An Extended Description Logic for Inconsistency-tolerant Ontological

Reasoning with Sequential Information

Norihiro Kamide

Department of Information and Electronic Engineering, Faculty of Science and Engineering, Teikyo University,

Toyosatodai 1-1, Utsunomiya, Tochigi, Japan

Keywords:

Description Logic, Inconsistency-tolerant Reasoning, Sequential Information, Embedding Theorem,

Decidability.

Abstract:

Description logics are a family of logic-based knowledge representation formalisms. Inconsistency-tolerant

description logics, which are extensions of standard description logics, have been studied to cope with incon-

sistencies that frequently occur in an open world. In this study, an extended inconsistency-tolerant description

logic with a sequence modal operator is introduced. The logic proposed is intended to appropriately han-

dle inconsistency-tolerant ontological reasoning with sequential information (i.e., information expressed as

sequences, such as time, action, and event sequences). A theorem for embedding the proposed logic into a

fragment of the logic is proved. The logic is shown to be decidable by using the proposed embedding theo-

rem. These results demonstrate that using the embedding theorem enables the reuse of previously developed

methods and algorithms for the standard description logic for the effective handling of inconsistent ontologies

with sequential information described by the proposed logic.

1 INTRODUCTION

In this study, we introduce an extended inconsistency-

tolerant description logic with a sequence modal op-

erator that we have named sequential inconsistency-

tolerant description logic (A LC P S ). This new

logic A LC P S is intended to appropriately handle

inconsistency-tolerant ontological reasoning with se-

quential information (i.e., information expressed as

sequences, such as time, data, action, event, and

agent-communication sequences). We then prove sev-

eral theorems for embedding ALC P S into some frag-

ments of A L C P S . Using one of these embedding

theorems, we show the decidability of ALC P S .

The aim of this study is to combine and inte-

grate an inconsistency-tolerant description logic and

a sequential description logic. Therefore, we be-

gin with a brief introduction to description logics,

inconsistency-tolerant description logics, and sequen-

tial description logics. Description logics (Baader

et al., 2003) are a family of logic-based knowledge

representation formalisms that were adopted as the

logical foundation of the W3C web ontology lan-

guage (OWL). Many of useful description logics in-

cluding the standard description logic A L C intro-

duced by Schmidt-Schauss and Smolka in (Schmidt-

Schauss and Smolka, 1991) have been extensively

studied. Inconsistency-tolerant description logics

(also referred to as paraconsistent description log-

ics) (Ma et al., 2007; Ma et al., 2008; Meghini and

Straccia, 1996; Meghini et al., 1998; Odintsov and

Wansing, 2003; Odintsov and Wansing, 2008; Patel-

Schneider, 1989; Straccia, 1997; Zhang and Lin,

2008; Zhang et al., 2009; Kamide, 2012; Kamide,

2013) are typical examples of such useful descrip-

tion logics. Inconsistency-tolerant description logics

have been studied to cope with inconsistencies that

frequently occur in an open world. For a brief sur-

vey of inconsistency-tolerant description logics, see

the last section of this paper. Sequential description

logics, that were obtained from ALC by adding a

sequence modal operator, were introduced and stud-

ied by Kamide in (Kamide, 2010; Kamide, 2011),

where he presented several embedding, decidability,

and Craig interpolation theorems for these logics. The

sequence modal operator is useful for representing se-

quential information (i.e., information expressed as

sequences) and has also been used to obtain expres-

sive and useful non-classical logics in several ﬁelds

of computer science. For more information on such

extended non-classical logics with a sequence modal

operator, see the last section of this paper.

Kamide, N.

An Extended Description Logic for Inconsistency-tolerant Ontological Reasoning with Sequential Information.

DOI: 10.5220/0008876403130321

In Proceedings of the 12th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2020) - Volume 2, pages 313-321

ISBN: 978-989-758-395-7; ISSN: 2184-433X

 2022 by SCITEPRESS – Science and Technology Publications, Lda. All rights reser ved

313

The sequence modal operator [b] used in A LC P S ,

where b is a sequence, is useful for representing

sequential information that is expressed as data se-

quences, action sequences, time sequences, event se-

quences, agent communication sequences, program-

execution sequences, word (character or alphabet) se-

quences, DNA sequences, etc. This is regarded as

plausible because a sequence structure gives a monoid

hM,;,

0i with the following informational interpreta-

tion (Wansing, 1993): (1) M is a set of sequences

(i.e., a set of pieces of ordered information); (2) ‘;’

is a concatenation operator on M (i.e., a binary op-

erator that combines two pieces of information); (3)

0 is the empty sequence (i.e., an empty piece of in-

formation). By the informational interpretation, the

intuitive meanings of the sequence modal operator

can be obtained as follows: A concept of the form

; b

;···; b

]C intuitively means that “C is true

based on a sequence b

; b

;···; b

of ordered pieces

of information.” Moreover, a concept of the form

[

0]C, which coincides with C, intuitively means that

“C is true without any information (i.e., it is an eter-

nal truth in the sense of classical description logic).”

We remark that [b] is regarded as a generalization of

the temporal next-time operator X of the linear-time

temporal logic LTL and the modal operator  of the

normal modal logic K. Actually, if we consider [b]

based on classical logic, then [b] is expressive than X

and .

In this study, we develop a sequential

inconsistency-tolerant description logic, ALC P S ,

which is a natural combination of sequential de-

scription logic and inconsistency-tolerant description

logic. To develop A LC P S , we overcame a technical

problem with the semantic interpretation for combin-

ing sequential and inconsistency-tolerant description

logics; namely, some existing inconsistency-tolerant

and sequential description logics have complex

multiple polarities or sequence-indexed interpretation

semantics. The presence of these complex interpre-

tation semantics makes it difﬁcult to combine these

two logics. This is one reason why such a combined

logic has not yet been developed. To overcome this

problem, we introduce a simple single interpretation

semantics that is compatible with the standard single

interpretation semantics of ALC . Using this simple

interpretation semantics, we can construct ALC P S

with the following technical merits: We can prove

a theorem for embedding ALC P S into the [b]-less

fragment of ALC P S and can simply formalize and

handle the operator [b].

