are not aware of any inconsistency measure
proposed in literature which satisfies dominance, but
does not satisfy strong dominance. Note that Mu et al.
(2011) show that Inc
MI
does not satisfy dominance,
hence it does not satisfy strong dominance.
The next result shows that the partial order of sets
based on  is also an order with respect to Inc if and
only if Inc satisfies strong dominance.
Proposition 6. Let Inc be an inconsistency measure.
Inc satisfies strong dominance if and only if for ev-
ery two knowledge bases Σ and Σ
0
, Σ  Σ
0
implies
Inc(Σ) ≥ Inc(Σ
0
).
Proof. Let us first show that if Inc satisfies strong
dominance then Σ  Σ
0
implies Inc(Σ) ≥ Inc(Σ
0
).
Let Inc satisfy strong dominance and let Σ  Σ
0
.
Denote Σ = {ϕ
1
,...,ϕ
n
} and Σ
0
= {ϕ
0
1
,...,ϕ
0
n
},
where ` ϕ
i
→ ϕ
0
i
for every i. From strong dom-
inance, we have that Σ  Inc({ϕ
0
1
,ϕ
2
,...,ϕ
n
}) ≥
Inc({ϕ
0
1
,ϕ
0
2
,ϕ
3
,...,ϕ
n
}) ≥ ... ≥ Inc(Σ
0
).
We now show that if Inc does not satisfy strong
dominance then there exist Σ,Σ
0
such that Σ  Σ
0
and
Inc(Σ) < Inc(Σ
0
). Since Inc does not satisfy strong
dominance then there exist Σ
0
,α,β such that α ` β
and Inc(Σ ∪{α}) < Inc(Σ ∪{β}). Letting Σ = Σ
0
∪
{α} and Σ
0
= Σ
0
∪{β} ends the proof.
5 STEPWISE VS DIRECT
INCONSISTENCY
RESOLUTION
Every inconsistency resolution approach based on
weakening changes the given set of formulae, and
consequently reduces the quality of information. We
consider two interrelated notions that every resolution
procedure should take into account:
1. Minimizing the change of the given knowledge
base.
2. Saving as much information as possible.
We show that stepwise resolution proposed in the
previous section correctly addresses the second item.
On the other hand, it can make some unnecessary
changes to the given knowledge base.
In order to measure the information of a knowl-
edge base, we use the definition by Hunter and
Konieczny (2005), which is the propositional ver-
sion of the definition proposed earlier in the literature
(Lozinskii, 1997).
Definition 7. A measure of quantity of information
is a function Inf from the set of knowledge bases to
[0,+∞), such that the following conditions hold:
1. Inf(
/
0) = 0
2. Inf(Var ∪{¬x | x ∈ Var}) = 0
3. If Σ 6` ⊥ and Σ ` ϕ, then Inf(Σ ∪{ϕ}) = Inf(Σ)
4. If Σ ∪{ϕ} 6` ⊥ and Σ 6` ϕ, then Inf(Σ ∪{ϕ}) >
Inf(Σ)
5. If Σ 6`⊥and Σ `ϕ, then Inf(Σ ∪{¬ϕ}) < Inf(Σ)
6. If Σ 6` ⊥ and for all x ∈ Var we have Σ ` x or
Σ ` ¬x, then for all Σ
0
Inf(Σ) ≥ Inf(Σ
0
)
HowtoDecreaseandResolveInconsistencyofaKnowledgeBase?
31

The item 3 of the previous definition, called
Closed (Grant and Hunter, 2011b), states that adding
a logical consequence of a knowledge base does not
change the quantity of information. The item 6 states
that a complete knowledge base, i.e. the set of formu-
lae that has a unique model, has the highest possible
information content.
Grant and Hunter introduce a weaker definition of
inconsistency measure, but additional useful condi-
tions are also considered (Grant and Hunter, 2011b).
One of them, named Equivalence, states that
Inf(Σ) = Inf(Σ
0
), whenever Σ 6` ⊥ and Σ ≡Σ
0
.
They show that Equivalence implies the item 3
from Definition 7. Actually, those two properties are
equivalent.
Lemma 1. Let f be a function from the set of knowl-
edge bases to [0, +∞). Function f satisfies Equiva-
lence if and only if it satisfies Closed.
Proof. As it is shown by Grant and Hunter (2011b),
if Inf satisfies Equivalence, then for any Σ such that
Σ 6` ⊥ and Σ ` ϕ we have that Σ ≡ Σ ∪{ϕ}. By the
assumption, Inf(Σ ∪{ϕ}) = Inf(Σ).
