Reasoning with Inconsistency-tolerant Fuzzy Description Logics
Norihiro Kamide
Teikyo University, Department of Information and Electronic Engineering, Tochigi, Japan
Keywords:
Inconsistency-tolerant Description Logic, Fuzzy Description Logic, Temporal Description Logic, Embedding.
Abstract:
An inconsistency-tolerant fuzzy description logic is introduced and a translation from this logic to a standard
fuzzy description logic is constructed. A theorem for embedding the proposed inconsistency-tolerant fuzzy
description logic into the standard fuzzy description logic is proven via this translation. A relative decidability
theorem for the inconsistency-tolerant fuzzy description logic w.r.t. the standard fuzzy description logic is
also proven using this embedding theorem. These proposed logic and translation are intended to effectively
handle inconsistent fuzzy knowledge bases. By using the translation, the previously developed algorithms and
methods for the standard fuzzy description logic can be re-purposed for appropriately handling inconsistent
fuzzy knowledge bases that are described by the proposed logic. Furthermore, an inconsistency-tolerant fuzzy
temporal next-time description logic is obtained from the inconsistency-tolerant fuzzy description logic by
adding a temporal next-time operator. Similar results as those for the inconsistency-tolerant fuzzy description
logic are also obtained for this temporal extension.
1 INTRODUCTION
Even though handling fuzzy (vague or imprecise)
concepts is well-known to be a significant issue in
knowledge representation (KR) in AI, inconsistency
handling is of growing importance in KR because in-
consistencies can frequently occur in the real world.
Thus, combining these issues is also regarded as a
significant issue in KR, especially for realizing smart
knowledge-based systems. Knowledge-based sys-
tems would be smarter, more robust, and more fine-
grained if they were capable of handling inconsistent
fuzzy knowledge bases. To effectively handle incon-
sistent fuzzy knowledge bases, this paper introduces
an inconsistency-tolerant fuzzy description logic, if-
ALC . This proposed logic is regarded as an exten-
sion of the standard fuzzy description logic f-ALC
originally introduced by Straccia in (Straccia, 2001),
although f-A L C was, however, not referred to as f-
ALC in the original paper. A translation from if-
ALC to f-A L C is defined and a theorem for embed-
ding if-ALC into f-ALC is proven using this transla-
tion. A relative decidability theorem for if-ALC w.r.t.
f-ALC is proven via this embedding theorem. This
relative decidability theorem shows that if a decision
problem for f-ALC is decidable, then the counter-
part decision problem for if-ALC is also decidable.
Based on the proposed translation, we can reuse the
previously developed algorithms and methods for f-
ALC in suitably handling inconsistent fuzzy knowl-
edge bases that are represented by if-ALC . Further-
more, in this study, an inconsistency-tolerant fuzzy
temporal next-time description logic, itf-A L C , is ob-
tained from if-ALC by adding the temporal next-time
operator that was originally introduced by Prior in
(Prior, A.N., 1957; Prior, A.N., 1967). Similar re-
sults as those for if-A L C are also obtained for itf-
ALC . Thus, we can also reuse the previously devel-
oped algorithms and methods for f-ALC in appropri-
ately handling inconsistent temporal fuzzy knowledge
bases that are described by itf-ALC .
As argued by Straccia in (Straccia, 1997a), com-
bining an inconsistency-tolerant (or paraconsistent)
logic with a fuzzy logic is important for handling
inconsistent vague information. For this purpose,
a four-valued (inconsistency-tolerant) fuzzy proposi-
tional logic, which is a combination of a four-valued
logic and a fuzzy propositional logic, was introduced
by Straccia in (Straccia, 1997a). It was shown in
(Straccia, 1997a) that this logic can effectively handle
both inconsistent and vague predicates with suitable
computational properties. Furthermore, in another
paper (Straccia, 2000), a four-valued (inconsistency-
tolerant) fuzzy description logic was introduced by
Kamide, N.
Reasoning with Inconsistency-tolerant Fuzzy Description Logics.
DOI: 10.5220/0010771200003116
In Proceedings of the 14th International Conference on Agents and Artificial Intelligence (ICAART 2022) - Volume 3, pages 63-74
ISBN: 978-989-758-547-0; ISSN: 2184-433X
Copyright
c
2022 by SCITEPRESS Science and Technology Publications, Lda. All rights reserved
63
Straccia to obtain a logical framework for multime-
dia information retrieval. This four-valued fuzzy de-
scription logic is essentially an extension of both the
fuzzy description logic f-ALC and a four-valued de-
scription logic that was also developed by Straccia in
another paper (Straccia, 1997b). In (Straccia, 2000), a
sequent calculus for the four-valued fuzzy description
logic was introduced, and the completeness theorem
with respect to this sequent calculus was proved. The
validity problem for the four-valued fuzzy descrip-
tion logic was also shown to be decidable in (Strac-
cia, 2000). The aim of this study is to advance, from
a purely logical point of view, the ideas proposed
by Straccia for combining an inconsistency-tolerant
logic with a fuzzy description logic. The proposed
logic if-ALC is a combination and extension of f-
ALC and an inconsistency-tolerant (or paraconsis-
tent) description logic, S ALC , which was introduced
in (Kamide, 2013).
The difference between the proposed logic if-
ALC and the four-valued fuzzy description logic pro-
posed by Straccia (Straccia, 2000) is explained as fol-
lows. Although these logics are essentially equiva-
lent, the main formal difference is that if-ALC has
a paraconsistent negation connective, but Straccia’s
logic has no such paraconsistent negation connective.
Using the paraconsistent negation connective in if-
ALC entails the following merits: (1) the inconsis-
tency in if-ALC can be handled explicitly by using
the paraconsistent negation connective (i.e., the no-
tion of paraconsistency is formally defined with re-
spect to the paraconsistent negation connective) and
(2) it is useful for handling some practical applica-
tions (e.g., as presented in (Wagner, 1991), a database
needs two kinds of negations including a paraconsis-
tent one). Another formal difference is that Straccia’s
logic uses two kinds of fuzzy valuations denoted as
| · |
t
and |· |
f
, but if-ALC has a single fuzzy valuation
that just coincides with a fuzzy interpretation func-
tion. This simplification of the fuzzy interpretation
function in if-A L C entails the following theoretical
merits: (1) we can show a theorem for embedding if-
ALC into f-A L C , (2) we can show the relative de-
cidability of if-ALC with respect to f-ALC , and (3)
we can obtain the temporal extension itf-A L C of if-
ALC . These theoretical results, which were not ob-
tained for Straccia’s logic, are the main contribution
of this study.
To clarify the construction of our proposed logic
if-ALC , we address a brief survey of closely re-
lated description logics as follows. Description log-
ics (Baader et al., 2003) are well-known to be a
family of logic-based knowledge representation for-
malisms with applications in various fields, such
as the field of developing web ontology languages.
