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

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

| · |

and |· |

, 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

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

: 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

and S

with respect to U:

1. µ

(u) := 1 − µ

(u),

2. µ

∩S

(u) := min{µ

(u),µ

(u)},

3. µ

∪S

(u) := max{µ

(u),µ

(u)}.

where S

is a fuzzy complement of S

in U , S

∩ S

denotes a fuzzy set intersection of S

and S

, and S

∪

denotes a fuzzy set union of S

and S

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 such that

1. ∆

is a non-empty set as for the crisp case,

2. ·

is a fuzzy interpretation function such that

(a) for any individual names a and b, we have

∈ ∆

such that a

6= b

if a 6= b,

(b) for any atomic concepts A, we have A

∆

→[0,1],

: ∆

× ∆

→[0,1].

The fuzzy interpretation function is inductively ex-

tended to concepts by the following clauses: for any

d ∈ ∆

1. (¬C)

(d) := 1 −C

(d),

2. (C u D)

(d) := min{C

(d), D

(d)},

3. (C t D)

(d) := max{C

(d), D

(d)},

4. (∀R.C)

(d) := in f

∈∆

{max{1 −R

(d, d

),C

)}},

5. (∃R.C)

(d) := sup

∈∆

{min{R

(d, d

),C

)}}.

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

) ≥ n

or R

) ≥ n, respectively. Similarly, a fuzzy in-

terpretation I satisﬁes a fuzzy assertion ha : C ≤ mi

or h(a, b) : R ≤ mi iff C

) ≤ m or R

) ≤ 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 ∈ ∆

(d) ≤

(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 ∈ ∆

(d) = D

(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

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

to denote the set

of assertions in Σ and we use an expression Σ

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

of terminological axioms (denoted

by C ≺

D) iff for every fuzzy interpretation I of Σ

∀d ∈ ∆

(d) ≤ D

(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

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],

(∼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

∈∆

I I

{max{1 − R

I I

(d, d

),C

I I

)}},

5. (∃R.C)

I I

(d) := sup

∈∆

I I

{min{R

I I

(d, d

),C

I I

)}},

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

∈∆

I I

{min{R

I I

(d, d

),(∼C)

I I

)}},

11. (∼∃R.C)

I I

(d) := in f

∈∆

I I

{max{1 − R

I I

(d, d

),(∼C)

I I

)}}.

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,

(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

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

to be a non-empty set of

atomic concepts, the symbol N

to be the set {A

| A ∈

} of atomic concepts, the symbol N

to be a non-

empty set of roles, and the symbol N

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

of if-ALC is de-

ﬁned using N

, N

, ∼, ¬, u, t, ∀R, and ∃R. The

language L of f-ALC is obtained from L

by adding

and deleting ∼. A mapping f from L

to L is

deﬁned inductively by:

1. for any R ∈ N

and any a ∈ N

, f (R) := R and

f (a) := a,

2. for any A ∈ N

, f (A) := A and f (∼A) := A

∈ N

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 of f-ALC

such that for any concept C in L

, C

I I

= f (C)

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 such that

1. ∆

is a non-empty set such that ∆

= ∆

I I

2. ·

is a fuzzy interpretation function where

(a) R

= R

I I

and a

= a

I I

for any R ∈ N

and any a ∈ N

(b) A

= A

I I

for any atomic concept A ∈ N

)

= (∼A)

I I

for any atomic concept A

∈ N

This lemma is then proved by induction on the

complexity of C.

• Base step:

1. Case C ≡ A ∈ N

: We obtain: A

I I

= A

= f (A)

(by the deﬁnition of f ).

2. Case C ≡ ∼A with A ∈ N

: We obtain: (∼A)

I I

)

= f (∼A)

(by the deﬁnition of f ).

• Induction step: We show some cases.

