Jul 18, 2014 - {rafael} ⥠hã. ãHappy â âhasFriend.Happy ⥠tã. hasFriend â hasFriendâ. Wien, July 18th, 2014. Fuzzy DLs over Finite Lattices with Nominals.
Institute of Theoretical Computer Science Chair of Automata Theory
FUZZY DLS OVER FINITE LATTICES WITH NOMINALS Stefan Borgwardt
Wien, July 18th, 2014
Introduction Fuzzy logics:
• fuzzy sets C : ∆ → [0, 1] replace crisp sets C ⊆ ∆
(Zadeh 1965)
Tall
• conjunction interpreted as minimum
Wien, July 18th, 2014
Fuzzy DLs over Finite Lattices with Nominals
1
Introduction Fuzzy logics:
• fuzzy sets C : ∆ → [0, 1] replace crisp sets C ⊆ ∆
(Zadeh 1965)
Tall
• conjunction interpreted as minimum • lattices and L-fuzzy sets C : ∆ → L • mathematical fuzzy logic with t-norms
Wien, July 18th, 2014
Fuzzy DLs over Finite Lattices with Nominals
(Goguen 1967) ´ (Hajek 2001)
1
Introduction Fuzzy logics:
• fuzzy sets C : ∆ → [0, 1] replace crisp sets C ⊆ ∆
(Zadeh 1965)
Tall
• conjunction interpreted as minimum • lattices and L-fuzzy sets C : ∆ → L • mathematical fuzzy logic with t-norms
(Goguen 1967) ´ (Hajek 2001)
Fuzzy description logics:
• Zadeh semantics • lattice-based semantics • t-norm-based semantics Wien, July 18th, 2014
Fuzzy DLs over Finite Lattices with Nominals
(Straccia 2001) (Straccia 2004b) ´ (Hajek 2005) 1
Complete Residuated De Morgan Lattices 1
1
complete distributive lattice (L, ∨, ∧, 0, 1): ∼a ∼b
∼c ⊆R
a
c
b 0
Wien, July 18th, 2014
Fuzzy DLs over Finite Lattices with Nominals
0
2
Complete Residuated De Morgan Lattices 1
1
complete distributive lattice (L, ∨, ∧, 0, 1):
• (generalized) t-norm ⊗ : L × L → L:
∼a ∼b
∼c ⊆R
associative, commutative, monotone, unit 1, (continuous) a
c
b 0
Wien, July 18th, 2014
Fuzzy DLs over Finite Lattices with Nominals
0
2
Complete Residuated De Morgan Lattices 1
1
complete distributive lattice (L, ∨, ∧, 0, 1):
• (generalized) t-norm ⊗ : L × L → L:
∼a ∼b
∼c ⊆R
associative, commutative, monotone, unit 1, (continuous)
• residuum ⇒ : L × L → L: (x ⊗ y) ≤ z iff y ≤ (x ⇒ z) • residual negation x = x ⇒ 0
a
c
b 0
Wien, July 18th, 2014
Fuzzy DLs over Finite Lattices with Nominals
0
2
Complete Residuated De Morgan Lattices 1
1
complete distributive lattice (L, ∨, ∧, 0, 1):
• (generalized) t-norm ⊗ : L × L → L:
∼a ∼b
∼c
∼x = 1−x
associative, commutative, monotone, unit 1, (continuous)
• residuum ⇒ : L × L → L: (x ⊗ y) ≤ z iff y ≤ (x ⇒ z) • residual negation x = x ⇒ 0 • involutive De Morgan negation ∼ : L → L
Wien, July 18th, 2014
Fuzzy DLs over Finite Lattices with Nominals
a
c
b 0
0
2
Complete Residuated De Morgan Lattices 1
1
complete distributive lattice (L, ∨, ∧, 0, 1):
• (generalized) t-norm ⊗ : L × L → L:
∼a ∼b
∼c
∼x = 1−x
associative, commutative, monotone, unit 1, (continuous)
• residuum ⇒ : L × L → L: (x ⊗ y) ≤ z iff y ≤ (x ⇒ z) • residual negation x = x ⇒ 0 • involutive De Morgan negation ∼ : L → L
a
c
b 0
0
Everyone is an expert ...
