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Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions PEAR: a Tool for Reasoning About Scenarios and Probabilities in Description Logics of Typicality Gian Luca Pozzato and Gabriele Soriano 1 1 Dipartimento di
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions
Gian Luca Pozzato and Gabriele Soriano1
1Dipartimento di Informatica, Universit´
a degli Studi di Torino, Italy
CILC 2019
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Introduction Prototypical Reasoning Description Logics of Typicality Monotonic semantics ALC + T The nonmonotonic semantics
PEAR Outline Extensions of DLs with Typicality and Probabilities:
Reasoning about ABox facts with probabilities of exceptions PEAR: a reasoner for DL + T + probabilities
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Introduction Prototypical Reasoning Description Logics of Typicality Monotonic semantics ALC + T The nonmonotonic semantics
Description Logics Description Logics Important formalisms of knowledge representation Two key advantages:
well-defined semantics based on first-order logic good trade-off between expressivity and complexity
at the base of languages for the semantic (e.g. OWL) Knowledge bases Two components:
TBox=inclusion relations among concepts
Dog ⊑ Mammal
ABox= instances of concepts and roles = properties and relations among individuals
Dog(saki)
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Introduction Prototypical Reasoning Description Logics of Typicality Monotonic semantics ALC + T The nonmonotonic semantics
Description Logics Description Logics Important formalisms of knowledge representation Two key advantages:
well-defined semantics based on first-order logic good trade-off between expressivity and complexity
at the base of languages for the semantic (e.g. OWL) Knowledge bases Two components:
TBox=inclusion relations among concepts
Dog ⊑ Mammal
ABox= instances of concepts and roles = properties and relations among individuals
Dog(saki)
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Introduction Prototypical Reasoning Description Logics of Typicality Monotonic semantics ALC + T The nonmonotonic semantics
Description Logics Description Logics Important formalisms of knowledge representation Two key advantages:
well-defined semantics based on first-order logic good trade-off between expressivity and complexity
at the base of languages for the semantic (e.g. OWL) Knowledge bases Two components:
TBox=inclusion relations among concepts
Dog ⊑ Mammal
ABox= instances of concepts and roles = properties and relations among individuals
Dog(saki)
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Introduction Prototypical Reasoning Description Logics of Typicality Monotonic semantics ALC + T The nonmonotonic semantics
Description Logics Description Logics Important formalisms of knowledge representation Two key advantages:
well-defined semantics based on first-order logic good trade-off between expressivity and complexity
at the base of languages for the semantic (e.g. OWL) Knowledge bases Two components:
TBox=inclusion relations among concepts
Dog ⊑ Mammal
ABox= instances of concepts and roles = properties and relations among individuals
Dog(saki)
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Introduction Prototypical Reasoning Description Logics of Typicality Monotonic semantics ALC + T The nonmonotonic semantics
Extensions of DLs DLs with nonmonotonic features need of representing prototypical properties and of reasoning about defeasible inheritance handle defeasible inheritance needs the integration of some kind of nonmonotonic reasoning mechanism
DLs + MKNF DLs + circumscription DLs + default
all these methods present some difficulties ...