Now suppose that Inf satisfies Closed. Let Σ =
{ϕ
1
,...,ϕ
n
}, Σ 6` ⊥ and Σ ≡ Σ
0
. We define Σ
0
= Σ
0
and Σ
i
= Σ
i−1
∪{ϕ
i
}, for i ∈ {1,. .., n}. Then we
have Σ
0
` ϕ
i
, so Σ
i
` ϕ
i
, for i ∈ {1,.. .,n}. By
the assumption, Inf(Σ
0
) = Inf(Σ
0
) = Inf(Σ
1
) =
··· = Inf(Σ
n
) = Inf(Σ
0
∪Σ). Similarly, Inf(Σ) =
Inf(Σ
0
∪Σ), so Inf(Σ) = Inf(Σ
0
).
As expected, our partial order on knowledge bases
agrees with information measures.
Proposition 7. Let Σ and Σ
0
be two consistent knowl-
edge bases and let Inf be an information measure. If
Σ  Σ
0
then Inf(Σ) ≥ Inf(Σ
0
).
Proof. Suppose that Σ consistent. If Σ  Σ
0
, then for
every ϕ
0
∈ Σ
0
there is ϕ ∈ Σ such that ` ϕ → ϕ
0
, so
Σ ∪Σ
0
≡ Σ. Let ϕ
Σ
be a formula such that {ϕ
Σ
} ≡ Σ.
Hence, {ϕ
Σ
}∪Σ
0
≡ Σ. Thus, {ϕ
Σ
}∪Σ
0
is also con-
sistent. By Lemma 1, Inf(Σ) = Inf({ϕ
Σ
} ∪ Σ
0
).
On the other hand, Inf({ϕ
Σ
} ∪ Σ
0
) ≥ Inf(Σ
0
), by
items 3 and 4 of Definition 7. Thus, we obtained
Inf(Σ) ≥ Inf(Σ
0
).
By item 6 of Definition 7 and the following propo-
sition, stepwise inconsistency resolution based on
minimal weakening (i.e. the process of obtaining a
consistent knowledge base by consecutively applying
minimal changes) leads to a maximally informative
consistent knowledge base.
Proposition 8. Let Σ be an inconsistent knowledge
base, and let Σ
0
be the knowledge base obtained from
Σ by stepwise inconsistency resolution based on min-
imal weakening. Then Σ
0
is complete.
Proof. If Σ
0
is a consistent knowledge base, obtained
from Σ by the stepwise inconsistency resolution based
on minimal weakening, i.e. Σ = Σ
0
 Σ
1
 ··· 
Σ
n−1
 Σ
n
= Σ
0
, then, by Proposition 2, there is a
co-atom that does not appear in any formula from
Σ
0
. Also, since Σ
n−1
is inconsistent, each co-atom
appears in at least one formula from Σ
n−1
. If Σ
0
=
Σ
n−1
(ϕ,del(α
i
)), then obviously α
i
is the only co-
atom that does not appear in any formula from Σ
0
. The
valuation v
α
i
defined in the proof of Proposition 2 is a
model of Σ
0
. Suppose that there is a valuation v 6= v
α
i
that is also a model of Σ
0
. We define the co-atom
α
v
=
_
x∈Var,v(x)= f alse
x ∨
_
x∈Var,v(x)=true
¬x.
If ψ ∈ Σ
0
is a formula such that α
v
appears in ψ, then
ψ is false under the valuation v, since
¬α
v
=
^
x∈Var,v(x)= f alse
¬x ∧
^
x∈Var,v(x)=true
x
is obviously true under v. Thus, the valuation v
α
i
is
the only model of Σ
0
.
Nevertheless, some of the changes made to the
knowledge bases might be unnecessary. Consider the
following example.
Example 3. Let Var = {x,y}, and let Σ =
{ϕ
1
,ϕ
2
,ϕ
3
,ϕ
4
}, where ϕ
1
= (x ∨y) ∧(x ∨¬y), ϕ
2
=
(¬x ∨y) ∧(¬x ∨¬y), ϕ
3
= (x ∨¬y) ∧ (¬x ∨y) and
ϕ
4
= (x ∨y) ∧(¬x ∨¬y). Let Σ
0
be obtained by re-
placing ϕ
1
with ϕ
0
1
= (x ∨y), i.e. ϕ
0
1
= ϕ
1
del(x∨¬y)
. By
Proposition 2, Σ
0
is inconsistent. If Σ
00
is obtained
from Σ
0
by replacing ϕ
2
with ϕ
0
2
= ϕ
2
del(¬x∨y)
, then
Σ
00
is also inconsistent. Let Σ
000
be obtained from Σ
00
by
replacing ϕ
3
with ϕ
0
3
= ϕ
3
del(x∨¬y)
. By Proposition 2,
Σ
000
= {ϕ
0
1
,ϕ
0
2
,ϕ
0
3
,ϕ
4
} is consistent, since x ∨¬y does
not appear in the formulae of Σ
000
. However, note that
the second step of the resolution was not necessary,
since the knowledge base {ϕ
0
1
,ϕ
2
,ϕ
0
3
,ϕ
4
} is also con-
sistent.