Wien, July 18th, 2014
Fuzzy DLs over Finite Lattices with Nominals
2
Complete Residuated De Morgan Lattices 1
1
complete distributive lattice (L, ∨, ∧, 0, 1):
• (generalized) t-norm ⊗ : L × L → L:
∼a ∼b
∼c
∼x = 1−x
associative, commutative, monotone, unit 1, (continuous)
• residuum ⇒ : L × L → L: (x ⊗ y) ≤ z iff y ≤ (x ⇒ z) • residual negation x = x ⇒ 0 • involutive De Morgan negation ∼ : L → L
a
c
b 0
0
Everyone is an expert ... to a certain degree:
doctor
textbook newspaper neighbor
Wien, July 18th, 2014
Fuzzy DLs over Finite Lattices with Nominals
2
Complete Residuated De Morgan Lattices 1
1
complete distributive lattice (L, ∨, ∧, 0, 1):
• (generalized) t-norm ⊗ : L × L → L:
∼a ∼b
∼c
∼x = 1−x
associative, commutative, monotone, unit 1, (continuous)
• residuum ⇒ : L × L → L: (x ⊗ y) ≤ z iff y ≤ (x ⇒ z) • residual negation x = x ⇒ 0 • involutive De Morgan negation ∼ : L → L
a
c
b 0
0
Everyone is an expert ... to a certain degree:
doctor
textbook newspaper neighbor
Wien, July 18th, 2014
Fuzzy DLs over Finite Lattices with Nominals
2
Complete Residuated De Morgan Lattices 1
1
complete distributive lattice (L, ∨, ∧, 0, 1):
• (generalized) t-norm ⊗ : L × L → L:
∼a ∼b
∼c
∼x = 1−x
associative, commutative, monotone, unit 1, (continuous)
• residuum ⇒ : L × L → L: (x ⊗ y) ≤ z iff y ≤ (x ⇒ z) • residual negation x = x ⇒ 0 • involutive De Morgan negation ∼ : L → L
a
c
b 0
0
Everyone is an expert ... to a certain degree:
doctor
textbook newspaper neighbor
Wien, July 18th, 2014
Fuzzy DLs over Finite Lattices with Nominals
2
Complete Residuated De Morgan Lattices 1
1
complete distributive lattice (L, ∨, ∧, 0, 1):
• (generalized) t-norm ⊗ : L × L → L:
∼a ∼b
∼c
∼x = 1−x
associative, commutative, monotone, unit 1, (continuous)
• residuum ⇒ : L × L → L: (x ⊗ y) ≤ z iff y ≤ (x ⇒ z) • residual negation x = x ⇒ 0 • involutive De Morgan negation ∼ : L → L
a
c
b 0
0
Everyone is an expert ... to a certain degree:
doctor
textbook newspaper neighbor
Wien, July 18th, 2014
Fuzzy DLs over Finite Lattices with Nominals
2
L-ISCHOI TallI : ∆I → L
Wien, July 18th, 2014
hasFriendI : ∆I × ∆I → L
Fuzzy DLs over Finite Lattices with Nominals
stefanI ∈ ∆I
3
L-ISCHOI TallI : ∆I → L p I (x) = p
Wien, July 18th, 2014
hasFriendI : ∆I × ∆I → L ( 1 if stefanI = x {stefan}I (x) = 0 otherwise
Fuzzy DLs over Finite Lattices with Nominals
stefanI ∈ ∆I (¬Tall)I (x) = ∼TallI (x)
3
L-ISCHOI TallI : ∆I → L p I (x) = p
hasFriendI : ∆I × ∆I → L ( 1 if stefanI = x {stefan}I (x) = 0 otherwise
(∃hasFriend.Tall)I (x) =
_
stefanI ∈ ∆I (¬Tall)I (x) = ∼TallI (x)
hasFriendI (x, y) ⊗ TallI (y)
r − , u, →, ∀
y∈∆I
Wien, July 18th, 2014
Fuzzy DLs over Finite Lattices with Nominals
3
L-ISCHOI TallI : ∆I → L
hasFriendI : ∆I × ∆I → L ( 1 if stefanI = x {stefan}I (x) = 0 otherwise
p I (x) = p
(∃hasFriend.Tall)I (x) =
_
stefanI ∈ ∆I (¬Tall)I (x) = ∼TallI (x)
hasFriendI (x, y) ⊗ TallI (y)