The structure of this paper is as follows: In Sec-

tion 2, we develop a basic inconsistency-tolerant de-

scription logic, ALC P , by adding a paraconsistent

negation connective ∼ to AL C . This logic A L C P

is roughly equivalent to the logic S AL C introduced

by Kamide in (Kamide, 2013), and is logically equiv-

alent to the logic PA LC introduced by Kamide in

(Kamide, 2012). The logic AL C P has a simple sin-

gle interpretation semantics and is shown to be em-

beddable into AL C by using the method presented

in (Kamide, 2012). Using this embedding theorem,

ALC P is also shown to be decidable. In Section 3,

we develop the sequential inconsistency-tolerant de-

scription logic ALC P S by extending ALC P with the

sequence modal operator [b]. This new logic ALC P S

also has a simple single interpretation semantics. A

translation function from ALC P S into ALC P is then

deﬁned, and a theorem for embedding A L C P S into

ALC P is proved. Using this embedding theorem,

we show that A LC P S is decidable. We also prove a

theorem for embedding ALC P S into A L C . In Sec-

tion 4, we present our conclusions and discuss related

work.

2 BASIC

INCONSISTENCY-TOLERANT

DESCRIPTION LOGIC

First, we introduce the inconsistency-tolerant descrip-

tion logic ALC P . The ALC P -concepts are con-

structed from atomic concepts, roles, ¬ (classical

negation or complement), ∼ (paraconsistent nega-

tion), u (intersection), t (union), ∀R (universal con-

cept quantiﬁcation) and ∃R (existential concept quan-

tiﬁcation). We use the letter A for atomic concepts,

the letter R for roles, and the letters C and D for con-

cepts. We use an expression C ≡ D to denote the

syntactical equivalence between C and D. We use

the symbol N

to denote a set of atomic concepts,

the symbol N

to denote the set {A

| A ∈ N

} of

atomic concepts, the symbol N

∼

to denote the set

{∼A | A ∈ N

} of negated atomic concepts, and the

symbol N

to denote a non-empty set of roles. We

remark that the symbol N

is not used for deﬁning

ALC P -concepts, but used for deﬁning a translation

function from the set of A LC P -concepts into the set

of A L C -concepts, where it is used for translating the

negated atomic ALC P -concepts to the corresponding

atomic ALC -concepts in N

Deﬁnition 2.1. Concepts C of ALC P are deﬁned by

the following grammar, assuming A represents atomic

concepts:

C ::= A | ¬C | ∼C | CuC | C tC | ∀R.C | ∃R.C

Deﬁnition 2.2. A paraconsistent interpretation P I is

a structure h∆

P I

,·

P I

i such that

ICAART 2020 - 12th International Conference on Agents and Artiﬁcial Intelligence

314

1. ∆

P I

is a non-empty set,

2. ·

P I

is an interpretation function which assigns to

every concept B ∈ N

∪N

∼

a set B

P I

⊆ ∆

P I

and to

every role R a binary relation R

P I

⊆ ∆

P I

× ∆

P I

The interpretation function is inductively extended

to concepts by the following conditions:

1. (¬C)

P I

:= ∆

P I

2. (C u D)

P I

:= C

P I

∩ D

P I

3. (C t D)

P I

:= C

P I

∪ D

P I

4. (∀R.C)

P I

:= {a ∈ ∆

P I

| ∀b [(a, b) ∈ R

P I

⇒ b ∈

P I

]},

5. (∃R.C)

P I

:= {a ∈ ∆

P I

| ∃b [(a,b) ∈ R

P I

∧ b ∈

P I

]},

6. (∼∼C)

P I

:= C

P I

7. (∼¬C)

P I

:= ∆

P I

\ (∼C)

P I

8. (∼(C u D))

P I

:= (∼C)

P I

∪ (∼D)

P I

9. (∼(C t D))

P I

:= (∼C)

P I

∩ (∼D)

P I

10. (∼∀R.C)

P I

:= {a ∈ ∆

P I

| ∃b [(a,b) ∈ R

P I

∧ b ∈

(∼C)

P I

]},

11. (∼∃R.C)

P I

:= {a ∈ ∆

P I

| ∀b [(a,b) ∈ R

P I

⇒ b ∈

(∼C)

P I

]}.

An expression P I |= C is deﬁned as C

P I

0. A

paraconsistent interpretation P I := h∆

P I

,·

P I

i is a

model of a concept C (denoted as P I |= C) if P I |=

C. A concept C is said to be satisﬁable in ALC P if

there exists a paraconsistent interpretation P I such

that P I |= C.

Next, we introduce the logic ALC (Schmidt-

Schauss and Smolka, 1991) as a sublogic of ALC P .

The ALC -concepts are constructed from atomic con-

cepts, roles, ¬, u, t, ∀R, and ∃R.

Deﬁnition 2.3. Concepts C of A L C are deﬁned by

the following grammar, assuming A represents atomic

concepts:

C ::= A | ¬C | C uC | C tC | ∀R.C | ∃R.C

Deﬁnition 2.4. An interpretation I is a structure

h∆

,·

i such that

1. ∆

is a non-empty set,

2. ·

is an interpretation function which assigns to

every concept A ∈ N

a set A

⊆ ∆

and to every

role R a binary relation R

⊆ ∆

× ∆

The interpretation function is extended to concepts

by the conditions 1-5 in Deﬁnition 2.2 by replacing

P I

with ·

An expression I |= C is deﬁned as C

0. An

interpretation I := h∆

,·

i is a model of a concept C

(denoted as I |= C) if I |= C. A concept C is said to

be satisﬁable in ALC if there exists an interpretation

I such that I |= C.

Remark 2.5. We make the following remarks.