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Introduction Prototypical Reasoning Description Logics of Typicality Monotonic semantics ALC + T The nonmonotonic semantics
DLs with typicality What are they? (with Laura Giordano, V. Gliozzi, N. Olivetti) Non-monotonic extensions of Description Logics for reasoning about prototypical properties and inheritance with exceptions
Basic idea: to extend DLs with a typicality operator T T(C) singles out the “most normal” instances of the concept C semantics of T defined by a set of postulates that are a restatement
Basic notions
A KB comprises assertions T(C) ⊑ D T(Dog) ⊑ Affectionate means “normally, dogs are affectionate” T is nonmonotonic
C ⊑ D does not imply T(C) ⊑ T(D)
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Introduction Prototypical Reasoning Description Logics of Typicality Monotonic semantics ALC + T The nonmonotonic semantics
DLs with typicality What are they? (with Laura Giordano, V. Gliozzi, N. Olivetti) Non-monotonic extensions of Description Logics for reasoning about prototypical properties and inheritance with exceptions
Basic idea: to extend DLs with a typicality operator T T(C) singles out the “most normal” instances of the concept C semantics of T defined by a set of postulates that are a restatement
Basic notions
A KB comprises assertions T(C) ⊑ D T(Dog) ⊑ Affectionate means “normally, dogs are affectionate” T is nonmonotonic
C ⊑ D does not imply T(C) ⊑ T(D)
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Introduction Prototypical Reasoning Description Logics of Typicality Monotonic semantics ALC + T The nonmonotonic semantics
The logic ALC + Tmin Example T(Pig) ⊑ ¬FireBreathing T(Pig ⊓ Pokemon) ⊑ FireBreathing Reasoning ABox:
Pig(tepig)
Expected conclusions:
¬FireBreathing(tepig)
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Introduction Prototypical Reasoning Description Logics of Typicality Monotonic semantics ALC + T The nonmonotonic semantics
The logic ALC + Tmin Example T(Pig) ⊑ ¬FireBreathing T(Pig ⊓ Pokemon) ⊑ FireBreathing Reasoning ABox:
Pig(tepig)
Expected conclusions:
¬FireBreathing(tepig)
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Introduction Prototypical Reasoning Description Logics of Typicality Monotonic semantics ALC + T The nonmonotonic semantics
The logic ALC + Tmin Example T(Pig) ⊑ ¬FireBreathing T(Pig ⊓ Pokemon) ⊑ FireBreathing Reasoning ABox:
Pig(tepig)
Expected conclusions:
¬FireBreathing(tepig)
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Introduction Prototypical Reasoning Description Logics of Typicality Monotonic semantics ALC + T The nonmonotonic semantics
The logic ALC + Tmin Example T(Pig) ⊑ ¬FireBreathing T(Pig ⊔ Pokemon) ⊑ FireBreathing Reasoning ABox:
Pig(tepig), Pokemon(tepig)
Expected conclusions:
FireBreathing(tepig)
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Introduction Prototypical Reasoning Description Logics of Typicality Monotonic semantics ALC + T The nonmonotonic semantics
The logic ALC + T Semantics M = ∆I, <, .I
additional ingredient: preference relation among domain elements < is an irreflexive, transitive, modular and well-founded relation over ∆I:
for all S ⊆ ∆I, for all x ∈ S, either x ∈ Min<(S) or ∃y ∈ Min<(S) such that y < x Min<(S) = {u : u ∈ S and ∄z ∈ S s.t. z < u}
Semantics of the T operator: (T(C))I = Min<(C I)
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Introduction Prototypical Reasoning Description Logics of Typicality Monotonic semantics ALC + T The nonmonotonic semantics
Weakness of monotonic semantics Logic ALC + T The operator T is nonmonotonic, but... The logic is monotonic
If KB | = F, then KB’ | = F for all KB’ ⊇ KB
Example in the KB of the previous slides:
if Pig(tepig) ∈ ABox, we are not able to:
assume that T(Pig)(tepig) infer that ¬FireBreathing(tepig)
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Introduction Prototypical Reasoning Description Logics of Typicality Monotonic semantics ALC + T The nonmonotonic semantics
Weakness of monotonic semantics Logic ALC + T The operator T is nonmonotonic, but... The logic is monotonic
If KB | = F, then KB’ | = F for all KB’ ⊇ KB
Example in the KB of the previous slides:
if Pig(tepig) ∈ ABox, we are not able to:
assume that T(Pig)(tepig) infer that ¬FireBreathing(tepig)
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Introduction Prototypical Reasoning Description Logics of Typicality Monotonic semantics ALC + T The nonmonotonic semantics
Weakness of monotonic semantics Logic ALC + T The operator T is nonmonotonic, but... The logic is monotonic
If KB | = F, then KB’ | = F for all KB’ ⊇ KB
Example in the KB of the previous slides:
if Pig(tepig) ∈ ABox, we are not able to:
assume that T(Pig)(tepig) infer that ¬FireBreathing(tepig)
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Introduction Prototypical Reasoning Description Logics of Typicality Monotonic semantics ALC + T The nonmonotonic semantics
The nonmonotonic logic ALC + Tmin Rational closure
Preference relation among models of a KB