r − , u, →, ∀
y∈∆I
hstefan:Happy ./ pi hSuccessful v ¬Happy ≥ pi
Wien, July 18th, 2014
HappyI (stefanI ) ./ p SuccessfulI (x) ⇒ (¬Happy)I (x) ≥ p
Fuzzy DLs over Finite Lattices with Nominals
3
L-ISCHOI TallI : ∆I → L
hasFriendI : ∆I × ∆I → L ( 1 if stefanI = x {stefan}I (x) = 0 otherwise
p I (x) = p
(∃hasFriend.Tall)I (x) =
_
stefanI ∈ ∆I (¬Tall)I (x) = ∼TallI (x)
hasFriendI (x, y) ⊗ TallI (y)
r − , u, →, ∀
y∈∆I
hstefan:Happy ./ pi hSuccessful v ¬Happy ≥ pi hasFriend v likes trans(likes)
Wien, July 18th, 2014
HappyI (stefanI ) ./ p SuccessfulI (x) ⇒ (¬Happy)I (x) ≥ p hasFriendI (x, y) ≤ likesI (x, y) likesI (x, y) ⊗ likesI (y, z) ≤ likesI (x, z)
Fuzzy DLs over Finite Lattices with Nominals
3
L-ISCHOI TallI : ∆I → L
hasFriendI : ∆I × ∆I → L ( 1 if stefanI = x {stefan}I (x) = 0 otherwise
p I (x) = p
(∃hasFriend.Tall)I (x) =
_
stefanI ∈ ∆I (¬Tall)I (x) = ∼TallI (x)
hasFriendI (x, y) ⊗ TallI (y)
r − , u, →, ∀
y∈∆I
hstefan:Happy ./ pi hSuccessful v ¬Happy ≥ pi hasFriend v likes trans(likes)
HappyI (stefanI ) ./ p SuccessfulI (x) ⇒ (¬Happy)I (x) ≥ p hasFriendI (x, y) ≤ likesI (x, y) likesI (x, y) ⊗ likesI (y, z) ≤ likesI (x, z)
subsumption: O |= hSuccessful v ¬Happy ≥ pi ?
Wien, July 18th, 2014
Fuzzy DLs over Finite Lattices with Nominals
3
L-ISCHOI TallI : ∆I → L
hasFriendI : ∆I × ∆I → L ( 1 if stefanI = x {stefan}I (x) = 0 otherwise
p I (x) = p
(∃hasFriend.Tall)I (x) =
_
stefanI ∈ ∆I (¬Tall)I (x) = ∼TallI (x)
hasFriendI (x, y) ⊗ TallI (y)
r − , u, →, ∀
y∈∆I
hstefan:Happy ./ pi hSuccessful v ¬Happy ≥ pi hasFriend v likes trans(likes)
HappyI (stefanI ) ./ p SuccessfulI (x) ⇒ (¬Happy)I (x) ≥ p hasFriendI (x, y) ≤ likesI (x, y) likesI (x, y) ⊗ likesI (y, z) ≤ likesI (x, z)
subsumption: O |= hSuccessful v ¬Happy ≥ pi iff O ∪ {ha:Successful → ¬Happy < pi} is inconsistent
Wien, July 18th, 2014
Fuzzy DLs over Finite Lattices with Nominals
3
Related Work
Consistency is ...
• undecidable in small extensions of L-EL over many infinite lattices L ˜ (Baader and Penaloza 2011; Cerami and Straccia 2013)
Wien, July 18th, 2014
Fuzzy DLs over Finite Lattices with Nominals
4
Related Work
Consistency is ...
• undecidable in small extensions of L-EL over many infinite lattices L ˜ (Baader and Penaloza 2011; Cerami and Straccia 2013)
• trivial in L-ISHOI without zero divisors and {≤, }-assertions ˜ (Borgwardt, Distel, and Penaloza 2012)
Wien, July 18th, 2014
Fuzzy DLs over Finite Lattices with Nominals
4
Related Work
Consistency is ...
• undecidable in small extensions of L-EL over many infinite lattices L ˜ (Baader and Penaloza 2011; Cerami and Straccia 2013)
• trivial in L-ISHOI without zero divisors and {≤, }-assertions ˜ (Borgwardt, Distel, and Penaloza 2012)
• decidable in L-ISCROIQ over finite total orders L (Bobillo et al. 2012; Straccia 2004a)
Wien, July 18th, 2014
Fuzzy DLs over Finite Lattices with Nominals
4
Related Work
Consistency is ...
