PEAR: a Tool for Reasoning About Scenarios and Probabilities in Description Logics … · 2019. 6. 25. · Gian Luca Pozzato. Description Logics of Typicality DLs with Typicality

Description Logics of TypicalityDLs with Typicality and Probabilities

PEARConclusions

PEAR: a Tool for Reasoning About Scenarios and

Probabilities in Description Logics of Typicality

Gian Luca Pozzato and Gabriele Soriano1

1Dipartimento di Informatica, Universita degli Studi di Torino, Italy

CILC 2019

Gian Luca Pozzato


PEARConclusions

IntroductionPrototypical ReasoningDescription Logics of TypicalityMonotonic semantics ALC + TThe 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


PEARConclusions


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 v Mammal

ABox= instances of concepts and roles = properties and relationsamong individuals

Dog(saki)

Gian Luca Pozzato


PEARConclusions


Description Logics

Description Logics


Two key advantages:




Knowledge bases


Dog v Mammal


Dog(saki)

Gian Luca Pozzato


PEARConclusions


Description Logics

Description Logics


Two key advantages:




Knowledge bases


Dog v Mammal


Dog(saki)

Gian Luca Pozzato


PEARConclusions


Description Logics

Description Logics


Two key advantages:




Knowledge bases


Dog v Mammal


Dog(saki)

Gian Luca Pozzato


PEARConclusions


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 ofnonmonotonic reasoning mechanism

DLs + MKNF

DLs + circumscription

DLs + default

all these methods present some difficulties ...

Gian Luca Pozzato


PEARConclusions


DLs with typicality

What are they?

(with Laura Giordano, V. Gliozzi, N. Olivetti)

Non-monotonic extensions of Description Logics for reasoning aboutprototypical 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

of Lehmann-Magidor axioms of rational logic R

Basic notions

A KB comprises assertions T(C) v D

T(Dog) v Affectionate means “normally, dogs are affectionate”

T is nonmonotonic

C v D does not imply T(C) v T(D)

Gian Luca Pozzato


PEARConclusions


DLs with typicality

What are they?

(with Laura Giordano, V. Gliozzi, N. Olivetti)

Non-monotonic extensions of Description Logics for reasoning aboutprototypical 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

of Lehmann-Magidor axioms of rational logic R

Basic notions

A KB comprises assertions T(C) v D

T(Dog) v Affectionate means “normally, dogs are affectionate”

T is nonmonotonic

C v D does not imply T(C) v T(D)

Gian Luca Pozzato


PEARConclusions


The logic ALC + Tmin

Example

T(Pig) v ¬FireBreathingT(Pig u Pokemon) v FireBreathing

Reasoning

ABox:

Pig(tepig)

Expected conclusions:

¬FireBreathing(tepig)

Gian Luca Pozzato


PEARConclusions



Example


Reasoning

ABox:

Pig(tepig)



Gian Luca Pozzato


PEARConclusions



Example


Reasoning

ABox:

Pig(tepig)



Gian Luca Pozzato


PEARConclusions



Example

T(Pig) v ¬FireBreathingT(Pig t Pokemon) v FireBreathing

Reasoning

ABox:

Pig(tepig),Pokemon(tepig)


FireBreathing(tepig)

Gian Luca Pozzato


PEARConclusions


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<(CI)

Gian Luca Pozzato


PEARConclusions


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


PEARConclusions



Logic ALC + T




Example




Gian Luca Pozzato


PEARConclusions



Logic ALC + T




Example




Gian Luca Pozzato


PEARConclusions


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


PEARConclusions



Rational closure




M′ <M

Minimal entailment


Nonmonotonic logic



Gian Luca Pozzato


PEARConclusions



Rational closure




M′ <M

Minimal entailment


Nonmonotonic logic



Gian Luca Pozzato


PEARConclusions

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


PEARConclusions



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


PEARConclusions



Introduction

ALC + TP: extension of ALC by typicality inclusions equipped by

probabilities of exceptionality

T(C ) vp D, where p ∈ (0, 1)

intuitive meaning: normally, C s are Ds, and the probability of having

exceptional C s not being Ds is 1− p

Example

T(Student) v0.3 SportLover

T(Student) v0.9 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


PEARConclusions



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


PEARConclusions



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 typicalityassumptions

A1 = {T(C1)(a1),T(C2)(a2), . . . ,T(Cn)(an)}A2 = { T(C1)(a1), T(C2)(a2), . . . ,T(Cn)(an)}A3 = {T(C1)(a1), T(C2)(a2), . . . ,T(Cn)(an)}A4 = { T(C1)(a1),T(C2)(a2), . . . ,T(Cn)(an) }. . .