1. The logic C A LC

introduced in (Odintsov and

Wansing, 2003) has the same interpretations for

A (atomic concept), ∼A (negated atomic concept),

u and t as in AL C P . Since C ALC

is construc-

tive, it has no classical negation, but has construc-

tive inclusion (constructive implication) ⊆

2. A L C P has the following equations with respect

to ∼:

(a) (∼∼C)

P I

= C

P I

(b) (∼¬C)

P I

= (¬∼C)

P I

= (∼C t ∼D)

P I

(d) (∼(C t D))

P I

= (∼C u ∼D)

P I

(e) (∼(∀R.C))

P I

= (∃R.∼C)

P I

(f) (∼(∃R.C))

P I

= (∀R.∼C)

P I

3. A L C P is regarded as a four-valued logic in the

following sense. For each concept C, we can take

one of the following cases:

(a) C is veriﬁed with respect to an element a of ∆

P I

(i.e., a ∈ C

P I

(b) C is falsiﬁed with respect to an element a of ∆

P I

(i.e., a ∈ (∼C)

P I

(d) C is neither veriﬁed nor falsiﬁed.

4. A semantic consequence relation |= is called para-

consistent with respect to a negation connective ∼

if there are formulas α and β such that {α,∼α} 6|=

β. In case of A LC P , assume a paraconsistent

interpretation P I = h∆

P I

,·

P I

i such that A

P I

⊆

∆

P I

, (∼A)

P I

⊆ ∆

P I

, and B

P I

⊆ ∆

P I

for a pair

of distinct atomic concepts A and B. Then, (A u

∼A)

P I

6⊆ B

P I

, and hence A LC P is paraconsis-

tent with respect to ∼. Note that ALC P is not

paraconsistent with respect to ¬.

5. A L C P and ALC can be extended to deal with an

ABox, a TBox, and a knowledge base by adding a

non-empty set of individual names. But, in this pa-

per, we do not deal with these constructors, since

we intend to concentrate the discussion on the es-

sential logical reasoning.

Next, we introduce a translation form A L C P into

ALC , and present a theorem for embedding A L C P

into ALC . By using this embedding theorem, we can

obtain the decidability for ALC P .

Deﬁnition 2.6. The language L

of A L C P is deﬁned

using N

, N

, ∼, ¬,u, t, ∀R and ∃R. The language

L of AL C is obtained from L

by adding N

and

deleting ∼.

A mapping f from L

to L is deﬁned inductively

1. for any R ∈ N

and any f (R) := R,

An Extended Description Logic for Inconsistency-tolerant Ontological Reasoning with Sequential Information

315

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) where ] ∈ {∀, ∃},

6. f (∼∼C) := f (C),

7. f (∼¬C) := ¬ f (∼C),

8. f (∼(C uD)) := f (∼C) t f (∼D),

9. f (∼(C tD)) := f (∼C) u f (∼D),

10. f (∼∀R.C) := ∃ f (R). f (∼C),

11. f (∼∃R.C) := ∀ f (R). f (∼C).

Theorem 2.7 (Embedding from ALC P into A L C ).

Let f be the mapping deﬁned in Deﬁnition 2.6. For

any concept C,

C is satisﬁable in ALC P iff f (C) is satisﬁable

in ALC .

Proof. Similar to the method presented in

(Kamide, 2012) for another inconsistency-tolerant de-

scription logic PALC or S ALC . Q.E.D.

Theorem 2.8 (Decidability for AL C P ). The concept

satisﬁability problem for ALC P is decidable.

Proof. The concept satisﬁability problem for

ALC is well known to be decidable (Baader et al.,

2003; Schmidt-Schauss and Smolka, 1991). By this

decidability for A L C , for each concept C of A LC P ,

it is possible to decide if f (C) is satisﬁable in ALC .

Then, by Theorem 2.7, the satisﬁability problem for

ALC P is decidable. Q.E.D.

Remark 2.9. We make the following remarks.

1. A similar translation as presented in Deﬁnition

2.6 has been used by Gurevich (Gurevich, 1977),

Rautenberg (Rautenberg, 1979), and Vorob’ev

(Vorob’ev, 1952) to embed Nelson’s construc-

tive logic (Almukdad and Nelson, 1984; Nelson,

1949) into intuitionistic logic.

2. The satisﬁability problems of a TBox, an ABox,

and a knowledge base for A L C P are also shown

to be decidable, since these problems can be re-

duced to those of ALC .

3. The complexities of the decision problems for

ALC P are also the same as those for ALC , since

the mapping f is a polynomial-time reduction.

3 SEQUENTIAL

INCONSISTENCY-TOLERANT

DESCRIPTION LOGIC

Next, we introduce the sequential inconsistency-

tolerant description logic ALC P S . The A LC P S -

concepts are constructed from the AL C P -concepts

by adding [b] (sequence modal operator) where b is a

sequence. Sequences are constructed from countable

atomic sequences,

0 (empty sequence) and ; (compo-

sition). We use lower-case letters b, c,... to denote

sequences, and the symbol SE to denote the set of

sequences (including

0). An expression [

0]C means

C, and expressions [

0 ; b]C and [b ;

0]C mean [b]C.

We use the symbol N

[d]

(d ∈ SE) to denote the set

{[d]B | B ∈ N

∪ N

∼

}, and the symbol N

(d ∈ SE)

to denote the set {B

| B ∈ N

∪ N

∼

} of atomic and

negated atomic concepts where we assume B

= B.

Note that N

[

= N

∪ N

∼

. Moreover, we also

take the following assumption: For any A ∈ N

and

any d ∈ SE,

(∼A)

= ∼(A

) (commutativity of ∼ and ·

This assumption will be used for proving Lemma 3.7.

Deﬁnition 3.1. Concepts C of A LC P S are deﬁned by

the following grammar, assuming A represents atomic

concepts and e represents atomic sequences:

C ::= A | ¬C | ∼C | C uC | C tC

| ∀R.C | ∃R.C | [b]C

b ::= e |

0 | b ; b

The symbol ω is used to represent the set of nat-

ural numbers. An expression [d] is used to repre-

sent [d

][d

]···[d

] with i ∈ ω, d

∈ SE and d

≡

Note that [d] can be the empty sequence. We remark

that [d] is not uniquely determined. For example,

if d ≡ d

; d

where d

, d

and d

are atomic

sequences, then [d] means [d

][d

], [d

; d

][d

; d

] or [d

; d

]. Note that [d] includes [d].