M1 < M2 if M1 contains less exceptional (not minimal) elements M minimal model of KB if there is no M′ model of KB such that M′ < M
Minimal entailment
KB | =min F if F holds in all minimal models of KB
Nonmonotonic logic
KB | =min F does not imply KB’ | =min F with KB’ ⊃ KB
Corresponds to a notion of rational closure of KB
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Introduction Prototypical Reasoning Description Logics of Typicality Monotonic semantics ALC + T The nonmonotonic semantics
The nonmonotonic logic ALC + Tmin Rational closure
Preference relation among models of a KB
M1 < M2 if M1 contains less exceptional (not minimal) elements M minimal model of KB if there is no M′ model of KB such that M′ < M
Minimal entailment
KB | =min F if F holds in all minimal models of KB
Nonmonotonic logic
KB | =min F does not imply KB’ | =min F with KB’ ⊃ KB
Corresponds to a notion of rational closure of KB
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Introduction Prototypical Reasoning Description Logics of Typicality Monotonic semantics ALC + T The nonmonotonic semantics
The nonmonotonic logic ALC + Tmin Rational closure
Preference relation among models of a KB
M1 < M2 if M1 contains less exceptional (not minimal) elements M minimal model of KB if there is no M′ model of KB such that M′ < M
Minimal entailment
KB | =min F if F holds in all minimal models of KB
Nonmonotonic logic
KB | =min F does not imply KB’ | =min F with KB’ ⊃ KB
Corresponds to a notion of rational closure of KB
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Typicalities and Probabilities
DLs + T and probabilities Introduction in the non-monotonic DL, all typicality assumptions that are consistent with the KB can be inferred counterintuitive, especially if we have hundreds of instances
they are all typical dogs!!!!
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Typicalities and Probabilities
DLs + T and probabilities Introduction in the non-monotonic DL, all typicality assumptions that are consistent with the KB can be inferred counterintuitive, especially if we have hundreds of instances
they are all typical dogs!!!!
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Typicalities and Probabilities
DLs + T and probabilities Introduction ALC + TP: extension of ALC by typicality inclusions equipped by probabilities of exceptionality T(C) ⊑p D, where p ∈ (0, 1) intuitive meaning: normally, Cs are Ds, and the probability of having exceptional Cs not being Ds is 1 − p Example
T(Student) ⊑0.3 SportLover T(Student) ⊑0.9 SocialNetworkUser sport lovers and social network users are both typical properties of students probability of not having exceptions is 30% and 90%, respectively
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Typicalities and Probabilities
DLs + T and probabilities Basic idea extensions of an ABox containing only some of the “plausible” typicality assertions of the rational closure of KB
each extension represents a scenario having a specific probability probability distribution among scenarios nonmonotonic entailment restricted to extensions whose probabilities belong to a given and fixed range reason about scenarios that are not necessarily the most probable
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Typicalities and Probabilities
DLs + T and probabilities Extensions of ABox typicality assumptions T(C1)(a1), T(C2)(a2), . . . , T(Cn)(an) inferred from ALC + Tmin extensions of ABox obtained by choosing some typicality assumptions
. . .
reasoning in the monotonic ALC + T considering TBox and ABox extended with chosen assumptions
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Typicalities and Probabilities
DLs + T and probabilities Extensions of ABox typicality assumptions T(C1)(a1), T(C2)(a2), . . . , T(Cn)(an) inferred from ALC + Tmin extensions of ABox obtained by choosing some typicality assumptions
. . .
reasoning in the monotonic ALC + T considering TBox and ABox extended with chosen assumptions
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Typicalities and Probabilities
DLs + T and probabilities Extensions of ABox typicality assumptions T(C1)(a1), T(C2)(a2), . . . , T(Cn)(an) inferred from ALC + Tmin extensions of ABox obtained by choosing some typicality assumptions
. . .
reasoning in the monotonic ALC + T considering TBox and ABox extended with chosen assumptions
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Typicalities and Probabilities
DLs + T and probabilities Extensions of ABox typicality assumptions T(C1)(a1), T(C2)(a2), . . . , T(Cn)(an) inferred from ALC + Tmin extensions of ABox obtained by choosing some typicality assumptions
. . .
reasoning in the monotonic ALC + T considering TBox and ABox extended with chosen assumptions
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Typicalities and Probabilities
DLs + T and probabilities Extensions of ABox typicality assumptions T(C1)(a1), T(C2)(a2), . . . , T(Cn)(an) inferred from ALC + Tmin extensions of ABox obtained by choosing some typicality assumptions
. . .