Now we propose Σ
del(α
i
)
as candidates for the re-
solving inconsistent knowledge base Σ. Note that, in
the case when we want to end up with a consistent
knowledge base, it can be argued that deleting a co-
atom α
i
in some but not in all formulae does not make
sense. Namely, whether a co-atom appears once or
ICAART2015-InternationalConferenceonAgentsandArtificialIntelligence
32

several times in Σ does not make any difference con-
cerning consistency of Σ. Thus, in a scenario where
we continue applying changes until obtaining a con-
sistent knowledge base, we should either remove a co-
atom from all formulae or not remove it at all (in order
to avoid information loss without consistency gain).
The next result shows that if we delete all occur-
rences of an arbitrary co-atom, Σ becomes not only
consistent, but also maximally informative, in the
sense that adding information (in terms of semantics)
would result in its inconsistency. Recall that we call a
knowledge base complete if and only if it has exactly
one model.
Proposition 9. If Σ is an inconsistent knowledge base
in CCNF then for every α
i
∈ Coatoms(Σ), Σ
del(α
i
)
is
complete.
Proof. If Σ is inconsistent, by Proposition 2, each co-
atom appears in at least one formula from Σ. Conse-
quently, α
i
is the only co-atom that does not appear in
any formula from Σ
del(α
i
)
. Similarly as in the proof of
Proposition 8, one can show that the valuation v
α
i
is
the only model of Σ
del(α
i
)
, so it is complete.
By the following proposition, however the resolu-
tion of inconsistency of Σ is conducted, if the obtained
set is not of the form Σ
del(α
i
)
, the inconsistency can be
resolved by applying less changes to Σ.
Proposition 10. Let Σ,Σ
0
be two knowledge bases in
CCNF. If Σ is inconsistent then:
1. If Σ  Σ
0
and Σ
0
is consistent, then there exists
α
i
∈ Coatoms(Σ) such that Σ  Σ
del(α
i
)
 Σ
0
.
2. For every α
i
∈ Coatoms(Σ), we have that if Σ 
Σ
0
 Σ
del(α
i
)
, then Σ
0
is inconsistent.
Proof.
1. Let Σ = {ϕ
1
,...,ϕ
n
} and Σ
0
= {ϕ
0
1
,...,ϕ
0
n
} such
that for every j, it holds that ` ϕ
j
→ ϕ
0
j
. Since
Σ
0
is consistent, there exists a co-atom α
i
that is
absent from all ϕ
0
j
. Thus, for every j, ` ϕ
j
→
ϕ
j
del(α
i
)
and `ϕ
j
del(α
i
)
→ϕ
0
j
hold. Consequently,
Σ  Σ
del(α
i
)
= {ϕ
1
del(α
i
)
,...,ϕ
n
del(α
i
)
}  Σ
0
.
2. First, recall that v
α
i
is the only model of Σ
del(α
i
)
.
From Σ
0
 Σ
del(α
i
)
we obtain that v is not a model
of Σ
0
, for every v 6= v
α
i
. If Σ = {ϕ
1
,...,ϕ
n
}
and Σ
0
= {ϕ
0
1
,...,ϕ
0
n
}, then for every j, ϕ
0
j
= ϕ
j
or ϕ
0
j
= ϕ
j
del(α
i
)
. From Σ
0
 Σ
del(α
i
)
we obtain
that there is k such that ϕ
0
k
6= ϕ
k
del(α
i
)
. Thus,
ϕ
0
k
= ϕ
k
del(α
i
)
∧α
i
. Then the valuation v
α
i
is not a
model of ϕ
0
k
, so Σ
0
is inconsistent.
The results from this section allow us to draw sev-
eral conclusions. First, we see that minimal ways to
weaken Σ are to apply deletion of exactly one co-
atom. If one weakens it more, then we lose more than
necessary in terms of information. Does stepwise in-
consistency resolution make sense if we want to end
up with a consistent knowledge base (given that it will
never yield a better result in terms of information loss
than one-step approach based on deleting one co-atom
from all the formulae)?
Indeed, one might just want to go directly to the
closest consistent knowledge base. But how to know
which knowledge base is the closest? We answer this
question in the next section.
6 DEFINING METRICS ON THE
SET OF KNOWLEDGE BASES
Relation  allows to compare knowledge bases and
enjoys some desirable properties related to inconsis-
tency and information measures. However, since this
relation is not a total order, some knowledge bases are
not comparable. Consider the following example.