Deﬁnition 3.2. A sequential paraconsistent interpre-

tation S P I is a structure h∆

S P I

,·

S P I

i such that

1. ∆

S P I

is a non-empty set,

2. ·

S P I

is an interpretation function which assigns to

every concept B ∈ N

∪ N

∼

∪ N

[d]

(d ∈ SE) a set

S P I

⊆ ∆

S P I

and to every atomic role R a binary

relation R

S P I

⊆ ∆

S P I

× ∆

S P I

3. For any A ∈ N

, ([d]∼A)

S P I

= (∼[d]A)

S P I

The interpretation function is inductively extended

to concepts by the following conditions:

1. ([d][b]C)

S P I

:= ([d ; b]C)

S P I

2. ([d]¬C)

S P I

:= ∆

S P I

\ ([d]C)

S P I

3. ([d](C uD))

S P I

:= ([d]C)

S P I

∩ ([d]D)

S P I

4. ([d](C tD))

S P I

:= ([d]C)

S P I

∪ ([d]D)

S P I

5. ([d]∀R.C)

S P I

:= {a ∈ ∆

S P I

| ∀b [(a,b) ∈ R

S P I

⇒

b ∈ ([d]C)

S P I

]},

6. ([d]∃R.C)

S P I

:= {a ∈ ∆

S P I

| ∃b [(a, b) ∈ R

S P I

∧

b ∈ ([d]C)

S P I

]},

ICAART 2020 - 12th International Conference on Agents and Artiﬁcial Intelligence

316

7. ([d]∼∼C)

S P I

:= ([d]C)

S P I

8. ([d]∼[b]C)

S P I

:= ([d ; b]∼C)

S P I

9. ([d]∼¬C)

S P I

:= ∆

S P I

\ ([d]∼C)

S P I

10. ([d]∼(C uD))

S P I

:= ([d]∼C)

S P I

∪ ([d]∼D)

S P I

11. ([d]∼(C tD))

S P I

:= ([d]∼C)

S P I

∩ ([d]∼D)

S P I

12. ([d]∼∀R.C)

S P I

:= {a ∈ ∆

S P I

| ∃b [(a,b) ∈

S P I

∧ b ∈ ([d]∼C)

S P I

]},

13. ([d]∼∃R.C)

S P I

:= {a ∈ ∆

S P I

| ∀b [(a,b) ∈

S P I

⇒ b ∈ ([d]∼C)

S P I

]},

14. (∼[d]∼C)

S P I

:= ([d]C)

S P I

15. (∼[d][b]C)

S P I

:= (∼[d ; b]C)

S P I

16. (∼[d]¬C)

S P I

:= ∆

S P I

\ (∼[d]C)

S P I

17. (∼[d](C u D))

S P I

:= (∼[d]C)

S P I

∪ (∼[d]D)

S P I

18. (∼[d](C t D))

S P I

:= (∼[d]C)

S P I

∩ (∼[d]D)

S P I

19. (∼[d]∀R.C)

S P I

:= {a ∈ ∆

S P I

| ∃b [(a,b) ∈

S P I

∧ b ∈ (∼[d]C)

S P I

]},

20. (∼[d]∃R.C)

S P I

:= {a ∈ ∆

S P I

| ∀b [(a,b) ∈

S P I

⇒ b ∈ (∼[d]C)

S P I

]}.

An expression S P I |= C is deﬁned as C

S P I

A sequential paraconsistent interpretation SP I :=

h∆

S P I

,·

S P I

i is a model of a concept C (denoted as

S P I |= C) if S P I |= C. A concept C is said to be sat-

isﬁable in A L C P S if there exists a sequential para-

consistent interpretation S P I such that SP I |= C.

Proposition 3.3. In A LC P S , we have the following

equivalence: For any concept C and any sequence d,

(

[d]∼C)

S P I

= (∼[d]C)

S P I

Proof. By induction on C. We show some cases.

• Base step:

Case C ≡ A ∈ N

: Obvious by the deﬁnition of

S P I

• Induction step:

1. Case C ≡ [b]D: ([d]∼[b]D)

S P I

= ([d ; b]∼D)

S P I

= (∼[d ; b]D)

S P I

(by induction hypothesis) =

(∼[d][b]D)

S P I

2. Case C ≡ ∼D: ([d]∼∼D)

S P I

= ([d]D)

S P I

(∼[d]∼D)

S P I

3. Case C ≡ ¬D: ([d]∼¬D)

S P I

= ∆

S P I

([d]∼D)

S P I

= ∆

S P I

\ (∼[d]D)

S P I

(by induction

hypothesis) = (∼[d]¬D)

S P I

4. Case C ≡ D

u D

: ([d]∼(D

u D

))

S P I

([d]∼D

)

S P I

∪ ([d]∼D

)

S P I

= (∼[d]D

)

S P I

∪ (∼[d]D

)

S P I

(by induction hypothesis) =

(∼[d](D

u D

))

S P I

5. Case ∀R.D:

([d]∼∀R.C)

S P I

= {a ∈ ∆

S P I

| ∃b [(a, b) ∈ R

S P I

∧ b ∈

([d]∼D)

S P I

]}

= {a ∈ ∆

S P I

| ∃b [(a, b) ∈ R

S P I

∧ b ∈

(∼[d]D)

S P I

]} (by induction hypothesis)

= (∼[d]∀R.C)

S P I

. Q.E.D.

Remark 3.4. We make the following remarks.