reasoning in the monotonic ALC + T considering TBox and ABox extended with chosen assumptions
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Typicalities and Probabilities
DLs + T and probabilities Extensions of ABox typicality assumptions T(C1)(a1), T(C2)(a2), . . . , T(Cn)(an) inferred from ALC + Tmin extensions of ABox obtained by choosing some typicality assumptions
. . .
reasoning in the monotonic ALC + T considering TBox and ABox extended with chosen assumptions
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Typicalities and Probabilities
Extensions of ABox and probabilities
T(C) v0.3 D T(C) v0.7 E T(F) v0.8 G T(C) v0.5 H T(C)(a) T(F)(a) T(C)(b)
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Typicalities and Probabilities
Extensions of ABox and probabilities
T(C) v0.3 D T(C) v0.7 E T(F) v0.8 G T(C) v0.5 H T(C)(a) T(F)(a) T(C)(b)
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Typicalities and Probabilities
Extensions of ABox and probabilities
T(C) v0.3 D T(C) v0.7 E T(F) v0.8 G T(C) v0.5 H T(C)(a) T(F)(a) T(C)(b) 0.3 × 0.7 × 0.5
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Typicalities and Probabilities
Extensions of ABox and probabilities
T(C) v0.3 D T(C) v0.7 E T(F) v0.8 G T(C) v0.5 H T(C)(a) T(F)(a) T(C)(b) 0.3 × 0.7 × 0.5 0.105
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Typicalities and Probabilities
Extensions of ABox and probabilities
T(C) v0.3 D T(C) v0.7 E T(F) v0.8 G T(C) v0.5 H T(C)(a) T(F)(a) T(C)(b) 0.3 × 0.7 × 0.5 0.105
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Typicalities and Probabilities
Extensions of ABox and probabilities
T(C) v0.3 D T(C) v0.7 E T(F) v0.8 G T(C) v0.5 H T(C)(a) T(F)(a) T(C)(b) 0.3 × 0.7 × 0.5 0.105 0.8
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Typicalities and Probabilities
Extensions of ABox and probabilities
T(C) v0.3 D T(C) v0.7 E T(F) v0.8 G T(C) v0.5 H T(C)(a) T(F)(a) T(C)(b) 0.3 × 0.7 × 0.5 0.105 0.8
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Typicalities and Probabilities
Extensions of ABox and probabilities
T(C) v0.3 D T(C) v0.7 E T(F) v0.8 G T(C) v0.5 H T(C)(a) T(F)(a) T(C)(b) 0.3 × 0.7 × 0.5 0.105 0.105 0.8
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Typicalities and Probabilities
Extensions of ABox and probabilities
T(C) v0.3 D T(C) v0.7 E T(F) v0.8 G T(C) v0.5 H T(C)(a) T(F)(a) T(C)(b) 0.3 × 0.7 × 0.5 0.105 0.105 0.8
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Typicalities and Probabilities
Extensions of ABox and probabilities
T(C) v0.3 D T(C) v0.7 E T(F) v0.8 G T(C) v0.5 H T(C)(a) T(F)(a) T(C)(b) 0.3 × 0.7 × 0.5 0.105 0.105 0.8 [0.105, 0.8, 0.105] f A1 T(C)(a) T(F)(a) T(C)(b)
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Typicalities and Probabilities
Extensions of ABox and probabilities
T(C) v0.3 D T(C) v0.7 E T(F) v0.8 G T(C) v0.5 H T(C)(a) T(F)(a) T(C)(b) 0.3 × 0.7 × 0.5 0.105 0.105 0.8 [0.105, 0.8, 0.105] f A1 f A2 [0.105, 0, 0] T(C)(a) T(F)(a) T(C)(b) T(C)(a) T(F)(a) T(C)(b)
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Typicalities and Probabilities
Extensions of ABox and probabilities
T(C) v0.3 D T(C) v0.7 E T(F) v0.8 G T(C) v0.5 H T(C)(a) T(F)(a) T(C)(b) 0.3 × 0.7 × 0.5 0.105 0.105 0.8 [0.105, 0.8, 0.105] f A1 f A2 f A3 [0.105, 0, 0] [0, 0.8, 0.105] T(C)(a) T(F)(a) T(C)(b) T(C)(a) T(F)(a) T(C)(b) T(F)(a) T(C)(a) T(C)(b)