Example 4. Suppose that Var = {x,y,z}. Then there
are 8 co-atoms, α
1
,α
2
,...,α
8
. Let
ϕ
1
= α
1
∧α
2
ϕ
2
= α
3
∧α
4
∧α
5
∧α
6
∧α
7
ϕ
3
= α
3
∧α
8
ϕ
4
= α
2
∧α
3
By Proposition 2, the knowledge base
Σ = {ϕ
1
,ϕ
2
,ϕ
3
,ϕ
4
} is inconsistent. By the pre-
vious section, minimal ways to weaken Σ are to
apply deletion of exactly one co-atom. Note that
Σ
del(α
1
)
can be obtained from Σ by only one step in
the stepwise inconsistency resolution (replacing ϕ
1
by ϕ
1
del(α
1
)
). On the other hand, we need three steps
to obtain Σ
del(α
3
)
from Σ, so Σ
del(α
1
)
is intuitively
closer to Σ. Nevertheless, the partial order  can
not capture this intuition, since Σ
del(α
1
)
and Σ
del(α
3
)
are incomparable, i.e. neither Σ
del(α
1
)
 Σ
del(α
3
)
nor
Σ
del(α
3
)
 Σ
del(α
1
)
.
Motivated by the previous example, we propose to
define a family of metrics to measure distance from a
given set to the closest consistent set. Let us recall the
definition of a metric.
Definition 8. A metric on a set S is any function d :
S
2
−→ R, such that for all a,b, c ∈ S the following
conditions hold:
1. d(a, b) ≥ 0
2. d(a, b) = 0 iff a = b
HowtoDecreaseandResolveInconsistencyofaKnowledgeBase?
33

3. d(a, b) = d(b, a)
4. d(a, b) + d(b, c) ≥ d(a, c).
The pair hS,di is called metric space. For A ⊆ S,
distance from element a to A is defined as d(a, A) =
inf
b∈A
d(a,b).
We are primarily interested in applying the met-
rics in order to analyze how the distance between a
knowledge base and the closest consistent knowledge
base changes when we apply the approach proposed
in Sections 4 and 5. Since our weakenings do not
change the cardinality of a knowledge base, as in the
case of partial order , it suffices to measure the dis-
tances between knowledge bases of the same cardi-
nality.
In the following definition, 4 denotes the sym-
metric difference of two sets
3
and Π(n) is the set of
permutations of the set {1,... ,n}. Furthermore, since
we work with models, we identify the formulae hav-
ing the same models. In other words, we do not dis-
tinguish between ϕ and ψ if {ϕ} ≡ {ψ}.
Definition 9. For a fixed n ∈ N and p ∈ [1,+∞), we
define the binary function d
p
on the set of knowledge
bases of cardinality n as follows. For Σ = {ϕ
1
,...ϕ
n
}
and Σ
0
= {ψ
1
,...ψ
n
} let
d
p
(Σ,Σ
0
) = min
π∈Π(n)
(
n
∑
i=1
|Mod(ϕ
i
) 4Mod(ψ
π(i)
)|
p
)
1
p
.
We also define
d
∞
(Σ,Σ
0
) = min
π∈Π(n)
max
i∈{1,...,n}
|Mod(ϕ
i
) 4Mod(ψ
π(i)
)|.
The previous definition uses a variant of p-norms.
Intuitively, it takes the minimum over all the permu-
tations of Σ
0
in order to “try to find” the alignment of
Σ
0
that is “the most similar” to Σ. The bigger the dif-
ference in number of models between ϕ and ϕ
0
, the
bigger the distance between the bases.
By [1,+∞] we denote the set [1, +∞) ∪{+∞}.
Proposition 11. For each p ∈ [1,+∞], d
p
is a metric.
Proof. The proofs of d
p
(Σ,Σ
0
) ≥ 0, d
p
(Σ,Σ
0
) = 0
iff Σ = Σ
0
, and d
p
(Σ,Σ
0
) = d
p
(Σ
0
,Σ) are trivial, so
we only prove d
p
(Σ,Σ
0
) + d
p
(Σ
0
,Σ
00
) ≥ d
p
(Σ,Σ
00
).
