How Fuzzy is my Fuzzy Description Logic? - Semantic Scholar

Institute of Theoretical Computer Science Chair of Automata Theory

HOW FUZZY IS MY FUZZY DESCRIPTION LOGIC? Stefan Borgwardt

Felix Distel

Manchester, June 28, 2012

˜ Rafael Penaloza

Overview

• fuzzy DLs can deal with vagueness: h¬Small v ∃parent.¬Smalli


How Fuzzy is my Fuzzy Description Logic?

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Overview

• fuzzy DLs can deal with vagueness: h¬Small v ∃parent.¬Small ≥ 0.4i What is the largest degree x to which ¬Small u ∀parent.Small is satisfiable?



2

Overview


• some fuzzy DLs cannot deal with vagueness: The answer is always x = 0.



2

Overview





2

Overview

• fuzzy DLs can deal with vagueness: h¬Small v ∃parent.¬Small ≥ 0.1i, hTall v ¬Small ≥ 1i What is the largest degree x to which ¬Small u ∀parent.Small is satisfiable?


• it’s not that bad: What is the largest degree x to which Tall v ∃parent.¬Small always holds?



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Overview

• fuzzy DLs can deal with vagueness: h¬Small v ∃parent.¬Small ≥ 0.1i, hTall v ¬Small ≥ 1i What is the largest degree x to which ¬Small u ∀parent.Small is satisfiable?


• it’s not that bad: What is the largest degree x to which Tall v ∃parent.¬Small always holds? x = 0.1



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Mathematical fuzzy logic t-norm ⊗ : [0, 1] × [0, 1] → [0, 1] generalizes ∧ : {0, 1} × {0, 1} → {0, 1}:

• • • • •

associative commutative monotone unit 1 (continuous)



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• • • • •


derived operators:

• residuum x ⇒ y (x ⊗ y ≤ z iff y ≤ x ⇒ z) • precomplement x := x ⇒ 0 • t-conorm x ⊕ y := 1 − ((1 − x) ⊗ (1 − y))



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• • • • •


derived operators:

• residuum x ⇒ y (x ⊗ y ≤ z iff y ≤ x ⇒ z) • precomplement x := x ⇒ 0 • t-conorm x ⊕ y := 1 − ((1 − x) ⊗ (1 − y)) ´ [Hajek 2001, 2005]



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⊗-ALC and ⊗-SHOI name concept name

syntax A ∈ NC

semantics AI : ∆I → [0, 1]

role name

r ∈ NR

r I : ∆I × ∆I → [0, 1]

individual name

a ∈ NI

a I ∈ ∆I



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syntax A ∈ NC


role name

r ∈ NR

r I : ∆I × ∆I → [0, 1]

individual name conjunction

a ∈ NI CuD

a I ∈ ∆I C I (x) ⊗ DI (x)

disjunction

CtD

C I (x) ⊕ DI (x)

¬C

C I (x)

C→D >/⊥

C I (x) ⇒ DI (x) 1/0

existential restriction

∃r.C

supy∈∆I r I (x, y) ⊗ C I (y)

value restriction

∀r.C

infy∈∆I r I (x, y) ⇒ C I (y)

negation implication top / bottom



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syntax A ∈ NC


role name

r ∈ NR

r I : ∆I × ∆I → [0, 1]


a ∈ NI CuD

a I ∈ ∆I C I (x) ⊗ DI (x)

disjunction

CtD

C I (x) ⊕ DI (x)

¬C

C I (x)

C→D >/⊥

C I (x) ⇒ DI (x) 1/0


∃r.C


value restriction

∀r.C


ha : C, ì

C I (aI ) ≥ `


concept assertion role assertion general concept inclusion (GCI)


h(a, b) : r, ì

r I (aI , bI ) ≥ `

hC v D, ì

C I (x) ⇒ DI (x) ≥ `


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syntax A ∈ NC


role name

r ∈ NR

r I : ∆I × ∆I → [0, 1]


a ∈ NI CuD

a I ∈ ∆I C I (x) ⊗ DI (x)

disjunction

CtD

C I (x) ⊕ DI (x)