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Typicalities and Probabilities
Extensions of ABox and probabilities
T(C) v0.3 D T(C) v0.7 E T(F) v0.8 G T(C) v0.5 H T(C)(a) T(F)(a) T(C)(b) 0.3 × 0.7 × 0.5 0.105 0.105 0.8 [0.105, 0.8, 0.105] f A1 f A2 f A3 f A8 . . . [0.105, 0, 0] [0, 0.8, 0.105] [0, 0, 0] T(C)(a) T(F)(a) T(C)(b) T(C)(a) T(F)(a) T(C)(b) T(F)(a) T(F)(a) T(C)(a) T(C)(a) T(C)(b) T(C)(b)
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Typicalities and Probabilities
Extensions of ABox and probabilities
T(C) v0.3 D T(C) v0.7 E T(F) v0.8 G T(C) v0.5 H T(C)(a) T(F)(a) T(C)(b) 0.3 × 0.7 × 0.5 0.105 0.105 0.8 [0.105, 0.8, 0.105] f A1 f A2 f A3 f A8 . . . [0.105, 0, 0] [0, 0.8, 0.105] [0, 0, 0] T(C)(a) T(F)(a) T(C)(b) T(C)(a) T(F)(a) T(C)(b) T(F)(a) T(F)(a) T(C)(a) T(C)(a) T(C)(b) T(C)(b) P f
A1 = 0.105 × 0.8 × 0.105
P f
A2 = 0.105 × (1 − 0.8) × (1 − 0.105)
P f
A3 = (1 − 0.105) × 0.8 × 0.105
P f
A8 = (1 − 0.105) × (1 − 0.8) × (1 − 0.105)
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Typicalities and Probabilities
DLs + T and probabilities Entailment Given KB=(T , A) and p, q ∈ (0, 1] E = { A1, A2, . . . , Ak} set of extensions of A whose probabilities are p ≤ P1 ≤ q, p ≤ P2 ≤ q, . . . , p ≤ Pk ≤ q T ′ = {T(C) ⊑ D | T(C) ⊑r D ∈ T } ∪ {C ⊑ D ∈ T } KB | =p,q
ALC+TP F
if F is C ⊑ D or T(C) ⊑ D, if (T ′, A) | =ALC+Tmin F if F is C(a), if (T ′, A ∪ Ai) | =ALC+T F for all Ai ∈ E
probability of F: P(F) =
k
Pi
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Typicalities and Probabilities
DLs + T and probabilities Entailment Given KB=(T , A) and p, q ∈ (0, 1] E = { A1, A2, . . . , Ak} set of extensions of A whose probabilities are p ≤ P1 ≤ q, p ≤ P2 ≤ q, . . . , p ≤ Pk ≤ q T ′ = {T(C) ⊑ D | T(C) ⊑r D ∈ T } ∪ {C ⊑ D ∈ T } KB | =p,q
ALC+TP F
if F is C ⊑ D or T(C) ⊑ D, if (T ′, A) | =ALC+Tmin F if F is C(a), if (T ′, A ∪ Ai) | =ALC+T F for all Ai ∈ E
probability of F: P(F) =
k
Pi
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Typicalities and Probabilities
DLs + T and probabilities Entailment Given KB=(T , A) and p, q ∈ (0, 1] E = { A1, A2, . . . , Ak} set of extensions of A whose probabilities are p ≤ P1 ≤ q, p ≤ P2 ≤ q, . . . , p ≤ Pk ≤ q T ′ = {T(C) ⊑ D | T(C) ⊑r D ∈ T } ∪ {C ⊑ D ∈ T } KB | =p,q
ALC+TP F
if F is C ⊑ D or T(C) ⊑ D, if (T ′, A) | =ALC+Tmin F if F is C(a), if (T ′, A ∪ Ai) | =ALC+T F for all Ai ∈ E
probability of F: P(F) =
k
Pi
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Typicalities and Probabilities
DLs + T and probabilities Entailment Given KB=(T , A) and p, q ∈ (0, 1] E = { A1, A2, . . . , Ak} set of extensions of A whose probabilities are p ≤ P1 ≤ q, p ≤ P2 ≤ q, . . . , p ≤ Pk ≤ q T ′ = {T(C) ⊑ D | T(C) ⊑r D ∈ T } ∪ {C ⊑ D ∈ T } KB | =p,q
ALC+TP F
if F is C ⊑ D or T(C) ⊑ D, if (T ′, A) | =ALC+Tmin F if F is C(a), if (T ′, A ∪ Ai) | =ALC+T F for all Ai ∈ E
probability of F: P(F) =
k
Pi
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Typicalities and Probabilities
DLs + T and probabilities Entailment Given KB=(T , A) and p, q ∈ (0, 1] E = { A1, A2, . . . , Ak} set of extensions of A whose probabilities are p ≤ P1 ≤ q, p ≤ P2 ≤ q, . . . , p ≤ Pk ≤ q T ′ = {T(C) ⊑ D | T(C) ⊑r D ∈ T } ∪ {C ⊑ D ∈ T } KB | =p,q
ALC+TP F
if F is C ⊑ D or T(C) ⊑ D, if (T ′, A) | =ALC+Tmin F if F is C(a), if (T ′, A ∪ Ai) | =ALC+T F for all Ai ∈ E