• undecidable in small extensions of L-EL over many infinite lattices L ˜ (Baader and Penaloza 2011; Cerami and Straccia 2013)
• trivial in L-ISHOI without zero divisors and {≤, }-assertions ˜ (Borgwardt, Distel, and Penaloza 2012)
• decidable in L-ISCROIQ over finite total orders L (Bobillo et al. 2012; Straccia 2004a)
• PSPACE/EXPTIME-complete in L-ISCHI over finite lattices L ˜ (Borgwardt and Penaloza 2013)
Wien, July 18th, 2014
Fuzzy DLs over Finite Lattices with Nominals
4
Related Work
Consistency is ...
• undecidable in small extensions of L-EL over many infinite lattices L ˜ (Baader and Penaloza 2011; Cerami and Straccia 2013)
• trivial in L-ISHOI without zero divisors and {≤, }-assertions ˜ (Borgwardt, Distel, and Penaloza 2012)
• decidable in L-ISCROIQ over finite total orders L (Bobillo et al. 2012; Straccia 2004a)
• PSPACE/EXPTIME-complete in L-ISCHI over finite lattices L
ë extension to L-ISCHOI:
˜ (Borgwardt and Penaloza 2013)
• construction for nominals • PSPACE results for two sublogics
Wien, July 18th, 2014
Fuzzy DLs over Finite Lattices with Nominals
4
Pre-completions t h f
Wien, July 18th, 2014
x ⊗y f h t
f f f f
h f f h
t f h t
x⇒y f h t
Fuzzy DLs over Finite Lattices with Nominals
f t h f
h t t h
t t t t
5
Pre-completions t
x ⊗y f h t
h f
hstefan:Happy ≥ hi
f f f f
h f f h
t f h t
hrafael:Happy ≥ hi
hHappy v ∀hasFriend.Happy ≥ ti
Wien, July 18th, 2014
x⇒y f h t
f t h f
h t t h
t t t t
hstefan:∃hasFriend.{rafael} ≥ hi hasFriend v hasFriend−
Fuzzy DLs over Finite Lattices with Nominals
5
Pre-completions t
x ⊗y f h t
h f
hstefan:Happy ≥ hi
f f f f
h f f h
t f h t
hrafael:Happy ≥ hi
hHappy v ∀hasFriend.Happy ≥ ti
x⇒y f h t
f t h f
h t t h
t t t t
hstefan:∃hasFriend.{rafael} ≥ hi hasFriend v hasFriend−
Pre-completion:
• partition of the individual names
Wien, July 18th, 2014
Fuzzy DLs over Finite Lattices with Nominals
5
Pre-completions t
x ⊗y f h t
h f
hstefan:Happy ≥ hi
f f f f
h f f h
t f h t
hrafael:Happy ≥ hi
hHappy v ∀hasFriend.Happy ≥ ti
x⇒y f h t
f t h f
h t t h
t t t t
hstefan:∃hasFriend.{rafael} ≥ hi hasFriend v hasFriend−
Pre-completion:
• partition of the individual names • Hintikka functions : sub(O) → L Hstefan : Happy, ∃hasFriend.{rafael}, ∀hasFriend.Happy, {stefan} Hrafael : Happy, ∀hasFriend.Happy, {rafael}
Wien, July 18th, 2014
Fuzzy DLs over Finite Lattices with Nominals
5
Pre-completions t
x ⊗y f h t
h f
hstefan:Happy ≥ hi
f f f f
h f f h
t f h t
hrafael:Happy ≥ hi
hHappy v ∀hasFriend.Happy ≥ ti
x⇒y f h t
f t h f
h t t h
t t t t
hstefan:∃hasFriend.{rafael} ≥ hi hasFriend v hasFriend−
Pre-completion:
• partition of the individual names • Hintikka functions : sub(O) → L Hstefan : Happy, ∃hasFriend.{rafael}, ∀hasFriend.Happy, {stefan} Hrafael : Happy, ∀hasFriend.Happy, {rafael}
• role connections: RhasFriend : (stefan, rafael), (rafael, stefan)
Wien, July 18th, 2014
Fuzzy DLs over Finite Lattices with Nominals
5
Pre-completions t
x ⊗y f h t
h f
hstefan:Happy ≥ hi
f f f f
h f f h
t f h t
hrafael:Happy ≥ hi
hHappy v ∀hasFriend.Happy ≥ ti
x⇒y f h t
f t h f
h t t h
t t t t
hstefan:∃hasFriend.{rafael} ≥ hi hasFriend v hasFriend−
Pre-completion:
• partition of the individual names • Hintikka functions : sub(O) → L Hstefan : Happy, ∃hasFriend.{rafael}, ∀hasFriend.Happy, {stefan} Hrafael : Happy, ∀hasFriend.Happy, {rafael}