We only consider the case when p ∈ [1,+∞);
the case when p = +∞ can be proved in
the similar way. Let Σ = {ϕ
1
,...,ϕ
n
},
Σ
0
= {ϕ
0
1
,...,ϕ
0
n
} and Σ
00
= {ϕ
00
1
,...,ϕ
00
n
}. Let
σ ∈ Π(n) be the permutation such that d
p
(Σ,Σ
0
) =
(
∑
n
i=1
|Mod(ϕ
i
) 4Mod(ϕ
0
σ(i)
)|
p
)
1
p
. Then d
p
(Σ,Σ
0
) +
3
Recall that symmetric difference of two sets S
1
and S
2
is defined as S
1
4S
2
= (S
1
\S
2
) ∪(S
2
\S
1
).
d
p
(Σ
0
,Σ
00
) = (
∑
n
i=1
|Mod(ϕ
i
) 4 Mod(ϕ
0
σ(i)
)|
p
)
1
p
+
min
π∈Π(n)
(
∑
n
i=1
|Mod(ϕ
0
i
) 4 Mod(ϕ
00
π(i)
)|
p
)
1
p
=
(
∑
n
i=1
|Mod(ϕ
i
) 4 Mod(ϕ
0
σ(i)
)|
p
)
1
p
+
min
π∈Π(n)
(
∑
n
i=1
|Mod(ϕ
0
σ(i)
) 4 Mod(ϕ
00
π(σ(i))
)|
p
)
1
p
=
min
π∈Π(n)
((
∑
n
i=1
|Mod(ϕ
i
) 4 Mod(ϕ
0
σ(i)
)|
p
)
1
p
+
(
∑
n
i=1
|Mod(ϕ
0
σ(i)
) 4 Mod(ϕ
00
π(σ(i))
)|
p
)
1
p
). From the
inequality
(
n
∑
i=1
a
p
i
)
1
p
+ (
n
∑
i=1
b
p
i
)
1
p
≥(
n
∑
i=1
(a
i
+ b
i
)
p
)
1
p
, a
i
,b
i
≥0
we obtain d
p
(Σ,Σ
0
) + d
p
(Σ
0
,Σ
00
) ≥
min
π∈Π(n)
(
∑
n
i=1
(|Mod(ϕ
i
) 4 Mod(ϕ
0
σ(i)
)| +
|Mod(ϕ
0
σ(i)
) 4Mod(ϕ
00
π(σ(i))
)|)
p
)
1
p
.
Note that, for any sets A, B and C,
(A 4B) ∪(B 4C) ⊇ A 4C, so |A 4B|+ |B 4C| ≥
|(A 4 B) ∪ (B 4 C)| ≥ |A 4 C|. Thus,
d
p
(Σ,Σ
0
)+d
p
(Σ
0
,Σ
00
) ≥min
π∈Π(n)
((
∑
n
i=1
|Mod(ϕ
i
)4
Mod(ϕ
00
π(σ(i))
)|
p
)
1
p
= min
π
0
∈Π(n)
((
∑
n
i=1
|Mod(ϕ
i
) 4
Mod(ϕ
00
π
0
(i)
)|
p
)
1
p
= d
p
(Σ,Σ
00
).
The following example illustrates that the met-
rics d
1
and d
2
are not compatible. Namely, it shows
that it can be the case that d
1
(Σ,Σ
0
) < d
1
(Σ,Σ
00
), but
d
2
(Σ,Σ
0
) > d
2
(Σ,Σ
00
).
Example 5. Suppose that Var = {x
1
,...,x
6
}. Then
there are 64 co-atoms. Let
ϕ
1
= α
1
∧α
2
∧···∧α
10
ϕ
2
= α
21
∧α
22
∧···∧α
30
ϕ
3
= α
21
∧α
22
∧···∧α
40
ϕ
4
= α
1
∧α
2
∧···∧α
15
ϕ
5
= α
21
∧α
22
∧···∧α
36
Let Σ = {ϕ
1
,ϕ
2
}, Σ
0
= {ϕ
1
,ϕ
3
} and Σ
00
= {ϕ
4
,ϕ
5
}.
Then
d
1
(Σ,Σ
0
) = |Mod(ϕ
1
)4Mod(ϕ
1
)|+|Mod(ϕ
2
)4
Mod(ϕ
3
)| = 0 + 10 = 10
d
2
(Σ,Σ
0
) = (|Mod(ϕ
1
) 4 Mod(ϕ
1
)|
2
+
|Mod(ϕ
2
) 4Mod(ϕ
3
)|
2
)
1
2
=
√
0
2
+ 10
2
= 10
d
1
(Σ,Σ
00
) = |Mod(ϕ
1
)4Mod(ϕ
4
)|+|Mod(ϕ
2
)4
Mod(ϕ
5
)| = 5 + 6 = 11
d
2
(Σ,Σ
00
) = (|Mod(ϕ
1
) 4 Mod(ϕ
4
)|
2
+
|Mod(ϕ
2
) 4Mod(ϕ
5
)|
2
)
1
2
=
√
5
2
+ 6
2
=
√
61
Thus, we obtain that d
1
(Σ,Σ
0
) < d
1
(Σ,Σ
00
), but
d
2
(Σ,Σ
0
) > d
2
(Σ,Σ
00
).