1. By using the condition ([d]∼C)

S P I

= (∼[d]C)

S P I

as shown in Proposition 3.3, we can derive the

condition ([d]∼A)

S P I

:= (∼[d]A)

S P I

(A ∈ N

)

and the conditions 14-20 in Deﬁnition 3.2. This

fact implies that we can deﬁne an alternative se-

mantics which is obtained from the semantics de-

ﬁned in Deﬁnition 3.2 by replacing the condition

([d]∼A)

S P I

:= (∼[d]A)

S P I

(A ∈ N

) and the con-

ditions 14-20 with the condition ([d]∼C)

S P I

(∼[d]C)

S P I

2. A L C P S has the following equations with respect

to [b]:

(a) ([b](C ] D))

S P I

= (([b]C) ] ([b]D))

S P I

where

] ∈ {u, t},

(b) ([b]]C)

S P I

= (][b]C)

S P I

where ] ∈ {¬,∀R., ∃R.},

S P I

= ([d]C)

S P I

3. Similar to ALC P , AL C P S is regarded as a four-

valued logic and a paraconsistent logic, and can

be extended to deal with ABox, TBox, and knowl-

edge base.

Deﬁnition 3.5. The language L

of A L C P S is de-

ﬁned using N

, N

, [b], ¬, ∼, u,t, ∀R, and ∃R.

The language L

of A LC P is obtained from L

adding N

(d ∈ SE) and deleting [b]. A mapping f

from L

to L

is deﬁned inductively by

1. for any R ∈ N

, f (R) := R,

2. for any A ∈ N

, f ([d]A) := A

∈ N

, especially,

f (A) := A,

3. f ([d]]C) := ] f ([d]C) where ] ∈ {¬,∼},

4. f ([d](C ] D)) := f ([d]C) ] f ([d]D) where ] ∈

{u,t},

5. f ([d]]R.C) := ] f (R). f ([d]C) where ] ∈ {∀,∃},

6. f ([d][b]C) := f ([d ; b]C), especially, f ([d]C) :=

f ([d]C).

Proposition 3.6. Let f be the mapping deﬁned in Def-

inition 3.5. The following conditions hold for any

concept C, and any sequences b, c,d, and k:

1. f ([d][b][c]C) = f ([d][b ; c]C),

2. f ([d][k]C) = f ([d][k]C),

3. f ([d]∼C) = f (∼[d]C).

An Extended Description Logic for Inconsistency-tolerant Ontological Reasoning with Sequential Information

317

Proof. We show (1) and (3) below.

1. Case (1): By using the condition 6 in Deﬁni-

tion 3.5 repeatedly, we obtain: f ([d][b][c]C) =

f ([d ; b][c]C) = f ([d ; b ; c]C) = f ([d][b ; c]C).

2. Case (3): By using the condition 3 in Deﬁnition

3.5 twice, we obtain: f ([d]∼C) = ∼ f ([d]C) =

f (∼[d]C). Q.E.D.

Lemma 3.7. Let f be the mapping deﬁned in Def-

inition 3.5. For any sequential paraconsistent in-

terpretation S P I := h∆

S P I

,·

S P I

i of A L C P S , we

can construct a paraconsistent interpretation P I :=

h∆

P I

,·

P I

i of A LC P such that for any concept C in

and any d ∈ SE,

([d]C)

S P I

= f ([d]C)

P I

Proof. Suppose that S P I is a sequential paracon-

sistent interpretation h∆

S P I

,·

S P I

i such that

1. ∆

S P I

is a non-empty set,

2. ·

S P I

is an interpretation function which assigns

to every concept B ∈

d∈SE

[d]

a set B

S P I

⊆

∆

S P I

and to every atomic role R a binary relation

S P I

⊆ ∆

S P I

× ∆

S P I

3. For any A ∈ N

, ([d]∼A)

S P I

= (∼[d]A)

S P I

We deﬁne a paraconsistent interpretation P I :=

h∆

P I

,·

P I

i such that

1. ∆

P I

is a non-empty set such that ∆

P I

= ∆

S P I

2. ·

P I

is an interpretation function which assigns to

every concept B ∈

d∈SE

a set B

P I

⊆ ∆

P I

and

to every atomic role R a binary relation R

P I

⊆

∆

P I

× ∆

P I

3. for any R ∈ N

, R

P I

= R

S P I

4. for any B ∈ N

∪N

∼

and any d ∈ SE, ([d]B)

S P I

)

P I

Then, the claim is proved by induction on the

complexity of C.

• Base step:

1. Case C ≡ A ∈ N

: We obtain: ([d]A)

S P I

= (A

)

P I

= f ([d]A)

P I

(by the deﬁnition of f ).

2. Case C ≡ ∼A ∈ N

∼

: We obtain: ([d]∼A)

S P I

= ((∼A)

)

P I

= ∼(A

)

P I

(by the assumption

(∼A)

= ∼(A

)) = ∼ f ([d]A)

P I

(by the deﬁnition

of f ) = f ([d]∼A)

P I

(by the deﬁnition of f ).

3. Case C ≡ [b]A where A ∈ N

: We obtain:

([d][b]A)

S P I

= (A

d ; b

)

P I

= f ([d ; b]A)

P I

(by the

deﬁnition of f ) = f ([d ; b]A)

P I

(by the deﬁnition

of f ) = f ([d][b]A)

P I

(by the deﬁnition of f ).

4. Case C ≡ [b]∼A where A ∈ N

: We ob-

tain: ([d][b]∼A)

S P I

= ((∼A)

d ; b

)

P I

∼(A

d ; b

)

P I

(by the commutativity of ∼ and

) = ∼ f ([d ; b]A)

P I

(by the deﬁnition of f )

= ∼ f ([d ; b]A)

P I

(by the deﬁnition of f )

= ∼ f ([d][b]A)

P I

(by the deﬁnition of f ) =

f ([d][b]∼A)

P I

(by the deﬁnition of f ).

• Induction step: We show some cases.

1. Case C ≡ [b]D: We obtain: ([d][b]D)

S P I

f ([d][b]D)

P I

(by induction hypothesis).

2. Case C ≡ ¬D: We obtain: ([d]¬D)

S P I

= ∆

S P I

([d]C)

S P I

= ∆

S P I

\ f ([d]C)

P I

(by induction hy-

pothesis) = ∆

P I

\ f ([d]C)

P I

(by the condition

∆

S P I

= ∆

P I

) = (¬ f ([d]D))

P I

= f ([d]¬D)

P I

(by

the deﬁnition of f ).