• role connections: RhasFriend : (stefan, rafael), (rafael, stefan)
• for EXPTIME and PSPACE upper bounds
Wien, July 18th, 2014
Fuzzy DLs over Finite Lattices with Nominals
5
Hintikka trees x ⊗y f h t
f f f f
h f f h
t f h t
x⇒y f h t
f t h f
h t t h
t t t t
hHappy v ∀hasFriend.Happy ≥ ti hasFriend v hasFriend−
Hstefan : Happy, ∃hasFriend.{rafael}, ∀hasFriend.Happy, {stefan} Hrafael : Happy, ∀hasFriend.Happy, {rafael} RhasFriend : (stefan, rafael), (rafael, stefan)
Wien, July 18th, 2014
Fuzzy DLs over Finite Lattices with Nominals
6
Hintikka trees x ⊗y f h t
f f f f
h f f h
t f h t
x⇒y f h t
f t h f
h t t h
t t t t
hHappy v ∀hasFriend.Happy ≥ ti hasFriend v hasFriend−
Hstefan : Happy, ∃hasFriend.{rafael}, ∀hasFriend.Happy, {stefan} Hrafael : Happy, ∀hasFriend.Happy, {rafael} RhasFriend : (stefan, rafael), (rafael, stefan) Hintikka trees: Hstefan
Wien, July 18th, 2014
Fuzzy DLs over Finite Lattices with Nominals
6
Hintikka trees x ⊗y f h t
f f f f
h f f h
t f h t
x⇒y f h t
f t h f
h t t h
t t t t
hHappy v ∀hasFriend.Happy ≥ ti hasFriend v hasFriend−
Hstefan : Happy, ∃hasFriend.{rafael}, ∀hasFriend.Happy, {stefan} Hrafael : Happy, ∀hasFriend.Happy, {rafael} RhasFriend : (stefan, rafael), (rafael, stefan) Hintikka trees: Hstefan hasFriend hasFriend− {rafael}
Wien, July 18th, 2014
Fuzzy DLs over Finite Lattices with Nominals
6
Hintikka trees x ⊗y f h t
f f f f
h f f h
t f h t
x⇒y f h t
f t h f
h t t h
t t t t
hHappy v ∀hasFriend.Happy ≥ ti hasFriend v hasFriend−
Hstefan : Happy, ∃hasFriend.{rafael}, ∀hasFriend.Happy, {stefan} Hrafael : Happy, ∀hasFriend.Happy, {rafael} RhasFriend : (stefan, rafael), (rafael, stefan) Hintikka trees: Hstefan hasFriend hasFriend− {rafael}, Happy, ∀hasFriend.Happy
Wien, July 18th, 2014
Fuzzy DLs over Finite Lattices with Nominals
6
Hintikka trees x ⊗y f h t
f f f f
h f f h
t f h t
x⇒y f h t
f t h f
h t t h
t t t t
hHappy v ∀hasFriend.Happy ≥ ti hasFriend v hasFriend−
Hstefan : Happy, ∃hasFriend.{rafael}, ∀hasFriend.Happy, {stefan} Hrafael : Happy, ∀hasFriend.Happy, {rafael} RhasFriend : (stefan, rafael), (rafael, stefan) Hintikka trees: Hstefan hasFriend hasFriend− {rafael}, Happy, ∀hasFriend.Happy
Wien, July 18th, 2014
Fuzzy DLs over Finite Lattices with Nominals
6
Hintikka trees x ⊗y f h t
f f f f
h f f h
t f h t
x⇒y f h t
f t h f
h t t h
t t t t
hHappy v ∀hasFriend.Happy ≥ ti hasFriend v hasFriend−
Hstefan : Happy, ∃hasFriend.{rafael}, ∀hasFriend.Happy, {stefan} Hrafael : Happy, ∀hasFriend.Happy, {rafael} RhasFriend : (stefan, rafael), (rafael, stefan) Hintikka trees: Hstefan hasFriend hasFriend− {rafael}, Happy, ∀hasFriend.Happy
Wien, July 18th, 2014
hasFriend hasFriend− Happy, ∀hasFriend.Happy
Fuzzy DLs over Finite Lattices with Nominals
6
Hintikka trees x ⊗y f h t
f f f f
h f f h
t f h t
x⇒y f h t
f t h f
h t t h
t t t t
hHappy v ∀hasFriend.Happy ≥ ti hasFriend v hasFriend−
Hstefan : Happy, ∃hasFriend.{rafael}, ∀hasFriend.Happy, {stefan} Hrafael : Happy, ∀hasFriend.Happy, {rafael} RhasFriend : (stefan, rafael), (rafael, stefan) Hintikka trees: Hstefan hasFriend hasFriend− {rafael}, Happy, ∀hasFriend.Happy .. . Wien, July 18th, 2014
hasFriend hasFriend− Happy, ∀hasFriend.Happy .. .