A number of useful description logics including a
standard description logic, A L C (Schmidt-Schauss
and Smolka, 1991), have been studied by many re-
searchers. Inconsistency-tolerant (or paraconsistent)
description logics and their neighbors have been stud-
ied by several researchers ((Ma et al., 2007; Ma et al.,
2008; Meghini and Straccia, 1996; Meghini et al.,
1998; Odintsov, S.P. and Wansing, 2003; Odintsov,
S.P. and Wansing, 2008; Patel-Schneider, Peter F.,
1989; Straccia, 1997b; Kaneiwa, 2007; Zhang and
Lin, 2008; Zhang et al., 2009; Kamide, 2011;
Kamide, 2012; Kamide, 2013; Kamide, 2020a)) to
cope with inconsistencies that frequently occur in the
real world. The inconsistency-tolerant description
logic S ALC (Kamide, 2013), which is a base logic
for if-ALC , is an extension and combination of both
ALC and Belnap and Dunn’s four-valued logic (also
referred to as a first degree entailment logic) (Belnap,
1977b; Belnap, 1977a; Dunn J.M., 1976). For a sur-
vey of inconsistency-tolerant description logics, see
(Kamide, 2013). On the other hand, fuzzy descrip-
tion logics and their applications have extensively
been studied by many researchers (e.g., (Yen, 1991;
Tresp and Molitor, 2018; Straccia, 1997a; H
´
ajek,
2005; Stoilos et al., 2007; Jiang et al., 2010; Strac-
cia, 2015; Baader et al., 2015; Baader et al., 2017;
Bobillo and Straccia, 2017; Borgwardt and Penaloza,
2017; Kamide, 2020b)) to deal with fuzzy knowledge
bases. The fuzzy description logic f-ALC introduced
in (Straccia, 2001), which was, however, not referred
to as f-ALC in the original paper, is regarded as a
result of combining (or integrating) the standard de-
scription logic ALC with Zadeh fuzzy logic, which is
based on the idea of fuzzy sets by Zadeh (Zadeh, L.A.,
1965). For a comprehensive survey of fuzzy descrip-
tion logics and their applications, see (Straccia, 2015;
Borgwardt and Penaloza, 2017).
The contents of this paper are presented as fol-
lows. In Section 2, f-A L C (Straccia, 2001) is ad-
dressed to develop if-ALC . In Section 3, if-ALC
is obtained from f-ALC by adding the paraconsis-
tent negation connective and a translation from if-
ALC to f-ALC is defined. A theorem for embed-
ding if-ALC into f-ALC is proven via this transla-
tion. A relative decidability theorem for if-ALC w.r.t.
f-ALC is also proven via this embedding theorem.
Furthermore, we show a relative complexity theorem
for if-ALC w.r.t. f-ALC . This relative complexity
theorem shows that if the underlying decision prob-
lem for if-ALC is decidable, then the complexity of
the decision procedure of the problem for if-ALC is
the same as the complexity of the decision procedure
of the counterpart problem for f-A L C . In Section 4,
ICAART 2022 - 14th International Conference on Agents and Artificial Intelligence
64
itf-ALC is obtained from if-ALC by adding the tem-
poral next-time operator X and a translation from itf-
ALC to if-ALC is defined. A theorem for embedding
itf-ALC into if-ALC is proven via this translation.
The relative decidability and complexity theorems for
itf-ALC w.r.t. if-A L C are proven via this embed-
ding theorem. Similarly, a theorem for embedding
itf-ALC into f-ALC and the relative decidability and
complexity theorems for itf-ALC w.r.t. f-ALC are
proven. In Section 5, this paper is concluded with
some remarks.
2 PRELIMINARIES: FUZZY
DESCRIPTION LOGIC
We introduce the fuzzy description logic f-ALC . The
concepts of f-A L C are constructed from atomic con-
cepts and roles by u (intersection), t (union), ¬ (com-
plement), R (universal concept quantification), and
R (existential concept quantification). We use the
letter A to denote atomic concepts, the letter R to de-
note roles, the letters C and D to denote concepts, and
the letters a and b to denote individual names. We use
the symbol to represent the equality of symbols.
The definition of fuzzy sets is as follows.
Definition 2.1. A fuzzy set S w.r.t. a universe U is
characterized by a mapping (referred to as member-
ship function) µ
S
: U[0, 1] where [0,1] is the closed
real unit interval. The membership function satisfies
the following restrictions: For any u U and any
fuzzy sets S
1
and S
2
with respect to U:
1. µ
S
1
(u) := 1 µ
S
1
(u),
2. µ
S
1
S
2
(u) := min{µ
S
1
(u),µ
S
2
(u)},
3. µ
S
1
S
2
(u) := max{µ
S
1
(u),µ
S
2
(u)}.
where S
1
is a fuzzy complement of S
1
in U , S
1
S
2
denotes a fuzzy set intersection of S
1
and S
2
, and S
1
S
2
denotes a fuzzy set union of S
1
and S
2
.
The definition of f-ALC is as follows.
Definition 2.2. A fuzzy interpretation (or fuzzy
model) I for f-ALC is a pair h
I
,·
I
i such that
1.
I
is a non-empty set as for the crisp case,
2. ·
I
is a fuzzy interpretation function such that
(a) for any individual names a and b, we have
a
I
,b
I
I
such that a
I
6= b
I
if a 6= b,
(b) for any atomic concepts A, we have A
I
:
I
[0,1],
(c) for any roles R, we have R
I
:
I
×
I
[0,1].
The fuzzy interpretation function is inductively ex-
tended to concepts by the following clauses: for any
d
I
,
1. (¬C)
I
(d) := 1 C
I
(d),
2. (C u D)
I
(d) := min{C
I
(d), D
I
(d)},
3. (C t D)
I
(d) := max{C
I
(d), D
I
(d)},
4. (R.C)
I
(d) := in f
d
0
I
{max{1 R
I
(d, d
0
),C
I
(d
0
)}},
5. (R.C)
I
(d) := sup
d
0
I
{min{R
I
(d, d
0
),C
I
(d
0
)}}.
Remark 2.3. The fuzzy interpretation defined in Def-
inition 2.2 is constructed on the basis of Zadeh logic.
We can consider other fuzzy interpretation functions
that are based on the following fuzzy logics: G
¨
odel
logic, Łukasiewicz logic, and product logic. But, we
do not discuss such interpretations in this study.
The definition of fuzzy assertion is as follows.
Definition 2.4. A crisp assertion (denoted by α) is an
expression of the form a : C or (a, b) : R where a and
b are individual names, C is a concept, and R is a
role. A crisp primitive assertion is either a crisp as-
sertion of the form a : A where A is an atomic con-
cept, or a crisp assertion of the form (a, b) : R where
R is a role. A fuzzy assertion (denoted as ψ) is an ex-
pression of the form hα ni or hα mi where α is
a crisp assertion, n (0, 1], and m [0, 1). A fuzzy
ABox is a finite set of fuzzy assertions. We frequently
use some similar expressions of the form hα > ni or
hα < ni. A fuzzy interpretation I satisfies a fuzzy as-
sertion ha : C ni or h(a, b) : R ni iff C
I
(a
I
) n
or R
I
(a
I
,b
I
) n, respectively. Similarly, a fuzzy in-
terpretation I satisfies a fuzzy assertion ha : C mi
or h(a, b) : R mi iff C
I
(a
I
) m or R
I
(a
I
,b
I
) m,
respectively. A fuzzy primitive assertion is a fuzzy as-
sertion involving a primitive assertion.
