Undecidability of Fuzzy Description Logics

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t-norm. Examples. G ödel min{x, y}. Łukasiewicz max{x + y − 1, 0}. Product x · y. 0. 1. |. Ł. |. Π. |. Ł(0,b). Rome, 12.6.2012. Undecidability of Fuzzy Description ...
Institute of Theoretical Computer Science Chair of Automata Theory

UNDECIDABILITY OF FUZZY DESCRIPTION LOGICS ˜ Stefan Borgwardt and Rafael Penaloza

Rome, 12.6.2012

Imprecise Knowledge Tall : ∆ → [0, 1]

How to combine these degrees?

Rome, 12.6.2012

Undecidability of Fuzzy Description Logics

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Interpreting Conjunctions binary operator ⊗ : [0, 1] × [0, 1] → [0, 1]:

• commutative, associative • monotonic • unit 1

t-norm

Examples ¨ Godel

min{x, y}

Łukasiewicz

max{x + y − 1, 0}

Product

x ·y

Ł

Π

0|

|

|1

Ł(0,b)

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Undecidability of Fuzzy Description Logics

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Fuzzy Description Logics

EL

A

AI : ∆ → [0, 1]

r

r I : ∆ × ∆ → [0, 1]

>

1

C1 u C2

C1I (x) ⊗ C2I (x)

∃r.C

max y∈∆I r I (x, y) ⊗ C I (y)

N

C

C I (x) ⇒ 0

C

¬C

1 − C I (x)

I

⊥ C→D

A

∀r.C

0 C I (x) ⇒ DI (x) min y∈∆I r I (x, y) ⇒ C I (y)

restriction: witnessed interpretations Rome, 12.6.2012

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Axioms

hC v D . `i

GCIs

ha : C . `i

assertions

h(a, b) : r . `i

C I (x) ⇒ DI (x) . ` C I (aI ) . ` r I (aI , bI ) . `

• .: binary relation on [0, 1] • ` ∈ [0, 1]

hC v D ≥ 1i

hC v Di C I (x) ≤ DI (x)

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Undecidability of Fuzzy Description Logics

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A Big Family

GCIs

Πw -ELC c≥ t-norm

models

assertions

constructors

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Undecidability Results

Ontology consistency is undecidable for:

• Πw -ALC c,> ≥

[Baader,P FuzzIEEE11]

• Πw(0,b) -IALc≥,=

[Baader,P FroCoS11]

• Łw -ALC c≥

[Cerami,Straccia 11]

.. .

Same basic idea, can we generalize it?

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PCP v1 , w1 Input: (v1 , w1 ), . . . , (vn , wn ) in {1, . . . , s}∗

r1

rn ...

v11 , w11

v1n , w1n

vνi = vν vi r1

r1

rn

v111 , w111

v11n , w11n

v1n1 , w1n1

v1nn , w1nn

...

...

...

...

...

Question: is there ν with v1ν = w1ν ?

rn ...

1. “encode” search tree 2. ensure v1ν 6= w1ν for all ν

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“Encoding” the Tree injective encoding function constant k

: Σ∗ → [0, 1],

Πw -ELC c≥ v = 2−v k = 21

v 6= w iff either v ≤ w ⊗ k or w ≤ v ⊗ k V = v1 , W = w1

...

rn

Rome, 12.6.2012

...

...

V = v111 , W = w111

rn

V = v1n1 , W = w1n1 V = v11n , W = w11n

V = v1n , W = w1n

r1

...

rn V = v1nn , W = w1nn

Undecidability of Fuzzy Description Logics

...

r1

...

...

r1

V = v11 , W = w11

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Root Initialization

V = v1 , W = w1

Vi = vi , Wi = wi

Property Pini exists OC(a)=u such that C I (aI ) = u

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Πw -ELC c≥ {ha : C ≥ ui,

Undecidability of Fuzzy Description Logics

ha : ¬C ≥ 1 − ui }

10

Creating Successors

r1

rn ...

rn

r1

rn

...

... ...

...

...

...

r1

Property P→ exists O∃r such that for every x, exists y with

Πw -ELC c≥ {h> v ∃r.>i }

r I (x, y) = 1 Rome, 12.6.2012

Undecidability of Fuzzy Description Logics

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Populating the Tree

V = vν

Vi = vi V 0 = vν vi

ri

V = vν vi

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Undecidability of Fuzzy Description Logics

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Concatenating Words

Property P◦ exists OC◦u such that if

• C I (x) = v, and • CuI (x) = u,

Πw -ELC c≥ |u|

{hDC◦u v C (s+1) |u|

hC (s+1)

u Cu i

u Cu v DC◦u i }

I (x) = vu then DC◦u

(C m )I (x) = 2n(−v)

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Undecidability of Fuzzy Description Logics

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Value Transfer

Property P exists OC

rD

Πw -ELC c≥

such that

{h∃r.¬D v ¬Ci h∃r.D v Ci }

if r I (x, y) = 1, then C I (x) = DI (y)

DI (y)

Rome, 12.6.2012



max r I (x, z) ⊗ DI (z)

=

(∃r.D)I (x)



C I (x)

z

Undecidability of Fuzzy Description Logics

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Canonical Model If Pini , P→ , P◦ , P , then exists OP such that every model “embeds”

V = v1 , W = w1

...

rn

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...

...

V = v111 , W = w111

rn

V = v1n1 , W = w1n1 V = v11n , W = w11n

V = v1n , W = w1n

r1

...

rn V = v1nn , W = w1nn

Undecidability of Fuzzy Description Logics

...

r1

...

...

r1

V = v11 , W = w11

15

Deciding Solutions

v 6= w iff v ≤ w ⊗ k or w ≤ v ⊗ k

Property P6=

Πw -ELC c≥

exists ontology O6= : V I (x) ≤ W I (x) ⊗ k, or

{hX v X u X i, hH v ¬Hi, h¬H v Hi,

W I (x) ≤ V I (x) ⊗ k

hX u V v X u W u Hi, h¬X u W v ¬X u V u Hi }

Theorem P has a solution iff OP ∪ O6= is inconsistent

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Undecidability of Fuzzy Description Logics

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Undecidable Logics

Ontology consistency is undecidable in any logic satisfying Pini , P→ , P◦ , P , P6=

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assertions

Results

crisp

constructors NEL IAL ELC Ł(0,b) Ł(0,b) Π Ł



Ł(0,b)

=

(0,b)

Ł

Ł(0,b)

G

G

G

• all GCIs crisp • w.r.t. (top-)witnessed models • w.r.t. general models if crisp roles

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Future Work

Generalize:

• general models • residuated lattice semantics

Study subsumption

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