3. Case C ≡ C

: We obtain: ([d](C

))

S P I

= ([d]C

)

S P I

∩ ([d]C

)

S P I

= f ([d]C

)

P I

∩

f ([d]C

)

P I

(by induction hypothesis) =

( f ([d]C

) u f ([d]C

))

P I

= f ([d](C

))

P I

(by

the deﬁnition of f ).

4. Case C ≡ ∀R.D: We obtain:

([d]∀R.D)

S P I

= {a ∈ ∆

S P I

| ∀b [(a, b) ∈ R

S P I

⇒ b ∈

([d]D)

S P I

]}

= {a ∈ ∆

P I

| ∀b [(a, b) ∈ R

P I

⇒ b ∈ ([d]D)

S P I

]}

(by the conditions ∆

S P I

= ∆

P I

and R

S P I

P I

)

= {a ∈ ∆

P I

| ∀b [(a,b) ∈ R

P I

⇒ b ∈ f ([d]D)

P I

]}

(by induction hypothesis)

= (∀R. f ([d]D))

P I

= (∀ f (R). f ([d]D))

P I

(by the deﬁnition of f )

= f ([d]∀R.D)

P I

(by the deﬁnition of f ).

5. Case C ≡ ∼∼D: We obtain: ([d]∼∼D)

S P I

([d]D)

S P I

= f ([d]D)

P I

(by induction hypothesis)

= (∼∼ f ([d]D))

P I

= f ([d]∼∼D)

P I

(by the deﬁ-

nition of f ).

6. Case C ≡ ∼¬D: We obtain: ([d]∼¬D)

S P I

∆

S P I

\ ([d]∼C)

S P I

= ∆

S P I

\ f ([d]∼C)

P I

(by in-

duction hypothesis) = ∆

P I

\ f ([d]∼C)

P I

(by the

condition ∆

S P I

= ∆

P I

) = (¬ f ([d]∼D))

P I

= (¬∼ f ([d]D))

P I

= (∼¬ f ([d]D))

P I

f ([d]∼¬D)

P I

(by the deﬁnition of f ).

7. Case C ≡ ∼(C

u C

): We obtain: ([d]∼(C

))

S P I

= ([d]∼C

)

S P I

∪ ([d]∼C

)

S P I

f ([d]∼C

)

P I

∪ f ([d]∼C

)

P I

(by induction hy-

pothesis) = ∼ f ([d]C

)

P I

∪ ∼ f ([d]C

) (by the

ICAART 2020 - 12th International Conference on Agents and Artiﬁcial Intelligence

318

deﬁnition of f ) = (∼( f ([d]C

) u f ([d]C

)))

P I

= ∼ f ([d](C

))

P I

(by the deﬁnition of f ) =

f ([d]∼(C

))

P I

(by the deﬁnition of f ).

8. Case C ≡ ∼∀R.D: We obtain:

([d]∼∀R.D)

S P I

= {a ∈ ∆

S P I

| ∃b [(a,b) ∈ R

S P I

∧ b ∈

([d]∼D)

S P I

]}

= {a ∈ ∆

P I

| ∃b [(a,b) ∈ R

P I

∧b ∈ ([d]∼D)

S P I

]}

(by the conditions ∆

S P I

= ∆

P I

and R

S P I

P I

)

= {a ∈ ∆

P I

| ∃b [(a,b) ∈ f (R)

P I

∧ b ∈

f (

[d]∼D)

P I

]} (by the condition f (R) = R and

induction hypothesis)

= (∼∀ f (R). f ([d]D))

P I

= (∼ f ([d]∀R.D))

P I

(by the deﬁnition of f )

= f ([d]∼∀R.D)

P I

(by the deﬁnition of f ).

Q.E.D.

Lemma 3.8. Let f be the mapping deﬁned in Def-

inition 3.5. For any sequential paraconsistent in-

terpretation S P I := h∆

S P I

,{·

S P I

i of AL C P S , we

can construct a paraconsistent interpretation P I :=

h∆

P I

,·

P I

i of A LC P such that for any concept C in

and any d ∈ SE,

S P I |= [d]C iff P I |= f ([d]C).

Proof. We obtain: S P I |= [d]C iff ([d]C)

S P I

iff f ([d]C)

P I

0 (by Lemma 3.7) iff P I |= f ([d]C).

Q.E.D.

Lemma 3.9. Let f be the mapping deﬁned in Deﬁni-

tion 3.5. For any paraconsistent interpretation P I :=

h∆

P I

,·

P I

i of ALC P , we can construct a sequential

paraconsistent interpretation S P I := h∆

S P I

,·

S P I

i of

ALC P S such that for any concept C in L

and any

d ∈ SE,

P I |= f ([d]C) iff SP I |= [d]C.

Proof. Similar to the proof of Lemma 3.8. Q.E.D.

Theorem 3.10 (Embedding from ALC P S into

ALC P ). Let f be the mapping deﬁned in Deﬁnition

3.5. For any concept C,

C is satisﬁable in AL C P S iff f (C) is satisﬁ-

able in ALC P .

Proof. By Lemmas 3.8 and 3.9. Q.E.D.

Theorem 3.11 (Decidability for ALC P S ). The con-

cept satisﬁability problem for ALC P S is decidable.

Proof. By Theorems 2.8 and 3.10. Q.E.D.

We can also obtain the following results.

Theorem 3.12 (Embedding from ALC P S into

ALC ). Let f be the composition of the mappings de-

ﬁned in Deﬁnitions 3.5 and 2.6. For any concept C,

C is satisﬁable in AL C P S iff f (C) is satisﬁ-

able in ALC .

Proof. By combining Theorems 2.7 and 3.10.

Q.E.D.

Remark 3.13. We make the following remarks.

1. The satisﬁability problems of a TBox, an ABox,

and a knowledge base for A L C P S are also

shown to be decidable, since these problems can

be reduced to those of ALC .

2. The complexities of the decision problems for

ALC P S are also the same as those for A LC ,

since the mapping (composition) used in Theorem

3.12 is a polynomial-time reduction.