Moreover, there are no distinct p
1
and p
2
such that
d
p
1
is compatible with d
p
2
for every Var.
ICAART2015-InternationalConferenceonAgentsandArtificialIntelligence
34

Proposition 12. For all p
1
,p
2
∈ [1,+∞] with
p
1
6= p
2
, there exist Var, Σ, Σ
0
and Σ
00
with P (Σ),P (Σ
0
),P (Σ
00
) ⊆ Var such that
d
p
1
(Σ,Σ
0
) < d
p
1
(Σ,Σ
00
) but d
p
2
(Σ,Σ
0
) > d
p
2
(Σ,Σ
00
).
Proof. Let p
1
< p
2
< +∞. Then 2
1
p
2
< 2
1
p
1
, so there
is m ∈ N such that m(2
1
p
1
−2
1
p
2
) > 1. Let us choose
n ∈ N such that m2
1
p
2
< n < m2
1
p
1
. We suppose that
the set of propositional letters is large enough, and we
denote co-atoms by α
i
and β
j
, assuming that α’s and
β’s are always different. Let
• Σ = {ϕ
1
,ϕ
2
}, where ϕ
1
=
V
k
i=1
α
i
and ϕ
2
=
V
k
i=1
β
i
• Σ = {ϕ
0
1
,ϕ
0
2
}, where ϕ
0
1
= ϕ
1
and ϕ
0
2
=
V
k+n
i=1
β
i
• Σ
00
= {ϕ
00
1
,ϕ
00
2
}, where ϕ
00
1
=
V
k+m
i=1
α
i
and ϕ
00
2
=
V
k+m
i=1
β
i
Then d
p
1
(Σ,Σ
0
) = (|Mod(ϕ
1
) 4 Mod(ϕ
0
1
)|
p
1
+
|Mod(ϕ
2
) 4Mod(ϕ
0
2
)|
p
1
)
1
p
1
= n, and d
p
2
(Σ,Σ
0
) = n.
Similarly, d
p
2
(Σ,Σ
00
) = (|Mod(ϕ
1
) 4Mod(ϕ
00
1
)|
p
2
+
|Mod(ϕ
2
) 4 Mod(ϕ
00
2
)|
p
2
)
1
p
2
= m2
1
p
1
, and
d
p
2
(Σ,Σ
00
) = m2
1
p
2
.
Thus, d
p
1
(Σ,Σ
0
) < d
p
1
(Σ,Σ
00
) and d
p
2
(Σ,Σ
0
) >
d
p
2
(Σ,Σ
00
).
In the case where p
1
= +∞ or p
2
= +∞, the proof
is similar.
Nevertheless, the next result shows that for every
metric d
p
, for each α
i
appearing in the least number
of formulae of Σ, the distance from Σ to the closest
consistent multi-set of the same cardinality is exactly
the distance from Σ to Σ
del(α
i
)
. In other words, in-
dependently of parameter p, using Σ
del(α
i
)
yields the
closest consistent knowledge base.
We first prove the following useful result showing
that the distance (with respect to metric d
p
) between
a knowledge base Σ and the knowledge base Σ
0
ob-
tained by deleting from Σ a co-atom α
i
that appears k
times in Σ is k
1
p
.
Lemma 2. Let Σ be a knowledge base in CCNF and
α
i
a co-atom such that |{ϕ ∈Σ |α
i
∈Coatoms(ϕ)}|=
k. Then
d
p
(Σ,Σ
del(α
i
)
) = k
1
p
Proof. Let Σ = {ϕ
1
,...,ϕ
n
} and, without loss of gen-
erality, suppose that α
i
appears in formulae ϕ
1
,...,ϕ
k
.
On one hand, by denoting Σ
0
= {ψ
1
,...,ψ
n
} with
ψ
i
= ϕ
i,del(α
i
)
and letting permutation π from Defi-
nition 9 be identity, we obtain that (
∑
n
i=1
|Mod(ϕ
i
) 4
Mod(ψ
π(i)
)|
p
)
1
p
= k
1
p
. On the other hand, since Σ
0
=
Σ
del(α
i
)
, then there exist k formulae in Σ that do not
appear in Σ
0
(those are formulae ϕ
1
,...,ϕ
k
). Each of
those k formulae must be matched with exactly one
formula ψ
q
from Σ
0
different from it. Thus, for ev-
ery permutation, there are at least k matched formu-
lae that differ, so d
p
(Σ,Σ
del(α
i
)
) ≥k
1
p
. From those two
facts, we conclude that d
p
(Σ,Σ
del(α
i
)
) = k
1
p
.