Reasoning with Inconsistency-tolerant Fuzzy Description Logics

1. Case C ≡ ¬D: We obtain:

(¬D)

I I

(d)

= 1 − D

I I

(d)

= 1 − f (D)

(d) (by induction hypothesis)

= (¬ f (D))

(d)

= f (¬D)

(d) (by the deﬁnition of f ).

2. Case C ≡ C

: We obtain:

)

I I

(d)

= min{C

I I

(d),C

I I

(d)}

= min{ f (C

)

(d), f (C

)

(d)} (by induction hy-

pothesis)

= ( f (C

) u f (C

))

(d)

= f (C

)

(d) (by the deﬁnition of f ).

3. Case C ≡ ∀R.D: We obtain:

(∀R.D)

I I

(d)

= in f

∈∆

I I

{max{1 − R

I I

(d, d

),D

I I

)}}

= in f

∈∆

I I

{max{1 − R

I I

(d, d

), f (D)

)}}

(by induction hypothesis)

= in f

∈∆

{max{1 − R

(d, d

), f (D)

)}} (by

∆

I I

= ∆

and R

I I

= R

)

= (∀R. f (D))

(d)

= f (∀R.D)

(d) (by the deﬁnition of f ).

4. Case C ≡ ∼∼D: We obtain:

(∼∼D)

I I

(d)

= D

I I

(d)

= f (D)

(d) (by induction hypothesis)

= f (∼∼D)

(d) (by the deﬁnition of f ).

5. Case C ≡ ∼¬D: We obtain:

(∼¬D)

I I

(d)

= 1 − (∼D)

I I

(d)

= 1 − f (∼D)

(d) (by induction hypothesis)

= (¬ f (∼D))

(d)

= f (∼¬D)

(d) (by the deﬁnition of f ).

6. Case C ≡ ∼(C

): We obtain:

(∼(C

))

I I

(d)

= max{(∼C

)

I I

(d), (∼C

)

I I

(d)}

= max{ f (∼C

)

(d), f (∼C

)

(d)} (by induction

hypothesis)

= ( f (∼C

) t f (∼C

))

(d)

= f (∼(C

))

(d) (by the deﬁnition of f ).

7. Case C ≡ ∼∀R.D: We obtain:

(∼∀R.D)

I I

(d)

= sup

∈∆

I I

{min{R

I I

(d, d

),(∼D)

I I

)}}

= sup

∈∆

I I

{min{R

I I

(d, d

), f (∼D)

)}} (by

induction hypothesis)

= sup

∈∆

{min{R

(d, d

), f (∼D)

)}} (by

∆

I I

= ∆

and R

I I

= R

)

= (∃R. f (∼D))

(d)

= f (∼∀R.D)

(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 of f-ALC

such that for any inconsistency-tolerant fuzzy asser-

tion ψ in L

, 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∆

,·

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

, 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

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

C with n ∈ ω by

C := C and X

n+1

C := XX

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∈ω

i where

1. ∆

I T I

is a non-empty set as for the crisp case,

2. for any i ∈ ω, ·

is an inconsistency-tolerant tem-

poral next-time fuzzy interpretation function such

that

(a) for any individual names a and b, we have

∈ ∆

I T I

such that a

6= b

if a 6= b,

(b) for any atomic concepts A, we have A

∆

I T I

→[0,1],

(∼A)

: ∆

I T I

→[0,1],

(d) for any roles R, we have R

: ∆

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

= a

and R

= R

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)

(d) := C

i+1

(d),

2. (¬C)

(d) := 1 −C

(d),

3. (C u D)

(d) := min{C

(d), D

(d)},

4. (C t D)

(d) := max{C

(d), D

(d)},

5. (∀R.C)

(d) := in f

∈∆

I T I

{max{1 − R

(d, d

),C

)}},

6. (∃R.C)

(d) := sup

∈∆

I T I

{min{R

(d, d

),C

)}},

7. (∼XC)

(d) := (∼C)

i+1

(d),

8. (∼∼C)

(d) := C

(d),

9. (∼¬C)

(d) := 1 − (∼C)

(d),

10. (∼(C u D))

(d) := max{(∼C)

(d), (∼D)

(d)},

11. (∼(C t D))

(d) := min{(∼C)

(d), (∼D)

(d)},

12. (∼∀R.C)

(d) := sup

∈∆

I T I

{min{R

(d, d

),(∼C)

)}},

13. (∼∃R.C)

(d) := in f

∈∆

I T I

{max{1 − R

(d, d

),(∼C)

)}}.