¬C

C I (x)

C→D >/⊥

C I (x) ⇒ DI (x) 1/0


∃r.C


value restriction

∀r.C


ha : C, ì

C I (aI ) ≥ `


concept assertion role assertion general concept inclusion (GCI)

h(a, b) : r, ì

r I (aI , bI ) ≥ `

hC v D, ì

C I (x) ⇒ DI (x) ≥ `

⊗-SHOI = ⊗-ALC + transitive roles, role hierarchy, nominals, inverse roles Manchester, June 28, 2012


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Fundamental continuous t-norms ¨ Godel: x ⊗ y = min(x, y)

0.8

x ⊗y

0.6

0.4

0.2

1 0.8

0

0.6

0 0.2

0.4

0.4 0.6 x


y

0.2 0.8

10


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Fundamental continuous t-norms ¨ Godel: x ⊗ y = min(x, y) Product: x ⊗ y = x · y

0.8

x ⊗y

0.6

0.4

0.2

1 0.8

0

0.6

0 0.2

0.4

0.4 0.6 x


y

0.2 0.8

10


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Fundamental continuous t-norms ¨ Godel: x ⊗ y = min(x, y) Product: x ⊗ y = x · y Łukasiewicz: x ⊗ y = max(0, x + y − 1)

0.8

x ⊗y

0.6

0.4

0.2

1 0.8

0

0.6

0 0.2

0.4

0.4 0.6 x


y

0.2 0.8

10


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Other continuous t-norms All continuous t-norms are (isomorphic to) ordinal sums of the fundamental t-norms.

x 0 Manchester, June 28, 2012

0.3

0.7


1

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Other continuous t-norms All continuous t-norms are (isomorphic to) ordinal sums of the fundamental t-norms. y 1

0.7

0.3

x

0 0 Manchester, June 28, 2012

0.3

0.7


1

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Other continuous t-norms All continuous t-norms are (isomorphic to) ordinal sums of the fundamental t-norms. y 1

0.7

0.3

x


0.3

0.7


1

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Other continuous t-norms All continuous t-norms are (isomorphic to) ordinal sums of the fundamental t-norms. y 1 ¨ Godel 0.7

Łukasiewicz

0.3 Product x


0.3

0.7


1

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Other continuous t-norms All continuous t-norms are (isomorphic to) ordinal sums of the fundamental t-norms. y 1 ¨ Godel

¨ Godel

0.7

Łukasiewicz

0.3 ¨ Godel

Product

x


0.3

0.7


1

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Other continuous t-norms All continuous t-norms are (isomorphic to) ordinal sums of the fundamental t-norms.

0.8

x ⊗y

0.6

0.4

0.2

1 0.8

0

0.6

0 0.2

0.4

0.4 0.6 x


y

0.2 0.8

10


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Zero divisors

⊗ has zero divisors if x ⊗ y = 0 for some x, y > 0. ( 1 if x = 0 ¨ Godel negation: x = 0 if x > 0



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Zero divisors

⊗ has zero divisors if x ⊗ y = 0 for some x, y > 0. ( 1 if x = 0 ¨ Godel negation: x = 0 if x > 0 Lemma: ⊗ has no zero divisors iff ¨ ⊗ has Godel negation



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Zero divisors

⊗ has zero divisors if x ⊗ y = 0 for some x, y > 0. ( 1 if x = 0 ¨ Godel negation: x = 0 if x > 0 Lemma: ⊗ has no zero divisors iff ¨ ⊗ has Godel negation iff ⊗ does not “start with Łukasiewicz”



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Consistency witnessed interpretation I: For every role r, concept C, and x ∈ ∆I , (∃r.C)I (x) = max r I (x, y) ⊗ C I (y) and y∈∆I


(∀r.C)I (x) = min r I (x, y) ⇒ C I (y).


y∈∆I

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(∀r.C)I (x) = min r I (x, y) ⇒ C I (y). y∈∆I

(witnessed) consistency of a finite set O of axioms:

NEL [u, >, ∃, ¬]

constructors IAL [u, >, ⊥, ∃, ∀, →]

ELC [u, >, ∃, ]

crisp (` = 1)