reasoning in the monotonic ALC + T considering TBox and ABox

extended with chosen assumptions

Gian Luca Pozzato


PEARConclusions



Extensions of ABox


from ALC + Tmin





Gian Luca Pozzato


PEARConclusions



Extensions of ABox


from ALC + Tmin





Gian Luca Pozzato


PEARConclusions



Extensions of ABox


from ALC + Tmin





Gian Luca Pozzato


PEARConclusions



Extensions of ABox


from ALC + Tmin





Gian Luca Pozzato


PEARConclusions



Extensions of ABox


from ALC + Tmin





Gian Luca Pozzato


PEARConclusions


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


PEARConclusions



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


PEARConclusions



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


PEARConclusions



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


PEARConclusions



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


PEARConclusions



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


PEARConclusions



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


PEARConclusions



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 0.1050.8

Gian Luca Pozzato


PEARConclusions



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 0.1050.8

Gian Luca Pozzato


PEARConclusions



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 0.1050.8

[0.105, 0.8, 0.105] fA1 T(C)(a) T(F )(a) T(C)(b)

Gian Luca Pozzato


PEARConclusions



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 0.1050.8

[0.105, 0.8, 0.105] fA1

fA2[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


PEARConclusions



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 0.1050.8

[0.105, 0.8, 0.105] fA1

fA2

fA3

[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


PEARConclusions



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 0.1050.8

[0.105, 0.8, 0.105] fA1

fA2

fA3

fA8

...

[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


PEARConclusions



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 0.1050.8

[0.105, 0.8, 0.105] fA1

fA2

fA3

fA8

...

[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)

PfA1= 0.105 ⇥ 0.8 ⇥ 0.105

PfA2= 0.105 ⇥ (1 � 0.8) ⇥ (1 � 0.105)

PfA3= (1 � 0.105) ⇥ 0.8 ⇥ 0.105

PfA8= (1 � 0.105) ⇥ (1 � 0.8) ⇥ (1 � 0.105)

Gian Luca Pozzato


PEARConclusions



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 ) v D | T(C ) vr D ∈ T } ∪ {C v D ∈ T }KB |=〈p,q〉

ALC+TP F

if F is C v D or T(C) v 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∑

i=1

Pi

Gian Luca Pozzato


PEARConclusions



Entailment



p ≤ P1 ≤ q, p ≤ P2 ≤ q, . . . , p ≤ Pk ≤ q


ALC+TP F




i=1

Pi

Gian Luca Pozzato


PEARConclusions



Entailment



p ≤ P1 ≤ q, p ≤ P2 ≤ q, . . . , p ≤ Pk ≤ q


ALC+TP F




i=1

Pi

Gian Luca Pozzato


PEARConclusions



Entailment



p ≤ P1 ≤ q, p ≤ P2 ≤ q, . . . , p ≤ Pk ≤ q


ALC+TP F




i=1

Pi

Gian Luca Pozzato


PEARConclusions



Entailment



p ≤ P1 ≤ q, p ≤ P2 ≤ q, . . . , p ≤ Pk ≤ q


ALC+TP F




i=1

Pi

Gian Luca Pozzato


PEARConclusions



TBox

PokemonCardPlayer v CardPlayer

T(CardPlayer) v0.85 ¬YoungPerson

T(PokemonCardPlayer) v0.7 YoungPerson

T(Student) v0.6 YoungPerson

T(Student) v0.8 InstagramUser

Inferences

T(CardPlayer u Italian) v ¬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


PEARConclusions



TBox

PokemonCardPlayer v CardPlayer

T(CardPlayer) v0.85 ¬YoungPerson

T(PokemonCardPlayer) v0.7 YoungPerson

T(Student) v0.6 YoungPerson

T(Student) v0.8 InstagramUser

Inferences

T(CardPlayer u Italian) v ¬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


PEARConclusions



New results

New results in two directions:with Antonio Lieto: semantics

probability as proportion:

| {x ∈ CI | x 6∈ (T(C))I and x ∈ (¬D)I} || CI |

≤ 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


PEARConclusions

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 ALCAvailable at http://di.unito.it/pear

Gian Luca Pozzato

http://di.unito.it/pear


PEARConclusions

Basic ideas

PEAR

Basic ideas

compute extensions A of the ABox and corresponding alternativescenarios 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


PEARConclusions

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 Godel-Lob modality 2:

T(C) defined as C u2¬C where the accessibility relation of 2 is the

preference relation < in ALC + T models

Gian Luca Pozzato


PEARConclusions

Basic ideas

PEAR

gives information to

computed by

manage order of executioncomputed by

manage information by

USER

Reasoning and query files

Auxiliary file

Text file

Gian Luca Pozzato


PEARConclusions

Basic ideas

PEAR

Gian Luca Pozzato


PEARConclusions

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


PEARConclusions

Future works

Any question?

Gian Luca Pozzato

Documents

PEAR: a Tool for Reasoning About Scenarios and Probabilities in Description Logics … · 2019. 6. 25. · Gian Luca Pozzato. Description Logics of Typicality DLs with Typicality