We use an expression I |= ψ to denote the fact that
a fuzzy interpretation I satisfies a fuzzy assertion ψ.
The definition of fuzzy terminological axiom is as
follows.
Definition 2.5. A fuzzy general concept inclusion is
an expression of the form C D where C and D are
concepts. A fuzzy interpretation I satisfies a fuzzy
general concept inclusion C D iff d
I
[C
I
(d)
D
I
(d)]. A fuzzy concept specialization is a restricted
fuzzy general concept inclusion of the form A C
where A is an atomic concept and C is a concept.
A fuzzy concept equivalence is an expression of the
form C : D where C and C are concepts. A fuzzy
interpretation I satisfies a fuzzy concept equivalence
C : D iff d
I
[C
I
(d) = D
I
(d)]. A fuzzy concept
definition is a restricted fuzzy concept equivalence of
the form A : C where A is an atomic concept and C
is a concept. A fuzzy terminological axiom (denoted
by υ) is either a fuzzy general concept inclusion or a
fuzzy concept equivalence. A fuzzy TBox is a finite
set of fuzzy terminological axioms.
Reasoning with Inconsistency-tolerant Fuzzy Description Logics
65
We use an expression I |= υ to denote the fact that
a fuzzy interpretation I satisfies a fuzzy terminologi-
cal axiom υ.
Remark 2.6. It was assumed in (Straccia, 2001) that
there is no cycle definition in a fuzzy TBox and there
is no fuzzy general concept inclusion in a fuzzy TBox
(i.e., atomic concept appears more than once on the
left hand side of a fuzzy terminological axiom).
The definitions of fuzzy knowledge base, fuzzy
entailment, and fuzzy subsumption are as follows.
Definition 2.7. A fuzzy knowledge base is a pair of a
fuzzy ABox and a fuzzy TBox. Let Σ be a fuzzy knowl-
edge base. We use an expression Σ
A
to denote the set
of assertions in Σ and we use an expression Σ
T
to de-
note the set of terminological axions in Σ. A fuzzy
interpretation I satisfies a fuzzy knowledge base Σ
iff I satisfies each element of Σ. A fuzzy knowledge
base Σ fuzzy entails a fuzzy assertion ψ (denoted by
Σ |= ψ) iff every fuzzy interpretation of Σ also satis-
fies ψ. A concept D fuzzy subsumes a concept C with
respect to a set Σ
T
of terminological axioms (denoted
by C
Σ
T
D) iff for every fuzzy interpretation I of Σ
T
,
d
I
[C
I
(d) D
I
(d)] holds.
We use an expression I |= Σ to denote the fact that
a fuzzy interpretation I satisfies a fuzzy knowledge
base Σ.
Definition 2.8. A fuzzy assertion ψ is called satisfi-
able in f-A L C iff it has a fuzzy interpretation I such
that I |= ψ. A fuzzy terminological axiom υ is called
satisfiable in f-A LC iff it has a fuzzy interpretation I
such that I |= υ. A fuzzy knowledge base Σ is called
satisfiable in f-A LC iff it has a fuzzy interpretation I
such that I |= γ for every element γ of Σ.
3 INCONSISTENCY-TOLERANT
FUZZY DESCRIPTION LOGIC
We introduce an inconsistency-tolerant fuzzy descrip-
tion logic, if-ALC . We also use the same notions and
terminologies for if-ALC as those for f-ALC . The
concepts of if-A L C are obtained from the concepts
of f-ALC by adding (paraconsistent negation).
The definition of if-ALC is as follows.
Definition 3.1. An inconsistency-tolerant fuzzy inter-
pretation (or inconsistency-tolerant fuzzy model) I I
for if-ALC is a pair h
I I
,·
I I
i where
1.
I I
is a non-empty set as for the crisp case,
2. ·
I I
is an inconsistency-tolerant fuzzy interpreta-
tion function where
(a) for any individual names a and b, we have
a
I I
,b
I I
I I
such that a
I I
6= b
I I
if a 6= b,
(b) for any atomic concepts A, we have A
I I
:
I I
[0,1],
(c) for any negated atomic concepts A, we have
(A)
I I
:
I I
[0,1],
(d) for any roles R, we have R
I I
:
I I
×
I I
[0,1].
The inconsistency-tolerant fuzzy interpretation
function is inductively extended to concepts by the fol-
lowing clauses: For any d
I I
,
1. (¬C)
I I
(d) := 1 C
I I
(d),
2. (C u D)
I I
(d) := min{C
I I
(d), D
I I
(d)},
3. (C t D)
I I
(d) := max{C
I I
(d), D
I I
(d)},
4. (R.C)
I I
(d) := in f
d
0
I I
{max{1 R
I I
(d, d
0
),C
I I
(d
0
)}},
5. (R.C)
I I
(d) := sup
d
0
I I
{min{R
I I
(d, d
0
),C
I I
(d
0
)}},
6. (∼∼C)
I I
(d) := C
I I
(d),
7. (∼¬C)
I I
(d) := 1 (C)
I I
(d),
8. ((C u D))
I I
(d) := max{(C)
I I
(d), (D)
I I
(d)},
9. ((C t D))
I I
(d) := min{(C)
I I
(d), (D)
I I
(d)},
10. (∼∀R.C)
I I
(d) := sup
d
0
I I
{min{R
I I
(d, d
0
),(C)
I I
(d
0
)}},
11. (∼∃R.C)
I I
(d) := in f
d
0
I I
{max{1 R
I I
(d, d
0
),(C)
I I
(d
0
)}}.
The following proposition shows that some prop-
erties concerning hold for if-ALC .
Proposition 3.2. For any concepts C and D and
any inconsistency-tolerant fuzzy interpretation func-
tion ·
I I
on if-ALC , we can obtain the following con-
ditions w.r.t. :
1. (∼∼C)
I I
(d) = C
I I
(d),
2. (∼¬C)
I I
(d) = (¬∼C)
I I
(d),
3. ((C u D))
I I
(d) = ((C) t (D))
I I
(d),
4. ((C t D))
I I
(d) = ((C) u (D))
I I
(d),
5. (∼∀R.C)
I I
(d) = (R.C)
I I
(d),
6. (∼∃R.C)
I I
(d) = (R.C)
I I
(d).
Proof. Straightforward by Definition 3.1.
Remark 3.3. We make the following remarks.
1. Intuitively speaking, if-ALC is extended based on
the following additional axiom schemes for :
(a) ∼∼C C,
(b) ∼¬C ¬∼C,
(c) (C u D) C t D,
(d) (C t D) C u D,
(e) (R.C) R.C,
(f) (R.C) R.C.
2. In general, a semantic consequence relation |= is
said to be paraconsistent with respect to a nega-
tion connective if there are formulas α and
β such that {α, α} 6|= β. We now consider
ICAART 2022 - 14th International Conference on Agents and Artificial Intelligence
66
the case of if-ALC . Assume an inconsistency-
tolerant fuzzy interpretation I I = h
I I
,·
I I
i such
that for a pair of distinct atomic concepts A and
B, A
I I
(d) = m, (A)
I I
(d) = m, and B
I I
(d) = n
where d
I I
and m > n with m,n [0,1]. Then,
(A u A)
I I
(d) B
I I
(d) does not hold. Thus, if-
ALC is regarded as paraconsistent with respect
to . Note that if-ALC is not paraconsistent with
respect to ¬.