Proposition 13. Denote by Cons(n) the set of all con-
sistent multi-sets of cardinality n. Then for all Σ in
CCNF, for all p ∈ [1,+∞], for all α
i
∈ Coatoms(Σ)
such that there exists no α
j
∈ Coatoms(Σ) such that
|{ϕ ∈ Σ | α
j
∈ Coatoms(ϕ)}| < |{ϕ ∈ Σ | α
i
∈
Coatoms(ϕ)}|, we have
d
p
(Σ,Cons(|Σ|)) = d
p
(Σ,Σ
del(α
i
)
).
Proof. If Σ is consistent, by Proposition 2 there is α
i
that is absent from every formula of Σ. Let Σ be in-
consistent and let p ∈[1,+∞) (the case when p = +∞
is similar).
First, by Proposition 2, Σ
del(α
i
)
is consistent, thus
d
p
(Σ,Cons(|Σ|)) ≤ d
p
(Σ,Σ
del(α
i
)
).
Second, by Lemma 2, we obtain d
p
(Σ,Σ
del(α
i
)
) =
k
1
p
.
Third, let us show that for every Σ
0
∈ Cons(|Σ|)
we have d
p
(Σ,Σ
0
) ≥ k
1
p
. By Proposition 2, there
exists a co-atom α
j
such that α
j
/∈ Coatoms(Σ
0
).
Suppose that α
j
appears in exactly l formulae of
Σ. For each formula ϕ ∈ Σ containing α
j
, we have
|Mod(ϕ) 4Mod(ψ)| ≥ 1 for every ϕ
0
∈ Σ
0
. Conse-
quently, d
p
(Σ,Σ
0
) ≥ l
1
p
≥ k
1
p
.
From the second and the third observation, we
have that d
p
(Σ,Cons(|Σ|)) ≥ d
p
(Σ,Σ
del(α
i
)
). This
fact, together with the first observation, ends the
proof.
The previous result confirms that our class of met-
rics is based on an intuitive distance. Namely, it
confirms the natural expectation that deleting a co-
atom that appears in the least number of formulae in
a knowledge base is the minimal change required to
obtain a consistent knowledge base. The following
statement is a direct consequence of Proposition 13.
Corollary 1. For all Σ, Σ
0
in CCNF such that
Σ  Σ
0
and for all p ∈ [1, +∞], it holds that
d
p
(Σ,Cons(|Σ|)) ≥ d
p
(Σ
0
,Cons(|Σ|)).
This result proves an important link between the
distances defined in Section 6 and the inconsistency
resolution method proposed in Sections 4 and 5.
Namely, it shows that applying minimal changes to
a knowledge base reduces its distance to consistency.
HowtoDecreaseandResolveInconsistencyofaKnowledgeBase?
35

7 CONCLUSION AND RELATED
WORK
This paper tackled the question of how to measure
inconsistency of a knowledge base, how to compare
knowledge bases in terms of inconsistency and infor-
mation and how to measure distance between knowl-
edge bases. Given an inconsistent knowledge base,
our goal is to decrease its inconsistency and/or find
the closest consistent knowledge base.
Namely, we proposed to use the operation ϕ
del(α
i
)
as a basic operation for decreasing inconsistency of a
knowledge base; its advantage is that it yields a mini-
mal change of a knowledge base. We defined a partial
order  on the set of knowledge bases and showed
its link with existing inconsistency and information
measures. We also introduced operations Σ
del(α
i
)
and
showed that they define the closest consistent knowl-
edge bases according to . Furthermore, we defined
a class of metrics that can be used to measure dis-
tances between two knowledge bases as well as the
distance from a knowledge base to the closest con-
sistent knowledge base. We formally showed that
our class of metrics agrees with an intuitive distance.
Namely, whatever the value of parameter p, deleting
a co-atom that appears in the least number of formu-
lae in a knowledge base is the minimal change (in the
sense of distance measured by d
p
) required to obtain
a consistent knowledge base. Also, we proved that
those metrics are compatible with , which means
that the distance between a knowledge base Σ and
consistency decreases whenever our minimal weak-
ening is applied to Σ.
The closest related paper is the work by Grant
and Hunter discussed in the introduction (Grant and
Hunter, 2011b). We are inspired by that paper but we
use only one operation instead of three. To be more
precise, Grant and Hunter proposed three classes of
operations, but given a concrete formula, say ϕ = x ∨y
in a concrete knowledge base Σ, they do not say how
to practically weaken ϕ. Our paper offers the answers
to this question. Also, we compare stepwise inconsis-
tency resolution and one-step approach where we go
directly to the closest consistent knowledge base.
There is another work that might seem very close
to the second part of our paper (Grant and Hunter,
2013). But indeed, there is a huge difference, since
they introduce metrics to measure distances between
models, which is not the case in our paper. We mea-
sure the distance between knowledge bases using a
variation of p-norms, applied on symmetric differ-
ence of models of formulae of two knowledge bases.
