t-norm. Examples. G ödel min{x, y}. Åukasiewicz max{x + y â 1, 0}. Product x · y. 0. 1. |. Å. |. Î . |. Å(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?
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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
Undecidability of Fuzzy Description Logics
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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 of Fuzzy Description Logics
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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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Undecidability of Fuzzy Description Logics
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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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Undecidability of Fuzzy Description Logics
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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
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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
9
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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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
14
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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