Some notions on if-ALC are defined as follows.
Definition 3.4. The notions of inconsistency-tolerant
fuzzy assertion, inconsistency-tolerant fuzzy prim-
itive assertion, inconsistency-tolerant fuzzy general
concept inclusion, inconsistency-tolerant fuzzy con-
cept specialization, inconsistency-tolerant fuzzy con-
cept equivalence, inconsistency-tolerant fuzzy con-
cept definition, inconsistency-tolerant fuzzy ABox,
inconsistency-tolerant fuzzy terminological axiom,
inconsistency-tolerant fuzzy TBox and inconsistency-
tolerant fuzzy knowledge base are defined in the same
manner as those in Definitions 2.4, 2.5, and 2.7. The
notion of the satisfiability of inconsistency-tolerant
fuzzy knowledge base is defined in the same manner
as that of defined in Definition 2.8.
The same notations that are introduced in Defini-
tions 2.4, 2.5, and 2.7 are also used for those of if-
ALC .
Next, we introduce a translation from if-ALC into
f-ALC . Using this translation, we prove a theorem
for embedding if-ALC into f-ALC . Using this em-
bedding theorem, we prove a relative decidability the-
orem for if-ALC with respect to f-ALC .
We use the symbol N
C
to be a non-empty set of
atomic concepts, the symbol N
0
C
to be the set {A
0
| A
N
C
} of atomic concepts, the symbol N
R
to be a non-
empty set of roles, and the symbol N
I
to be a non-
empty set of individual names.
The definition of the translation form if-ALC to
f-ALC is as follows.
Definition 3.5. The language L
n
of if-ALC is de-
fined using N
C
, N
R
, N
I
, , ¬, u, t, R, and R. The
language L of f-ALC is obtained from L
n
by adding
N
0
C
and deleting . A mapping f from L
n
to L is
defined inductively by:
1. for any R N
R
and any a N
I
, f (R) := R and
f (a) := a,
2. for any A N
C
, f (A) := A and f (A) := A
0
N
0
C
,
3. f (¬C) := ¬ f (C),
4. f (C ] D) := f (C) ] f (D) where ] {u, t},
5. f (]R.C) := ] f (R). f (C) = ]R. f (C) where ] {∀, ∃},
6. f (∼∼C) := f (C),
7. f (∼¬C) := ¬ f (C),
8. f ((C u D)) := f (C) t f (D),
9. f ((C t D)) := f (C) u f (D),
10. f (∼∀R.C) := f (R). f (C) = R. f (C),
11. f (∼∃R.C) := f (R). f (C) = R. f (C).
Analogous notations and expressions like f (ψ),
f (τ), and f (Σ) are also adopted for an inconsistency-
tolerant fuzzy assertion ψ, an inconsistency-tolerant
fuzzy terminological axiom τ, and an inconsistency-
tolerant fuzzy knowledge base Σ, respectively. For
example, we use an expression f (Σ) to denote the re-
sult of replacing every occurrence of a concept (or a
role) β in Σ by an occurrence of f (β). We also use
analogous notations for the other mappings discussed
later.
Remark 3.6. The translation function introduced
above is a modification of the translation function
introduced by Ma et al. (Ma et al., 2007) to em-
bed A L C 4 into A L C . A similar translation function
has been used by Gurevich (Gurevich, 1977), Raut-
enberg (Rautenberg, 1979) and Vorob’ev (Vorob’ev,
N.N, 1952) to embed Nelson’s constructive logic (Al-
mukdad and Nelson, 1984; Nelson, 1949) into intu-
itionistic logic. Some similar translations have also
been used, for example, in (Kamide and Shramko,
2017; Kamide and Zohar, 2019) to embed some para-
consistent logics into classical logic.
Prior to show the embedding theorem, we need to
prove some lemmas.
Lemma 3.7. Let f be the mapping defined in Def-
inition 3.5. For any inconsistency-tolerant fuzzy in-
terpretation I I := h
I I
,·
I I
i of if-ALC , we can con-
struct a fuzzy interpretation I := h
I
,·
I
i of f-ALC
such that for any concept C in L
n
, C
I I
= f (C)
I
.
Proof. Let I I be an inconsistency-tolerant fuzzy in-
terpretation h
I I
,·
I I
i. We now construct a fuzzy in-
terpretation I := h
I
,·
I
i such that
1.
I
is a non-empty set such that
I
=
I I
,
2. ·
I
is a fuzzy interpretation function where
(a) R
I
= R
I I
and a
I
= a
I I
for any R N
R
and any a N
I
.
(b) A
I
= A
I I
for any atomic concept A N
C
.
(c) (A
0
)
I
= (A)
I I
for any atomic concept A
0
N
0
C
.
This lemma is then proved by induction on the
complexity of C.
Base step:
1. Case C A N
C
: We obtain: A
I I
= A
I
= f (A)
I
(by the definition of f ).
2. Case C A with A N
C
: We obtain: (A)
I I
=
(A
0
)
I
= f (A)
I
(by the definition of f ).
Induction step: We show some cases.
Reasoning with Inconsistency-tolerant Fuzzy Description Logics
67
1. Case C ¬D: We obtain:
(¬D)
I I
(d)
= 1 D
I I
(d)
= 1 f (D)
I
(d) (by induction hypothesis)
= (¬ f (D))
I
(d)
= f (¬D)
I
(d) (by the definition of f ).
2. Case C C
1
uC
2
: We obtain:
(C
1
uC
2
)
I I
(d)
= min{C
I I
1
(d),C
I I
2
(d)}
= min{ f (C
1
)
I
(d), f (C
2
)
I
(d)} (by induction hy-
pothesis)
= ( f (C
1
) u f (C
2
))
I
(d)
= f (C
1
uC
2
)
I
(d) (by the definition of f ).
3. Case C R.D: We obtain:
(R.D)
I I
(d)
= in f
d
0
I I
{max{1 R
I I
(d, d
0
),D
I I
(d
0
)}}
= in f
d
0
I I
{max{1 R
I I
(d, d
0
), f (D)
I
(d
0
)}}
(by induction hypothesis)
= in f
d
0
I
{max{1 R
I
(d, d
0
), f (D)
I
(d
0
)}} (by
I I
=
I
and R
I I
= R
I
)
= (R. f (D))
I
(d)
= f (R.D)
I
(d) (by the definition of f ).
4. Case C ∼∼D: We obtain:
(∼∼D)
I I
(d)
= D
I I
(d)
= f (D)
I
(d) (by induction hypothesis)
= f (∼∼D)
I
(d) (by the definition of f ).
5. Case C ∼¬D: We obtain:
(∼¬D)
I I
(d)
= 1 (D)
I I
(d)
= 1 f (D)
I
(d) (by induction hypothesis)
= (¬ f (D))
I
(d)
= f (∼¬D)
I
(d) (by the definition of f ).