Z G

Z G

Π Ł G

≥`

Z G

Z G

G G

assertions

(undecidable) (decidable)


=` Z G G G (Z . . . all t-norms with zero divisors)


G G

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(∀r.C)I (x) = min r I (x, y) ⇒ C I (y). y∈∆I

(witnessed) consistency of a finite set O of axioms:

NEL [u, >, ∃, ¬]


ELC [u, >, ∃, ]

crisp (` = 1)

Z G

Z G

Π Ł G

≥`

Z G

Z G

G G

assertions


=` Z G G G (Z . . . all t-norms with zero divisors)

G G

˜ ˜ [Baader, Penaloza 2011], [Cerami, Straccia 2011], [Borgwardt, Penaloza 2012] ´ [Bobillo, Delgado, Gomez-Romero, Straccia 2009] Manchester, June 28, 2012


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T-norms without zero divisors ( ¨ Godel negation x =


1 0

if x = 0 if x > 0


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1 0

if x = 0 if x > 0 does not distinguish between values > 0.



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1 0


We will show that every consistent O has a crisp model (using only 0 and 1). ( 1 if x > 0 1(x) := x = 0 if x = 0



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1 0



• 1( x) = x = 1(x)


X


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1 0



• 1( x) = x = 1(x) X • 1(x ⊗ y) = 1(x) ⊗ 1(y) since x ⊗ y = 0 iff x = 0 or y = 0



X

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1 0



• • • •

1( x) = x = 1(x) X 1(x ⊗ y) = 1(x) ⊗ 1(y) since x ⊗ y = 0 iff x = 0 or y = 0 1(x ⊕ y) = 1(x) ⊕ 1(y) X 1(x ⇒ y) = 1(x) ⇒ 1(y) X



X

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1 0



• • • • • •

1( x) = x = 1(x) X 1(x ⊗ y) = 1(x) ⊗ 1(y) since x ⊗ y = 0 iff x = 0 or y = 0 1(x ⊕ y) = 1(x) ⊕ 1(y) X 1(x ⇒ y) = 1(x) ⇒ 1(y) X 1(sup{x | x ∈ X }) = sup{1(x) | x ∈ X } X 1(min{x | x ∈ X }) = min{1(x) | x ∈ X } X



X

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Crisp models Let I be a witnessed model of O and x, y ∈ ∆I . aJ := aI ,


AJ (x) := 1 AI (x) ,

r J (x, y) := 1 r I (x, y)


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AJ (x) := 1 AI (x) ,

r J (x, y) := 1 r I (x, y)

C J (x) = 1 C I (x)



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AJ (x) := 1 AI (x) ,

r J (x, y) := 1 r I (x, y)

C J (x) = 1 C I (x)

ha : C, ì ∈ O hC v D, ì ∈ O

C J (aJ ) = 1 ≥ ` since C I (aI ) ≥ ` > 0 C J (x) ⇒ DJ (x) = 1 C I (x) ⇒ DI (x) = 1 ≥ ` since C I (x) ⇒ DI (x) ≥ ` > 0



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AJ (x) := 1 AI (x) ,

r J (x, y) := 1 r I (x, y)

C J (x) = 1 C I (x)

ha : C, ì ∈ O hC v D, ì ∈ O




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AJ (x) := 1 AI (x) ,

r J (x, y) := 1 r I (x, y)

C J (x) = 1 C I (x)

ha : C, ì ∈ O hC v D, ì ∈ O


O has a model iff crisp(O) has a crisp model. replace every ` in O by 1(`)

ë



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Results

Deciding consistency in ⊗-SHOI is EXPTIME-complete.



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Results

Deciding consistency in ⊗-SHOI is EXPTIME-complete. ⊗-SHOI and its sublogics have the finite model property.



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Results

Deciding consistency in ⊗-SHOI is EXPTIME-complete. ⊗-SHOI and its sublogics have the finite model property. Contradicts previous result that Π-ALC does not have the finite model property. [Bobillo, Bou, Straccia 2011]



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Results

Deciding consistency in ⊗-SHOI is EXPTIME-complete. ⊗-SHOI and its sublogics have the finite model property. Contradicts previous result that Π-ALC does not have the finite model property. [Bobillo, Bou, Straccia 2011] The fuzzy DL ⊗-SHOI is not fuzzy if ⊗ does not have zero divisors!