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},

2. itf-ALC is an extension of if-A L C , because the

zero interpretation function ·

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

= R

and a

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

Deﬁnition 4.4. An inconsistency-tolerant tempo-

ral next-time fuzzy interpretation function I

sat-

isﬁes an inconsistency-tolerant temporal next-time

fuzzy assertion ha : C ≥ ni or h(a,b) : R ≥ ni

iff C

) ≥ n or R

) ≥ n, respectively.

Similarly, an inconsistency-tolerant temporal next-

time fuzzy interpretation function I

satisﬁes an

inconsistency-tolerant temporal next-time fuzzy as-

sertion ha : C ≤ mi or h(a,b) : R ≤ mi iff C

) ≤

m or R

) ≤ 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

) ≥ n or R

) ≥ 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

) ≤ m

or R

) ≤ m, respectively.

We use an expression I

|= ψ to denote the

fact that an inconsistency-tolerant temporal next-

time fuzzy interpretation function 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

|= 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

, N

, and N

as those

used for if-ALC . Furthermore, we use the new sym-

bol N

to denote the set {A

| A ∈ N

} of atomic con-

cepts where A

= A (i.e., N

= N

The deﬁnition of the translation from itf-ALC to

if-ALC is as follows.

Deﬁnition 4.5. The language L

of itf-ALC is de-

ﬁned using N

, N

, ∼, ¬, u, t, ∀R, ∃R, and X.

The language L

of if-ALC is obtained from L

adding

i∈ω

and deleting X. A mapping g from L

to L

is deﬁned inductively by:

1. for any R ∈ N

and any a ∈ N

, g(R) := R and g(a) := a,

2. for any A ∈ N

, g(X

A) := A

∈ N

(esp. g(A) := A),

3. g(X

]C) := ]g(X

C) where ] ∈ {¬, ∼},

4. g(X

(C ] D)) := g(X

C) ] g(X

D) where ] ∈ {u, t},

5. g(X

]R.C) := ]R.g(X

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∈ω

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

and any i ∈ ω, C

= g(X

I I

Proof. Suppose that I T I is an inconsistency-

tolerant temporal next-time fuzzy interpreta-

tion h∆

I T I

,{·

}

i∈ω

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

and a

I I

= a

for any R ∈ N

and any

a ∈ N

(b) (A

)

I I

= A

(esp. A

I I

= A

) for any atomic con-

cept A

∈ N

)

I I

= (∼A)

(esp. (∼A)

I I

= (∼A)

) for any

negated atomic concept ∼A

with A

∈ N

Then, this lemma is proved by induction on the

complexity of C.

• Base step:

1. Case C ≡ A ∈ N

: We obtain: (A)

= (A

)

I I

g(X

I I

(by the deﬁnition of g).

2. Case C ≡ ∼A with A ∈ N

: We obtain: (∼A)

(∼A

)

I I

= (∼g(X

A))

I I

(by the deﬁnition of g)

= g(X

∼A)

I I

(by the deﬁnition of g).

• Induction step: We show some cases.

1. Case C ≡ XD: We obtain:

(XD)

(d)

= D

i+1

(d)

= g(X

i+1

I I

(d) (by induction hypothesis)

= g(X

XD)

I I

(d).

2. Case C ≡ ¬D: We obtain:

(¬D)

(d)

= 1 − D

(d)

= 1 − g(X

I I

(d) (by induction hypothesis)

= (¬g(X

D))

I I

(d)

= g(X

¬D)

I I

(d) (by the deﬁnition of g).