6. Case C (C
1
uC
2
): We obtain:
((C
1
uC
2
))
I I
(d)
= max{(C
1
)
I I
(d), (C
2
)
I I
(d)}
= max{ f (C
1
)
I
(d), f (C
2
)
I
(d)} (by induction
hypothesis)
= ( f (C
1
) t f (C
2
))
I
(d)
= f ((C
1
uC
2
))
I
(d) (by the definition of f ).
7. Case C ∼∀R.D: We obtain:
(∼∀R.D)
I I
(d)
= sup
d
0
I I
{min{R
I I
(d, d
0
),(D)
I I
(d
0
)}}
= sup
d
0
I I
{min{R
I I
(d, d
0
), f (D)
I
(d
0
)}} (by
induction hypothesis)
= sup
d
0
I
{min{R
I
(d, d
0
), f (D)
I
(d
0
)}} (by
I I
=
I
and R
I I
= R
I
)
= (R. f (D))
I
(d)
= f (∼∀R.D)
I
(d) (by the definition of f ).
Lemma 3.8. Let f be the mapping defined in Def-
inition 3.5. For any inconsistency-tolerant fuzzy in-
terpretation I I := h
I I
,·
I I
i of if-ALC , we can con-
struct a fuzzy interpretation I := h
I
,·
I
i of f-ALC
such that for any inconsistency-tolerant fuzzy asser-
tion ψ in L
n
, I I |= ψ iff I |= f (ψ).
Proof. By using Lemma 3.7.
Lemma 3.9. Let f be the mapping defined in Defi-
nition 3.5. For any fuzzy interpretation I := h
I
,·
I
i
of f-A L C , we can construct an inconsistency-tolerant
fuzzy interpretation I I := h
I I
,·
I I
i of if-A L C such
that for any inconsistency-tolerant fuzzy assertion ψ
in L
n
, I I |= ψ iff I |= f (ψ).
Proof. Similar to the proof of Lemma 3.8.
The theorem for embedding if-ALC into f-ALC
is presented as follows.
Theorem 3.10. Let f be the mapping defined in Defi-
nition 3.5. For any inconsistency-tolerant fuzzy asser-
tion ψ, we have: ψ is satisfiable in if-ALC iff f (ψ)
is satisfiable in f-ALC .
Proof. By using Lemmas 3.8 and 3.9.
The theorem for relative decidability of if-ALC
w.r.t. f-ALC is presented as follows.
Theorem 3.11. If the satisfiability problem of a fuzzy
knowledge base (with or without some modifications
or restrictions) for f-ALC is decidable, then the sat-
isfiability problem of the counterpart inconsistency-
tolerant fuzzy knowledge base for if-ALC is also
decidable. Similarly, if the fuzzy entailment prob-
lem for f-ALC and the fuzzy subsumption prob-
lem for f-A L C are decidable, then the counter-
part inconsistency-tolerant entailment problem for if-
ALC and the counterpart inconsistency-tolerant sub-
sumption problem for if-ALC are also decidable.
Proof. Using Definition 3.5 and Theorem 3.10, we
can transform the problems of if-ALC into those of f-
ALC . Thus, if the problems of f-ALC are decidable,
then those of if-ALC are also decidable.
The theorem for relative complexity of if-A LC
w.r.t. f-ALC is presented as follows.
Theorem 3.12. If a decision problem for if-ALC
(e.g., one of the decision problems that are addressed
in Theorem 3.11) is decidable, then the complexity of
the decision problem of if-ALC is the same as that of
the counterpart decision problem of f-ALC .
Proof. By using Definition 3.5, Theorem 3.10, and
the fact that the mapping f defined in Definition 3.5
is a polynomial-time reduction.
ICAART 2022 - 14th International Conference on Agents and Artificial Intelligence
68
Remark 3.13. It is known that some significant deci-
sion problems for fuzzy description logics (especially
with some general concept inclusions) have not yet
been solved. It is also known that if some general con-
cept inclusions are available in some fuzzy descrip-
tion logics, then the corresponding decision problems
for the fuzzy description logics are undecidable. For
more information on the undecidability, decidability,
and complexity of fuzzy description logics, see, for ex-
ample, (Baader et al., 2015; Baader et al., 2017) and
the references therein. If some open decision prob-
lems for fuzzy description logics will be solved, then
Theorems 3.11 and 3.12 will be useful.
4 INCONSISTENCY-TOLERANT
TEMPORAL NEXT-TIME
FUZZY DESCRIPTION LOGIC
We introduce an inconsistency-tolerant temporal
next-time fuzzy description logic, itf-A L C . For itf-
ALC , we use the same terminologies and notions as
those for f-ALC and if-A L C . The concepts of itf-
ALC are obtained from the concepts of if-ALC by
adding X (next-time operator). We use the symbol
ω to represent the set of natural numbers. We in-
ductively define an expression X
n
C with n ω by
X
0
C := C and X
n+1
C := XX
n
C.
The definition of itf-ALC is as follows.
Definition 4.1. An inconsistency-tolerant tempo-
ral next-time fuzzy interpretation (or inconsistency-
tolerant temporal next-time fuzzy model) I T I for itf-
ALC is a pair h
I T I
,
I
i
}
iω
i where
1.
I T I
is a non-empty set as for the crisp case,
2. for any i ω, ·
I
i
is an inconsistency-tolerant tem-
poral next-time fuzzy interpretation function such
that
(a) for any individual names a and b, we have
a
I
i
,b
I
i
I T I
such that a
I
i
6= b
I
i
if a 6= b,
(b) for any atomic concepts A, we have A
I
i
:
I T I
[0,1],
(c) for any negated atomic concepts A, we have
(A)
I
i
:
I T I
[0,1],
(d) for any roles R, we have R
I
i
:
I T I
×
I T I
[0,1],
3. for any individual name a, any role R, and any
j, k ω, we have a
I
j
= a
I
k
and R
I
j
= R
I
k
.
The inconsistency-tolerant fuzzy temporal next-
time interpretation functions are inductively extended
to concepts by the following clauses: For any d
I T I
and any i ω,
1. (XC)
I
i
(d) := C
I
i+1
(d),
2. (¬C)
I
i
(d) := 1 C
I
i
(d),
3. (C u D)
I
i
(d) := min{C
I
i
(d), D
I
i
(d)},
4. (C t D)
I
i
(d) := max{C
I
i
(d), D
I
i
(d)},
5. (R.C)
I
i
(d) := in f
d
0
I T I
{max{1 R
I
i
(d, d
0
),C
I
i
(d
0
)}},
6. (R.C)
I
i
(d) := sup
d
0
I T I
{min{R
I
i
(d, d
0
),C
I
i
(d
0
)}},
7. (XC)
I
i
(d) := (C)
I
i+1
(d),
8. (∼∼C)
I
i
(d) := C
I
i
(d),
9. (∼¬C)
I
i
(d) := 1 (C)
I
i
(d),
10. ((C u D))
I
i
(d) := max{(C)
I
i
(d), (D)
I
i
(d)},
11. ((C t D))
I
i
(d) := min{(C)
I
i
(d), (D)
I
i
(d)},
12. (∼∀R.C)
I
i
(d) := sup
d
0
I T I
{min{R
I
i
(d, d
0
),(C)
I
i
(d
0
)}},
13. (∼∃R.C)
I
i
(d) := in f
d
0
I T I
{max{1 R
I
i
(d, d
0
),(C)
I
i
(d
0
)}}.