probability of F: P(F) =
k
Pi
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Typicalities and Probabilities
DLs + T and probabilities TBox
PokemonCardPlayer ⊑ CardPlayer T(CardPlayer) ⊑0.85 ¬YoungPerson T(PokemonCardPlayer) ⊑0.7 YoungPerson T(Student) ⊑0.6 YoungPerson T(Student) ⊑0.8 InstagramUser
Inferences
T(CardPlayer ⊓ Italian) ⊑ ¬YoungPerson is entailed in ALC + TP if A = {PokemonCardPlayer(lollo), Student(lollo), Student(poz)}: YoungPerson(lollo) has probability 70% InstagramUser(poz) has probability 48%
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Typicalities and Probabilities
DLs + T and probabilities TBox
PokemonCardPlayer ⊑ CardPlayer T(CardPlayer) ⊑0.85 ¬YoungPerson T(PokemonCardPlayer) ⊑0.7 YoungPerson T(Student) ⊑0.6 YoungPerson T(Student) ⊑0.8 InstagramUser
Inferences
T(CardPlayer ⊓ Italian) ⊑ ¬YoungPerson is entailed in ALC + TP if A = {PokemonCardPlayer(lollo), Student(lollo), Student(poz)}: YoungPerson(lollo) has probability 70% InstagramUser(poz) has probability 48%
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Typicalities and Probabilities
DLs + T and probabilities New results New results in two directions:
with Antonio Lieto: semantics
probability as proportion: | {x ∈ C I | x ∈ (T(C))I and x ∈ (¬D)I} | | C I | ≤ 1 − p probability as degree of belief: distributed semantics inspired by DISPONTE (Bellodi, Cota, Riguzzi, Zese)
submitted at the special issue of CILC 2018
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Basic ideas
PEAR Probability of Exceptions and typicAlity Reasoner Python implementation of the reasoning services provided by the logic ALC + TP Makes use of owlready2 to rely on HermiT Exploits the translation into standard ALC Available at http://di.unito.it/pear
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Basic ideas
PEAR Basic ideas compute extensions A of the ABox and corresponding alternative scenarios Ai
very expensive...optimizations needed
compute probabilities of each scenario select the extensions whose probabilities belong to a given range p, q check whether a query F is entailed from all the selected extensions in the monotonic logic ALC + T
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Basic ideas
PEAR Translation PEAR relies on a polynomial encoding of ALC + T into ALC (Giordano, Gliozzi, Olivetti) exploits the definition of T in terms of a G¨
T(C) defined as C ⊓ ¬C where the accessibility relation of is the preference relation < in ALC + T models
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Basic ideas
PEAR
gives information to computed by manage order of execution computed by manage information by USER Reasoning and query files Auxiliary file Text file
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Basic ideas
PEAR
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Future works
Beyond COCOS Future works Reasoning in real application domains:
which range of probabilities?
Extension to other DLs Learning of a knowledge base with DLs + T + probabilities Optimization of PEAR by the application of techniques introduced by (Alberti, Bellodi, Cota, Lamma, Riguzzi, Zese)
Gian Luca Pozzato
Description Logics of Typicality DLs with Typicality and Probabilities PEAR Conclusions Future works
Gian Luca Pozzato