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It’s not that bad

Also (`-)satisfiability in ⊗-SHOI is in EXPTIME.



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It’s not that bad

Also (`-)satisfiability in ⊗-SHOI is in EXPTIME. However, for entailment (subsumption and instance checking), it is not as easy.



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It’s not that bad

Also (`-)satisfiability in ⊗-SHOI is in EXPTIME. However, for entailment (subsumption and instance checking), it is not as easy. Subsumption: Does C I (x) ⇒ DI (x) ≥ ` hold in all models I of O?



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It’s not that bad

Also (`-)satisfiability in ⊗-SHOI is in EXPTIME. However, for entailment (subsumption and instance checking), it is not as easy. Subsumption: Does C I (x) ⇒ DI (x) ≥ ` hold in all models I of O? Crisp models are not enough!



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It’s not that bad

Also (`-)satisfiability in ⊗-SHOI is in EXPTIME. However, for entailment (subsumption and instance checking), it is not as easy. Subsumption: Does C I (x) ⇒ DI (x) ≥ ` hold in all models I of O? Crisp models are not enough! Consider O = {h> v A, 0.5i}, C = > , D = A, and ` = 1. In every crisp model J of O, AJ (x) = 1 holds. However, AI (x) = ` is also possible. Thus, AI (x) ≥ 1 in every crisp model, but not in every model.



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It’s not that bad

Also (`-)satisfiability in ⊗-SHOI is in EXPTIME. However, for entailment (subsumption and instance checking), it is not as easy. Subsumption: Does C I (x) ⇒ DI (x) ≥ ` hold in all models I of O? Crisp models are not enough! Consider O = {h> v A, 0.5i}, C = > , D = A, and ` = 1. In every crisp model J of O, AJ (x) = 1 holds. However, AI (x) = ` is also possible. Thus, AI (x) ≥ 1 in every crisp model, but not in every model.



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Conclusions • consistency/satisfiability w.r.t. witnessed models in ⊗-SHOI is decidable

NEL [u, >, ∃, ¬]


ELC [u, >, ∃, ]

crisp (` = 1)

Z G

Z G

Π Ł G

≥`

Z G

Z G

G G

=`

Z G

G G

G G

assertions




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assertions

(undecidable) (decidable) crisp (` = 1)


NEL [u, >, ∃, ¬]


ELC [u, >, ∃, ]

Z Z

Z Z

Π Ł G

≥`

Z Z

Z Z

G G

=`

Z G

G G

G G


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assertions

(undecidable) (decidable) crisp (` = 1)

NEL [u, >, ∃, ¬]


ELC [u, >, ∃, ]

Z Z

Z Z

Π Ł G

≥`

Z Z

Z Z

G G

=`

Z G

G G

G G

• similar results for general models (without ∀) ˜ [Borgwardt, Distel, Penaloza DL’12]

• subsumption/instance checking still open • decidability of consistency still unknown for some fuzzy DLs



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Thank You

´ Fernando Bobillo, Miguel Delgado, Juan Gomez-Romero, and Umberto Straccia. ¨ Fuzzy description logics under Godel semantics. International Journal of Approximate Reasoning, 50(3):494–514, 2009. ´ Bou, and Umberto Straccia. Fernando Bobillo, Felix On the failure of the finite model property in some fuzzy description logics. Fuzzy Sets and Systems, 172(1):1–12, 2011. ˜ Stefan Borgwardt and Rafael Penaloza. Undecidability of fuzzy description logics. In Proc. KR’12. AAAI Press, 2012. ˜ Stefan Borgwardt, Felix Distel, and Rafael Penaloza. ¨ Godel negation makes unwitnessed consistency crisp. In Yevgeny Kazakov, Domenico Lembo, and Frank Wolter, editors, Proc. DL’12, volume 846 of CEUR Workshop Proceedings, pages 103–113, 2012.



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