The following proposition shows that some
clauses concerning X hold for itf-ALC .
Proposition 4.2. For any concepts C and D and any
inconsistency-tolerant temporal next-time fuzzy inter-
pretation function ·
I T I
on itf-ALC , we can obtain the
following clauses with respect to X:
1. (X]C)
I T I
(d) = (]XC)
I T I
(d) where ] {∼, ¬},
2. (X(C ] D))
I T I
(d) = ((XC) ] (XD))
I T I
(d) where
] {u, t},
3. (X]R.C)
I T I
(d) = (]R.XC)
I T I
(d) where ]
{∀,∃}.
Proof. Straightforward by Definition 4.1.
Remark 4.3. We make the following remarks.
1. Intuitively speaking, itf-A L C is extended based
on the following additional axiom schemes for X:
(a) X]C ]XC where ] {∼, ¬},
(b) X(C]D) (XC)](XD) where ] {u, t},
(c) X(]R.C) ]R.(XC) where ] {∀, ∃}.
2. itf-ALC is an extension of if-A L C , because the
zero interpretation function ·
I
0
includes ·
I I
.
3. itf-ALC uses the following constant domain as-
sumption: It has the single common domain
I T I
.
4. itf-ALC uses the following rigid role and name
assumption: For any role R, any individual name
a and any i, j ω, we have R
I
i
= R
I
j
and a
I
i
=
a
I
j
.
5. The temporal fragment tf-ALC of itf-ALC can be
obtained from itf-ALC by deleting all the parts
concerning .
Similar notions and notations as those defined in
Definitions 2.4, 2.5, and 2.7 are also used for those of
itf-ALC . However, the notions of satisfiability need
some specific definitions. These notions are defined
as follows.
Reasoning with Inconsistency-tolerant Fuzzy Description Logics
69
Definition 4.4. An inconsistency-tolerant tempo-
ral next-time fuzzy interpretation function I
i
sat-
isfies an inconsistency-tolerant temporal next-time
fuzzy assertion ha : C ni or h(a,b) : R ni
iff C
I
i
(a
I
i
) n or R
I
i
(a
I
i
,b
I
i
) n, respectively.
Similarly, an inconsistency-tolerant temporal next-
time fuzzy interpretation function I
i
satisfies an
inconsistency-tolerant temporal next-time fuzzy as-
sertion ha : C mi or h(a,b) : R mi iff C
I
i
(a
I
i
)
m or R
I
i
(a
I
i
,b
I
i
) m, respectively. Moreover, an
inconsistency-tolerant temporal next-time fuzzy in-
terpretation I T I satisfies an inconsistency-tolerant
temporal next-time fuzzy assertion ha : C ni or
h(a,b) : R ni iff C
I
0
(a
I
0
) n or R
I
0
(a
I
0
,b
I
0
) n,
respectively. Similarly, an inconsistency-tolerant tem-
poral next-time fuzzy interpretation I T I satisfies an
inconsistency-tolerant temporal next-time fuzzy as-
sertion ha : C mi or h(a, b) : R mi iff C
I
0
(a
I
0
) m
or R
I
0
(a
I
0
,b
I
0
) m, respectively.
We use an expression I
i
|= ψ to denote the
fact that an inconsistency-tolerant temporal next-
time fuzzy interpretation function I
i
satisfies an
inconsistency-tolerant temporal next-time fuzzy as-
sertion ψ. We also use an expression I T I |= ψ to
denote the fact that an inconsistency-tolerant tempo-
ral next-time fuzzy interpretation I T I satisfies an
inconsistency-tolerant temporal next-time fuzzy as-
sertion ψ. Note that I T I |= ψ is equivalent to I
0
|= ψ.
Next, we introduce a translation from itf-ALC to
if-ALC . Using this translation, we prove a theorem
for embedding itf-A L C into if-A L C . Using this em-
bedding theorem, we prove a relative decidability the-
orem for itf-ALC w.r.t. if-A L C .
We use the same symbols N
C
, N
R
, and N
I
as those
used for if-ALC . Furthermore, we use the new sym-
bol N
i
C
to denote the set {A
i
| A N
C
} of atomic con-
cepts where A
0
= A (i.e., N
0
C
= N
C
).
The definition of the translation from itf-ALC to
if-ALC is as follows.
Definition 4.5. The language L
t
of itf-ALC is de-
fined using N
C
, N
R
, N
I
, , ¬, u, t, R, R, and X.
The language L
n
of if-ALC is obtained from L
t
by
adding
S
iω
N
i
C
and deleting X. A mapping g from L
t
to L
n
is defined inductively by:
1. for any R N
R
and any a N
I
, g(R) := R and g(a) := a,
2. for any A N
C
, g(X
i
A) := A
i
N
i
C
(esp. g(A) := A),
3. g(X
i
]C) := ]g(X
i
C) where ] , ∼},
4. g(X
i
(C ] D)) := g(X
i
C) ] g(X
i
D) where ] {u, t},
5. g(X
i
]R.C) := ]R.g(X
i
C) where ] {∀, ∃}.
Prior to prove the embedding theorem, we need to
show some lemmas.
Lemma 4.6. Let g be the mapping defined in
Definition 4.5. For any inconsistency-tolerant
temporal next-time fuzzy interpretation I T I :=
h
I T I
,
I
i
}
iω
i of itf-A L C , we can construct an
inconsistency-tolerant fuzzy interpretation I I :=
h
I I
,·
I I
i of if-A LC such that for any concept C in
L
t
and any i ω, C
I
i
= g(X
i
C)
I I
.
Proof. Suppose that I T I is an inconsistency-
tolerant temporal next-time fuzzy interpreta-
tion h
I T I
,
I
i
}
iω
i. Then, we construct an
inconsistency-tolerant fuzzy interpretation I I :=
h
I I
,·
I I
i where
1.
I I
is a non-empty set such that
I I
=
I T I
,
2. ·
I I
is an inconsistency-tolerant fuzzy interpreta-
tion function such that
(a) R
I I
= R
I
i
and a
I I
= a
I
i
for any R N
R
and any
a N
I
.
(b) (A
i
)
I I
= A
I
i
(esp. A
I I
= A
I
0
) for any atomic con-
cept A
i
N
i
C
.
(c) (A
i
)
I I
= (A)
I
i
(esp. (A)
I I
= (A)
I
0
) for any
negated atomic concept A
i
with A
i
N
i
C
.
Then, this lemma is proved by induction on the
complexity of C.
Base step:
1. Case C A N
C
: We obtain: (A)
I
i
= (A
i
)
I I
=
g(X
i
A)
I I
(by the definition of g).
2. Case C A with A N
C
: We obtain: (A)
I
i
=
(A
i
)
I I
= (g(X
i
A))
I I
(by the definition of g)
= g(X
i
A)
I I
(by the definition of g).