4 CONCLUSIONS AND RELATED

WORKS

In this study, we introduced the sequential

inconsistency-tolerant description logic A LC P S

that can appropriately handle inconsistency-tolerant

ontological reasoning with sequential information.

We proved the theorems for embedding ALC P S

into the fragments A L C P and A LC of ALC P S .

Using one of these embedding theorems, we proved

the decidability of the satisﬁability problem for

ALC P S . These results demonstrate that the existing

framework for the standard description logic A LC

can be extended to handle the useful constructors ∼

(paraconsistent negation) and [b] (sequence modal

operator). Namely, these results demonstrate that

using the embedding theorem enables the reuse of

previously developed methods and algorithms for

ALC for the effective handling of inconsistency-

tolerant ontologies with sequential information

described by ALC P S . In the following paragraphs,

we discuss related work in the literature.

Inconsistency-tolerant description logics obtained

from standard description logics by adding ∼ have

been studied. An inconsistency-tolerant four-valued

terminological logic, which is regarded as the original

inconsistency-tolerant description logic, was intro-

duced by Patel-Schneider in (Patel-Schneider, 1989).

A sequent calculus for reasoning in four-valued de-

scription logics was introduced by Straccia in (Strac-

cia, 1997). An application of four-valued description

logic to information retrieval was studied by Megh-

ini et al. in (Meghini and Straccia, 1996; Megh-

ini et al., 1998). Three inconsistency-tolerant con-

An Extended Description Logic for Inconsistency-tolerant Ontological Reasoning with Sequential Information

319

structive description logics, which are based on con-

structive logic, were studied by Odintsov and Wans-

ing in (Odintsov and Wansing, 2003; Odintsov and

Wansing, 2008). Kaneiwa (Kaneiwa, 2007) studied

ALC

∼

, which is an extended description logics with

contraries, contradictories, and subcontraries that

does not strictly qualify as an inconsistency-tolerant

description logic. Some paraconsistent four-valued

description logics, including A L C 4, were studied by

Ma et al. in (Ma et al., 2007; Ma et al., 2008).

Some quasi-classical description logics were studied

by Zhang et al. in (Zhang and Lin, 2008; Zhang

et al., 2009). An inconsistency-tolerant description

logic, PA L C , was studied by Kamide in (Kamide,

2012). In almost all these logics, except the logics

of Odintsov and Wansing, some dual interpretation

semantics were used. An inconsistency-tolerant de-

scription logic, S A LC , that is logically equivalent to

PALC was introduced by Kamide in (Kamide, 2013)

by using a simple single interpretation semantics that

is similar to those found in the logics of Odintsov and

Wansing. The logic A L C P that is introduced in this

paper is a slight modiﬁcation of S ALC .

Sequential description logics that were obtained

from ALC by adding [b] were introduced by Kamide

in (Kamide, 2010; Kamide, 2011), where he pre-

sented embedding and interpolation theorems for

these logics. However, these logics were constructed

based on complex multiple interpretation semantics.

The ∼-free fragment of A L C P S introduced in this

paper is not logically equivalent to the sequential de-

scription logic developed in (Kamide, 2010; Kamide,

2011). The condition of the interpretation function

with respect to the classical negation connective ¬ is

different. Although no other sequential description

logic equipped with [b], some extended non-classical

logics with [b] have been studied in some applica-

tions. An extended sequential paraconsistent compu-

tation tree logic, SPCTL, was introduced by Kamide

in (Kamide, 2015), and this logic was used for the ver-

iﬁcation of clinical reasoning. A sequence-indexed

linear-time temporal logic, SLTL, was introduced

by Kaneiwa and Kamide in (Kaneiwa and Kamide,

2010), and this logic was used for describing security

issues with agent communication. An extended linear

logic with [b], called a sequence-indexed linear logic,

was developed by Kamide and Kaneiwa in (Kamide

and Kaneiwa, 2013), and this logic was used for

formalizing resource-sensitive reasoning with agent

communication and event sequences. An extended

full computation tree logic with [b] was developed

by Kaneiwa and Kamide in (Kaneiwa and Kamide,

2011), and this logic was used for conceptual mod-

eling in domain ontologies. Some temporal logics

with [b] have recently been introduced by Kamide and

Yano in (Kamide and Yano, 2017; Kamide, 2018),

and these logics were used for the logical foundation

of hierarchical model checking. In these non-classical

logics, which are not description logics, [b] was used

for expressing hierarchies, data sequences, event se-

quences, action sequences, and agent communication

sequences.

In addition to the aforementioned studies

on inconsistency-tolerant description logics for

inconsistency-tolerant ontological reasoning (without

handling sequential information), there is another

direction of promising studies on inconsistency-

tolerant ontological reasoning based on description

logics. Such studies do not introduce a new

inconsistency-tolerant description logic, but, sev-

eral inconsistency-tolerant semantics, which are

not a semantics for a description logic itself, are

introduced and investigated for query answering in

description logic knowledge bases (Lembo et al.,

2010). For more information on recent developments

of inconsistency-tolerant semantics for query an-

swering in description logic knowledge bases, see

e.g., (Lembo et al., 2010; Bienvenu et al., 2014;

Lukasiewicz et al., 2019) and the references therein.

ACKNOWLEDGEMENTS

We would like to thank the anonymous referees for

their valuable comments and suggestions. This re-

search was supported by JSPS KAKENHI Grant

Numbers JP18K11171, JP16KK0007, and JSPS

Core-to-Core Program (A. Advanced Research Net-

works).

REFERENCES

Almukdad, A. and Nelson, D. (1984). Constructible falsity

and inexact predicates. Journal of Symbolic Logic,

49:231–233.

Baader, F., Calvanese, D., McGuinness, D., Nardi, D.,

and Patel-Schneider, P. (2003). The description logic

handbook: Theory, implementation and applications.

Cambridge University Press.

Bienvenu, M., Bourgaux, C., and Goasdou

e, F. (2014).

Querying inconsistent description logic knowledge

bases under preferred repair semantics. In Proceed-

ings of the 28th AAAI Conference on Artiﬁcial Intelli-

gence (AAAI 2014), pages 996–1002.