There is also related work on inconsistency
measurement for probabilistic logics (Picado-Mui
˜
no,
2011; Thimm, 2009), where p-norms are used to
calculate measure of inconsistency of a probabilistic
knowledge base. But unlike in our paper, in those pa-
pers the p-norms are used to determine the distance
between all the functions (which are not probability
measures) satisfying the (inconsistent) set of proba-
bility requirements from the knowledge base, and the
closest probability measure.
ACKNOWLEDGEMENTS
Dragan Doder was supported by the National Re-
search Fund (FNR) of Luxembourg through project
PRIMAT. The authors would like to thank S
´
ebastien
Konieczny for his help.
REFERENCES
Doder, D., Raskovic, M., Markovic, Z., and Ognjanovic,
Z. (2010). Measures of inconsistency and de-
faults. International Journal of Approximate Reason-
ing, 51(7):832–845.
Gabbay, D. M. and Hunter, A. (1991). Making incon-
sistency respectable: a logical framework for incon-
sistency in reasoning. In Fundamentals of Artificial
Intelligence Research, International Workshop (FAIR
’91), volume 535 of Lecture Notes in Computer Sci-
ence, pages 19–32. Springer.
Gabbay, D. M. and Hunter, A. (1993). Making incon-
sistency respectable: Part 2 - meta-level handling
of inconsistency. In Symbolic and Quantitative Ap-
proaches to Reasoning and Uncertainty, European
Conference (ECSQARU’93), volume 747 of Lecture
Notes in Computer Science, pages 129–136. Springer.
Grant, J. and Hunter, A. (2006). Measuring inconsistency in
knowledgebases. J. Intell. Inf. Syst., 27(2):159–184.
Grant, J. and Hunter, A. (2011a). Measuring consistency
gain and information loss in stepwise inconsistency
resolution. In Proceedings of the 7th European Con-
ference on Symbolic and Quantitative Approaches to
Reasoning with Uncertainty (ECSQARU’11), pages
362–373.
Grant, J. and Hunter, A. (2011b). Measuring the good and
the bad in inconsistent information. In Proceedings of
the 22nd International Joint Conference on Artificial
Intelligence (IJCAI’11), pages 2632–2637.
Grant, J. and Hunter, A. (2013). Distance-based measures
of inconsistency. In 12th Europen Conference on Sym-
bolic and Quantitative Approaches to Reasoning with
Uncertainty (ECSQARU’13), volume 7958 of Lecture
Notes in Computer Science, pages 230–241. Springer.
Hunter, A. and Konieczny, S. (2005). Approaches to
measuring inconsistent information. In Inconsistency
Tolerance [result from a Dagstuhl seminar], volume
3300 of Lecture Notes in Computer Science, pages
191–236. Springer.
ICAART2015-InternationalConferenceonAgentsandArtificialIntelligence
36

Hunter, A. and Konieczny, S. (2008). Measuring incon-
sistency through minimal inconsistent sets. In Pro-
ceedings of the 9th International Conference on the
Principles of Knowledge Representation and Reason-
ing (KR’08), pages 358–366.
Hunter, A. and Konieczny, S. (2010). On the measure of
conflicts: Shapley inconsistency values. Artificial In-
telligence Journal, 174(14):1007–1026.
Knight, K. (2002). Measuring inconsistency. Journal of
Philosophical Logic, 31(1):77–91.
Konieczny, S., Lang, J., and Marquis, P. (2003). Quanti-
fying information and contradiction in propositional
logic through test actions. In Proceedings of Interna-
tional Joint Conference on Artificial Intelligence (IJ-
CAI’03), pages 106–111.
Lozinskii, E. (1997). On knowledge evolution: Acquisi-
tion, revision, contraction. Journal of Applied Non-
Classical Logic, 7(2):177–211.
Mu, K., Liu, W., Jin, Z., and Bell, D. (2011). A syntax-
based approach to measuring the degree of inconsis-
tency for belief bases. International Journal of Ap-
proximate Reasoning, 52(7):978–999.
Picado-Mui
˜
no, D. (2011). Measuring and repairing incon-
sistency in probabilistic knowledge bases. Int. J. Ap-
prox. Reasoning, 52(6):828–840.
Sorensen, R. (1988). Blindspots. Claredon Press, Oxford.
Thimm, M. (2009). Measuring inconsistency in probabilis-
tic knowledge bases. In UAI, pages 530–537.
Whitesitt, J. (1995). Boolean Algebra and its applications.
Dover Publications.
HowtoDecreaseandResolveInconsistencyofaKnowledgeBase?
37