Induction step: We show some cases.
1. Case C XD: We obtain:
(XD)
I
i
(d)
= D
I
i+1
(d)
= g(X
i+1
D)
I I
(d) (by induction hypothesis)
= g(X
i
XD)
I I
(d).
2. Case C ¬D: We obtain:
(¬D)
I
i
(d)
= 1 D
I
i
(d)
= 1 g(X
i
D)
I I
(d) (by induction hypothesis)
= (¬g(X
i
D))
I I
(d)
= g(X
i
¬D)
I I
(d) (by the definition of g).
3. Case C C
1
uC
2
: We obtain:
(C
1
uC
2
)
I
i
(d)
= min{C
I
i
1
(d),C
I
i
2
(d)}
= min{g(X
i
C
1
)
I I
(d), g(X
i
C
2
)
I I
(d)} (by induc-
tion hypothesis)
ICAART 2022 - 14th International Conference on Agents and Artificial Intelligence
70
= (g(X
i
C
1
) u g(X
i
C
2
))
I I
(d)
= g(X
i
(C
1
uC
2
))
I I
(d) (by the definition of g).
4. Case C R.D: We obtain:
(R.D)
I
i
(d)
= in f
d
0
I T I
{max{1 R
I
i
(d, d
0
),D
I
i
(d
0
)}}
= in f
d
0
I T I
{max{1 R
I
i
(d, d
0
),g(X
i
D)
I I
(d
0
)}} (by
induction hypothesis)
= in f
d
0
I I
{max{1 R
I I
(d, d
0
),g(X
i
D)
I I
(d
0
)}}
(by
I T I
=
I I
and R
I
i
= R
I I
)
= (R.g(X
i
D))
I I
(d)
= g(X
i
R.D)
I I
(d) (by the definition of g).
5. Case C XD: We obtain:
(XD)
I
i
(d)
= (D)
I
i+1
(d)
= g(X
i+1
D)
I I
(d) (by induction hypothesis)
= (g(X
i+1
D))
I I
(d) (by the definition of g)
= (g(X
i
XD))
I I
(d)
= g(X
i
XD)
I I
(d) (by the definition of g).
6. Case C ∼∼D: We obtain:
(∼∼D)
I
i
(d)
= D
I
i
(d)
= g(X
i
D)
I I
(d) (by induction hypothesis)
= (∼∼g(X
i
D))
I I
(d)
= g(X
i
∼∼D)
I I
(d) (by the definition of g).
7. Case C ∼¬D: We obtain:
(∼¬D)
I
i
(d)
= 1 (D)
I
i
(d)
= 1 g(X
i
D)
I I
(d) (by induction hypothesis)
= (¬g(X
i
D))
I I
(d)
= (¬∼g(X
i
D))
I I
(d)
= (∼¬g(X
i
D))
I I
(d)
= g(X
i
∼¬D)
I I
(d) (by the definition of g).
8. Case C (C
1
uC
2
): We obtain:
((C
1
uC
2
))
I
i
(d)
= max{(C
1
)
I
i
(d), (C
2
)
I I
i
(d)}
= max{g(X
i
C
1
)
I I
(d), g(X
i
C
2
)
I I
(d)} (by in-
duction hypothesis)
= max{(g(X
i
C
1
))
I I
(d), (g(X
i
C
2
))
I I
(d)} (by
the definition of g)
= ((g(X
i
C
1
) u g(X
i
C
2
)))
I I
(d)
= (g(X
i
(C
1
uC
2
)))
I I
(d) (by the definition of g)
= g(X
i
(C
1
uC
2
))
I I
(d) (by the definition of g).
9. Case C ∼∀R.D: We obtain:
(X
i
∼∀R.D)
I
i
(d)
= sup
d
0
I T I
{min{R
I
i
(d, d
0
),(D)
I
i
(d
0
)}}
= sup
d
0
I T I
{min{R
I
i
(d, d
0
),g(X
i
D)
I I
(d
0
)}}
(by induction hypothesis)
= sup
d
0
I I
{min{R
I I
(d, d
0
),g(X
i
D)
I I
(d
0
)}}
(by
I T I
=
I I
and R
I
i
= R
I I
)
= sup
d
0
I I
{min{R
I I
(d, d
0
),(g(X
i
D))
I I
(d
0
)}}
(by the definition of g).
= (∼∀R.g(X
i
D))
I I
(d)
= (g(X
i
R.D))
I I
(d) (by the definition of g)
= g(X
i
∼∀R.D)
I I
(d) (by the definition of g).
We use an expression X
i
ψ to denote the fact that
the concept C appearing in an inconsistency-tolerant
temporal next-time fuzzy assertion ψ is replaced with
X
i
C.
Lemma 4.7. Let g be the mapping defined in
Definition 4.5. For any inconsistency-tolerant
temporal next-time fuzzy interpretation I T I :=
h
I T I
,
I
i
}
iω
i of itf-A L C , we can construct an
inconsistency-tolerant fuzzy interpretation I I :=
h
I I
,·
I I
i of if-A L C such that for any inconsistency-
tolerant temporal next-time fuzzy assertion ψ in L
t
and any i ω, I
i
|= ψ iff I I |= g(X
i
ψ).
Proof. By using Lemma 4.6.
Lemma 4.8. Let g be the mapping defined in Defini-
tion 4.5. For any inconsistency-tolerant fuzzy inter-
pretation I I := h
I I
,·
I I
i of if-ALC , we can con-
struct an inconsistency-tolerant temporal next-time
fuzzy interpretation I T I := h
I T I
,
I
i
}
iω
i of itf-
ALC such that for any inconsistency-tolerant tempo-
ral next-time fuzzy assertion ψ in L
t
and any i ω,
I
i
|= ψ iff I I |= g(X
i
ψ).
Proof. Similar to the proof of Lemma 4.7.
The theorem for embedding itf-ALC into if-ALC
is presented as follows.
Theorem 4.9. Let g be the mapping defined in Def-
inition 4.5. For any inconsistency-tolerant temporal
next-time fuzzy assertion ψ, ψ is satisfiable in itf-
ALC iff g(ψ) is satisfiable in if-A LC .
Proof. By using Lemmas 4.7 and 4.8.
The theorem for relative decidability of itf-AL C
w.r.t. if-AL C is presented as follows.
Theorem 4.10. If the satisfiability problem of an
inconsistency-tolerant fuzzy knowledge base (with
or without some modifications or restrictions) for
if-ALC is decidable, then the satisfiability prob-
lem of the counterpart inconsistency-tolerant tem-
poral next-time fuzzy knowledge base for itf-ALC
is also decidable. Similarly, if an inconsistency-
tolerant fuzzy entailment problem for if-ALC and
Reasoning with Inconsistency-tolerant Fuzzy Description Logics
71
an inconsistency-tolerant fuzzy subsumption prob-
lem for if-ALC are decidable, then the counterpart
inconsistency-tolerant temporal next-time fuzzy en-
tailment problem for itf-AL C and the counterpart
inconsistency-tolerant temporal next-time fuzzy sub-
sumption problem for itf-ALC are also decidable.
Proof. By Definition 4.5 and Theorem 4.9.