Gurevich, Y. (1977). Intuitionistic logic with strong nega-

tion. Studia Logica, 36:49–59.

Kamide, N. (2010). Sequential description logic. Far East

Journal of Applied Mathematics, 47 (1):205–214.

ICAART 2020 - 12th International Conference on Agents and Artiﬁcial Intelligence

320

Kamide, N. (2011). Interpolation theorems for some ex-

tended description logics. In Proceedings of the

15th International Conference on Knowledge-Based

and Intelligent Information and Engineering Systems

(KES 2011), Lecture Notes in Artiﬁcial Intelligence,

volume 6882, pages 246–255.

Kamide, N. (2012). Embedding-based approaches to para-

consistent and temporal description logics. Journal of

Logic and Computation, 22 (5):1097–1124.

Kamide, N. (2013). A comparison of paraconsistent de-

scription logics. International Journal of Intelligence

Science, 3 (2):99–109.

Kamide, N. (2015). Inconsistency-tolerant temporal rea-

soning with hierarchical information. Information Sci-

ences, 320:140–155.

Kamide, N. (2018). Logical foundations of hierarchical

model checking. Data Technology and Applications,

52 (4):539–563.

Kamide, N. and Kaneiwa, K. (2013). Reasoning about

resources and information: A linear logic approach.

Fundamenta Informaticae, 125 (1):51–70.

Kamide, N. and Yano, R. (2017). Logics and translations for

hierarchical model checking. In Proceedings of the

21st International Conference on Knowledge-Based

and Intelligent Information & Engineering Systems,

Procedia Computer Science, volume 112, pages 31–

40.

Kaneiwa, K. (2007). Description logics with contraries,

contradictories, and subcontraries. New Generation

Computing, 25 (4):443–468.

Kaneiwa, K. and Kamide, N. (2010). Sequence-indexed

linear-time temporal logic: Proof system and applica-

tion. Applied Artiﬁcial Intelligence, 24 (10):896–913.

Kaneiwa, K. and Kamide, N. (2011). Conceptual modeling

in full computation-tree logic with sequence modal

operator. International Journal of Intelligent Systems,

26 (7):636–651.

Lembo, D., Lenzerini, M., Rosati, R., Ruzzi, M., and Savo,

D. F. (2010). Inconsistency-tolerant semantics for de-

scription logics. In Proceedings of the International

Conference on Web Reasoning and Rule Systems (RR

2010), Lecture Notes in Computer Science, volume

6333, pages 103–117.

Lukasiewicz, T., Malizia, E., and Vaicenavi

clus, A. (2019).

Complexity of inconsistency-tolerant query answer-

ing in datalog+/- under cardinality-based repairs. In

Proceedings of the 23rd AAAI Conference on Artiﬁ-

cial Intelligence (AAAI 2019), pages 2962–2969.

Ma, Y., Hitzler, P., and Lin, Z. (2007). Algorithms for

paraconsistent reasoning with owl. In Proceedings of

the 4th European Semantic Web Conference (ESWC

2007), Lecture Notes in Computer Science, volume

4519, pages 399–413.

Ma, Y., Hitzler, P., and Lin, Z. (2008). Paraconsistent rea-

soning for expressive and tractable description logics.

In Proceedings of the 21st International Workshop on

Description Logic (DL 2008), Sun SITE Central Eu-

rope (CEUR) Electronic Workshop Proceedings, vol-

ume 553.

Meghini, C., Sebastiani, F., and Straccia, U. (1998). Mirlog:

A logic for multimedia information retrieval. In Un-

certainty and Logics: Advanced Models for the Rep-

resentation and Retrieval of Information, pages 151–

185.

Meghini, C. and Straccia, U. (1996). A relevance termino-

logical logic for information retrieval. In Proceedings

of the 19th Annual International ACM SIGIR Con-

ference on Research and Development in Information

Retrieval, pages 197–205.

Nelson, D. (1949). Constructible falsity. Journal of Sym-

bolic Logic, 14:16–26.

Odintsov, S. and Wansing, H. (2003). Inconsistency-

tolerant description logic: Motivation and basic sys-

tems. In V.F. Hendricks and J. Malinowski, Editors,

Trends in Logic: 50 Years of Studia Logica, pages

301–335.

Odintsov, S. and Wansing, H. (2008). Inconsistency-

tolerant description logic. Part II: Tableau algorithms.

Journal of Applied Logic, 6:343–360.

Patel-Schneider, P. F. (1989). A four-valued semantics for

terminological logics. Artiﬁcial Intelligence, 38:319–

351.

Rautenberg, W. (1979). Klassische und nicht-klassische

Aussagenlogik. Vieweg, Braunschweig.

Schmidt-Schauss, M. and Smolka, G. (1991). Attributive

concept descriptions with complements. Artiﬁcial In-

telligence, 48:1–26.

Straccia, U. (1997). A sequent calculus for reasoning in

four-valued description logics. In Proceedings of In-

ternational Conference on Automated Reasoning with

Analytic Tableaux and Related Methods (TABLEAUX

1997), Lecture Notes in Computer Science, volume

1227, pages 343–357.

Vorob’ev, N. (1952). A constructive propositional calculus

with strong negation (in Russian). Doklady Akademii

Nauk SSR, 85:465–468.

Wansing, H. (1993). The logic of information structures.

In Lecture Notes in Computer Science, volume 681,

pages 1–163.

Zhang, X. and Lin, Z. (2008). Paraconsistent reasoning

with quasi-classical semantics in ALC. In Proceedings

of the 2nd International Conference on Web Reason-

ing and Rule Systems (RR 2008), volume 5341, pages

222–229.

Zhang, X., Qi, G., Ma, Y., and Lin, Z. (2009). Quasi-

classical semantics for expressive description logics.

In Proceedings of the 22nd International Workshop on

Description Logic (DL 2009), Sun SITE Central Eu-

rope (CEUR) Electronic Workshop Proceedings, vol-

ume 477.

An Extended Description Logic for Inconsistency-tolerant Ontological Reasoning with Sequential Information

321