Fuzzy DLs over Finite Lattices with Nominals
6
PSPACE on-the-fly constructions
Looping tree automata using Hintikka functions as states
Wien, July 18th, 2014
Fuzzy DLs over Finite Lattices with Nominals
in EXPTIME
7
PSPACE on-the-fly constructions
Looping tree automata using Hintikka functions as states PSPACE on-the-fly constructions:
• • • •
in EXPTIME
˜ (Baader, Hladik, and Penaloza 2008)
polynomial arity states of polynomial size polynomial initial and transition conditions polynomial blocking
Wien, July 18th, 2014
Fuzzy DLs over Finite Lattices with Nominals
7
PSPACE on-the-fly constructions
Looping tree automata using Hintikka functions as states PSPACE on-the-fly constructions:
• • • • •
in EXPTIME
˜ (Baader, Hladik, and Penaloza 2008)
polynomial arity states of polynomial size polynomial initial and transition conditions polynomial blocking can be achieved for L-IALCHO and L-ISCOc with acyclic TBoxes
Wien, July 18th, 2014
Fuzzy DLs over Finite Lattices with Nominals
in PSPACE
7
PSPACE/EXPTIME boundary SHOI
ALCHOI
ALCOI
SOI
ALCHO
ALCO
SHO
ALCHI
ALCI
SO
ALCH
SHI
SI
SH
S
ALC
Wien, July 18th, 2014
Fuzzy DLs over Finite Lattices with Nominals
8
PSPACE/EXPTIME boundary SHOI
ALCHOI
ALCOI
SOI
ALCHO
ALCO
SHO
ALCHI
ALCI
SO
ALCH
SHI
SI
SH
S
ALC (Horrocks 1997; Horrocks, Sattler, and Tobies 2000; Tobies 2000)
Wien, July 18th, 2014
Fuzzy DLs over Finite Lattices with Nominals
8
PSPACE/EXPTIME boundary SHOI
ALCHOI
ALCOI
SOI
ALCHO
ALCO
SHO
ALCHI
ALCI
SO
ALCH
SHI
SI
SH
S
ALC (Horrocks 1997; Horrocks, Sattler, and Tobies 2000; Tobies 2000) ˜ (Borgwardt and Penaloza 2013) Wien, July 18th, 2014
Fuzzy DLs over Finite Lattices with Nominals
8
PSPACE/EXPTIME boundary SHOI
ALCHOI
ALCOI
SOI
ALCHO
ALCO
SHO
ALCHI
ALCI
SO
ALCH
SHI
SI
SH
S
ALC (Horrocks 1997; Horrocks, Sattler, and Tobies 2000; Tobies 2000) ˜ (Borgwardt and Penaloza 2013) (Borgwardt 2014) Wien, July 18th, 2014
Fuzzy DLs over Finite Lattices with Nominals
8
Conclusions Summary:
• General finite-valued fuzzy semantics for SHOI • Same complexity as classical reasoning • PSPACE upper bound for ALCHO and SO with acyclic TBoxes
Wien, July 18th, 2014
Fuzzy DLs over Finite Lattices with Nominals
9
Conclusions Summary:
• General finite-valued fuzzy semantics for SHOI • Same complexity as classical reasoning • PSPACE upper bound for ALCHO and SO with acyclic TBoxes Future Work:
• Fuzzy role axioms htrans(r) ≥ pi • More expressivity (SHIQ, SHOQ, SROIQ, concrete domains) • Implement efficient reasoner for finite-valued fuzzy DLs
Wien, July 18th, 2014
Fuzzy DLs over Finite Lattices with Nominals
9
Conclusions Summary:
• General finite-valued fuzzy semantics for SHOI • Same complexity as classical reasoning • PSPACE upper bound for ALCHO and SO with acyclic TBoxes Future Work:
• Fuzzy role axioms htrans(r) ≥ pi • More expressivity (SHIQ, SHOQ, SROIQ, concrete domains) • Implement efficient reasoner for finite-valued fuzzy DLs
Thank you
Wien, July 18th, 2014
Fuzzy DLs over Finite Lattices with Nominals
9
References I • Baader, Franz, Jan Hladik, and Rafael Penaloza ˜ (2008). “Automata Can Show PSPACE Results for Description Logics”. In: Inform. Comput. 206.9-10, pages 1045–1056. • Baader, Franz and Rafael Penaloza ˜ (2011). “On the Undecidability of Fuzzy Description Logics with GCIs and Product T-norm”. In: Proc. FroCoS’11. Volume 6989. LNCS, pages 55–70. • Bobillo, Fernando, Miguel Delgado, Juan Gomez-Romero, ´ and Umberto Straccia (2012). “Joining ¨ Godel and Zadeh Fuzzy Logics in Fuzzy Description Logics”. In: Int. J. Uncertain. Fuzz. 20.4, pages 475–508. • Borgwardt, Stefan (2014). “Fuzzy Description Logics with General Concept Inclusions”. ¨ Dresden. PhD thesis. Technische Universitat • Borgwardt, Stefan, Felix Distel, and Rafael Penaloza ˜ (2012). “How Fuzzy is my Fuzzy Description Logic?” In: Proc. IJCAR’12. Volume 7364. LNAI, pages 82–96. • Borgwardt, Stefan and Rafael Penaloza ˜ (2013). “The Complexity of Lattice-Based Fuzzy Description Logics”. In: J. Data Semant. 2.1, pages 1–19. • Cerami, Marco and Umberto Straccia (2013). “On the (Un)decidability of Fuzzy Description Logics under Łukasiewicz t-norm”. In: Inform. Sciences 227, pages 1–21. • Goguen, Joseph A. (1967). “L-Fuzzy Sets”. In: J. Math. Anal. Appl. 18.1, pages 145–174. • Hajek, ´ Petr (2001). Metamathematics of Fuzzy Logic (Trends in Logic). • — (2005). “Making Fuzzy Description Logic more General”. In: Fuzzy Set. Syst. 154.1, pages 1–15. • Horrocks, Ian (1997). “Optimising Tableaux Decision Procedures for Description Logics”. PhD thesis. University of Manchester.
Wien, July 18th, 2014
Fuzzy DLs over Finite Lattices with Nominals
10
References II
• Horrocks, Ian, Ulrike Sattler, and Stephan Tobies (2000). “Practical Reasoning for Very Expressive Description Logics”. In: L. J. IGPL 8.3, pages 239–263. • Straccia, Umberto (2001). “Reasoning within Fuzzy Description Logics”. In: J. Artif. Intell. Res. 14, pages 137–166. • — (2004a). “Transforming Fuzzy Description Logics into Classical Description Logics”. In: Proc. JELIA’04. Volume 3229. LNCS, pages 385–399. • — (2004b). “Uncertainty in Description Logics: A Lattice-based Approach”. In: Proc. IPMU’04, pages 251–258. • Tobies, Stephan (2000). “The Complexity of Reasoning with Cardinality Restrictions and Nominals in Expressive Description Logics”. In: J. Artif. Intell. Res. 12, pages 199–217. • Zadeh, Lotfi A. (1965). “Fuzzy Sets”. In: Inform. Control 8.3, pages 338–353.
Wien, July 18th, 2014
Fuzzy DLs over Finite Lattices with Nominals
11