The theorem for relative complexity of itf-ALC
w.r.t. if-AL C is presented as follows.
Theorem 4.11. If a decision problem for itf-AL C is
decidable, then the complexity of the decision prob-
lem for itf-ALC is the same as that of the counterpart
decision problem for if-ALC .
Proof. By using Definition 4.5, Theorem 4.9, and the
fact that the mapping g defined in Definition 4.5 is a
polynomial-time reduction.
The theorem for embedding itf-ALC into f-ALC
is presented as follows.
Theorem 4.12. Let h be the composition g f of the
mappings f and g defined in Definitions 3.5 and
4.5. For any inconsistency-tolerant temporal next-
time fuzzy assertion ψ, ψ is satisfiable in itf-ALC
iff h(ψ) is satisfiable in f-ALC .
Proof. By combining Theorems 3.10 and 4.9.
The theorem for relative decidability of itf-AL C
w.r.t. f-AL C is presented as follows.
Theorem 4.13. If the satisfiability problem of a fuzzy
knowledge base (with or without some modifications
or restrictions) for f-ALC is decidable, then the sat-
isfiability problem of the counterpart inconsistency-
tolerant temporal next-time fuzzy knowledge base for
itf-ALC is also decidable. Similarly, if a fuzzy en-
tailment problem for f-ALC and a fuzzy subsumption
problem for f-ALC are decidable, then the counter-
part inconsistency-tolerant temporal next-time fuzzy
entailment problem for itf-ALC and the counterpart
inconsistency-tolerant temporal next-time fuzzy sub-
sumption problem for itf-ALC are also decidable.
Proof. By combining Theorems 3.11 and 4.10.
The theorem for relative complexity of itf-ALC
w.r.t. f-AL C is presented as follows.
Theorem 4.14. If a decision problem for itf-AL C is
decidable, then the complexity of the decision prob-
lem for itf-ALC is the same as that of the counterpart
decision problem for f-ALC .
Proof. By combining Theorems 3.12 and 4.11.
Remark 4.15. We make the following remarks.
1. We can introduce a translation function from tf-
ALC to f-AL C and a translation function from
itf-ALC to tf-ALC . By using these translation
functions, we can show the theorem for embed-
ding tf-ALC into f-ALC and the theorem for em-
bedding itf-ALC into tf-ALC .
2. Similarly, we can show the relative decidability
and complexity theorems for tf-ALC w.r.t. f-ALC
and the relative decidability and complexity theo-
rems for itf-A LC w.r.t. tf-A L C .
5 CONCLUDING REMARKS
In this study, we introduced the inconsistency-tolerant
fuzzy description logic if-ALC , which is an extension
of the standard fuzzy description logic f-ALC orig-
inally developed in (Straccia, 2001). A translation
from if-ALC to f-A L C was defined and a theorem
for embedding if-ALC into f-ALC was proven using
this translation. The relative decidability and com-
plexity theorems for if-A L C with respect to f-AL C
were also proven using this embedding theorem. The
relative decidability theorem states that if a decision
problem for f-ALC is decidable, then the counterpart
decision problem for if-ALC is also decidable. The
relative complexity theorem states that if the under-
lying decision problem for if-AL C is decidable, then
the complexity of the decision procedure of the un-
derlying problem for if-ALC is the same as the com-
plexity of the decision procedure of the counterpart
problem for f-AL C . It was thus shown in this study
that by using the embedding theorem, the previously
developed algorithms and methods for f-A L C can be
re-purposed for appropriately handling inconsistent
fuzzy knowledge-bases that are described by if-ALC .
Furthermore, in this study, the inconsistency-tolerant
fuzzy temporal next-time description logic itf-AL C
was obtained from if-ALC by adding the temporal
next-time operator X. Similar results as those for if-
ALC were also obtained for itf-ALC . It was thus
also shown in this study that the previously devel-
oped algorithms and methods for f-AL C can be re-
purposed for appropriately handling inconsistent tem-
poral fuzzy knowledge bases that are described by itf-
ALC .
Finally, we address a brief survey of
inconsistency-tolerant description logics, because
inconsistency-tolerant description logics are not very
popular. An inconsistency-tolerant four-valued ter-
minological logic was introduced by Patel-Schneider
(Patel-Schneider, Peter F., 1989). A sequent calculus
for reasoning in four-valued description logics was
developed by Straccia (Straccia, 1997b). Application
of a four-valued description logic to information re-
trieval was proposed by Meghini et al. (Meghini and
Straccia, 1996; Meghini et al., 1998). Three kinds of
ICAART 2022 - 14th International Conference on Agents and Artificial Intelligence
72
inconsistency-tolerant constructive description logics
were introduced by Odintsov and Wansing (Odintsov,
S.P. and Wansing, 2003; Odintsov, S.P. and Wansing,
2008). These inconsistency-tolerant constructive
description logics are based on single-interpretation
semantics, which can be seen as a description logic
version of the single-consequence Kripke semantics
for Nelson’s paraconsistent four-valued logic N4
(Almukdad and Nelson, 1984; Nelson, 1949). Sev-
eral paraconsistent four-valued description logics
including AL C 4 were developed by Ma et al. (Ma
et al., 2007; Ma et al., 2008). The logic AL C 4 is
based on four-valued semantics, and can be effec-
tively translated to ALC . By using this translation,
the satisfiability problem for AL C 4 was shown
to be decidable. However, ALC 4 and its variants
have no classical negation (or complement). Several
quasi-classical description logics were developed
by Zhang et al. (Zhang and Lin, 2008; Zhang et al.,
2009). These quasi-classical description logics are
based on quasi-classical semantics and have the
classical negation. However, translations of the
quasi-classical description logics to the counterpart
standard description logics were not proposed. An
inconsistency-tolerant description logic, PA L C , was
introduced by Kamide (Kamide, 2012) based on the
idea of a multiple-interpretation description logic,
ALC
n
, proposed by Kaneiwa (Kaneiwa, 2007).
The logic PALC is constructed on the basis of
dual-interpretation semantics and has both the bene-
fits of A L C 4 and quasi-classical description logics
(i.e., it has the faithful translation and the classical
negation connective). The semantics of PA L C is
constructed on the basis of the dual-consequence
Kripke semantics for N4. A comparison among some
previously developed inconsistency-tolerant descrip-
tion logics was addressed by Kamide in (Kamide,
2013), wherein a simple system S A L C , which is
logically equivalent to PALC , was developed using
a simple single interpretation semantics. Some
interpolation theorems were proven by Kamide in
(Kamide, 2011) for two extended inconsistency-
tolerant and temporal description logics using some
theorems for embedding these logics into a standard
description logic. An extended inconsistency-tolerant
description logic with a sequence modal operator
has recently been introduced by Kamide in (Kamide,
2020a), wherein it was intended to suitably handle
inconsistency-tolerant ontological reasoning with
sequential information that is expressed as sequences.
This extended inconsistency-tolerant description
logic was shown in (Kamide, 2020a) to be decidable
via an embedding theorem.
ACKNOWLEDGEMENTS
This research was supported by JSPS KAKENHI
Grant Numbers JP18K11171 and JP16KK0007.
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