3. Case C ≡ C

: We obtain:

)

(d)

= min{C

(d),C

(d)}

= min{g(X

)

I I

(d), g(X

)

I I

(d)} (by induc-

tion hypothesis)

ICAART 2022 - 14th International Conference on Agents and Artiﬁcial Intelligence

= (g(X

) u g(X

))

I I

(d)

= g(X

))

I I

(d) (by the deﬁnition of g).

4. Case C ≡ ∀R.D: We obtain:

(∀R.D)

(d)

= in f

∈∆

I T I

{max{1 − R

(d, d

),D

)}}

= in f

∈∆

I T I

{max{1 − R

(d, d

),g(X

I I

)}} (by

induction hypothesis)

= in f

∈∆

I I

{max{1 − R

I I

(d, d

),g(X

I I

)}}

(by ∆

I T I

= ∆

I I

and R

= R

I I

)

= (∀R.g(X

D))

I I

(d)

= g(X

∀R.D)

I I

(d) (by the deﬁnition of g).

5. Case C ≡ ∼XD: We obtain:

(∼XD)

(d)

= (∼D)

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

XD))

I I

(d)

= g(X

∼XD)

I I

(d) (by the deﬁnition of g).

6. Case C ≡ ∼∼D: We obtain:

(∼∼D)

(d)

= D

(d)

= g(X

I I

(d) (by induction hypothesis)

= (∼∼g(X

D))

I I

(d)

= g(X

∼∼D)

I I

(d) (by the deﬁnition of g).

7. Case C ≡ ∼¬D: We obtain:

(∼¬D)

(d)

= 1 − (∼D)

(d)

= 1 − g(X

∼D)

I I

(d) (by induction hypothesis)

= (¬g(X

∼D))

I I

(d)

= (¬∼g(X

D))

I I

(d)

= (∼¬g(X

D))

I I

(d)

= g(X

∼¬D)

I I

(d) (by the deﬁnition of g).

8. Case C ≡ ∼(C

): We obtain:

(∼(C

))

(d)

= max{(∼C

)

(d), (∼C

)

I I

(d)}

= max{g(X

∼C

)

I I

(d), g(X

∼C

)

I I

(d)} (by in-

duction hypothesis)

= max{(∼g(X

))

I I

(d), (∼g(X

))

I I

(d)} (by

the deﬁnition of g)

= (∼(g(X

) u g(X

)))

I I

(d)

= (∼g(X

)))

I I

(d) (by the deﬁnition of g)

= g(X

∼(C

))

I I

(d) (by the deﬁnition of g).

9. Case C ≡ ∼∀R.D: We obtain:

∼∀R.D)

(d)

= sup

∈∆

I T I

{min{R

(d, d

),(∼D)

)}}

= sup

∈∆

I T I

{min{R

(d, d

),g(X

∼D)

I I

)}}

(by induction hypothesis)

= sup

∈∆

I I

{min{R

I I

(d, d

),g(X

∼D)

I I

)}}

(by ∆

I T I

= ∆

I I

and R

= R

I I

)

= sup

∈∆

I I

{min{R

I I

(d, d

),(∼g(X

D))

I I

)}}

(by the deﬁnition of g).

= (∼∀R.g(X

D))

I I

(d)

= (∼g(X

∀R.D))

I I

(d) (by the deﬁnition of g)

= g(X

∼∀R.D)

I I

(d) (by the deﬁnition of g).

We use an expression X

ψ to denote the fact that

the concept C appearing in an inconsistency-tolerant

temporal next-time fuzzy assertion ψ is replaced with

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∈ω

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

and any i ∈ ω, I

|= ψ iff I I |= g(X

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∈ω

i of itf-

ALC such that for any inconsistency-tolerant tempo-

ral next-time fuzzy assertion ψ in L

and any i ∈ ω,

|= ψ iff I I |= g(X

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

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

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

∼

, 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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