Logics for Data and Knowledge Representation Exercises: Modeling Fausto Giunchiglia, Rui Zhang and...

Logics for Data and KnowledgeRepresentation

Exercises: Modeling

Fausto Giunchiglia, Rui Zhang and Vincenzo Maltese

Outline Modeling Logical Modeling

Exercises with intensional models Forest

Exercises with extensional models Classroom Family My friends Databases

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Modeling: from the world to its representation

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World

LanguageL

TheoryT

DomainD

ModelM

DataKnowledge

Meaning

MentalModel

SEMANTICGAP

MODELING :: LOGICAL MODELING :: INTENTIONAL MODELS :: EXTENSIONAL MODELS

What and How World: the phenomenon we want to describe Domain: the abstract relevant elements in the real world Mental Model: what we have in mind. It is the first abstraction of

the world (subject to the semantic gap) Language: the set of words and rules we use to build sentences

used to express our mental model Model: the formalization of the mental model, i.e. the set of true

facts in the language, in agreement with the theory Theory: the set of sentences (constraints) about the world

expressed in the language that limit the possible models

NOTE: this does not necessarily need to be in formal semantics

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Example of informal Modeling

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World

MentalModel

SEMANTICGAP

ModelM

L: Informal description in NL

D: {monkey, banana, tree}

T: If the monkey climbs on the tree, he can get the banana

M: The monkey actually climbs on the tree and gets the banana

TheoryT

NOTE: a database can be seen as an informal model

LanguageL

DomainD


Logical Modeling

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Modeling

Realization

World

LanguageL

TheoryT

DomainD

ModelM

DataKnowledge

Meaning

MentalModel

SEMANTICGAP

Inte

rpre

tatio

n

I

En

tailm

en

t⊨

NOTE: the key point is that in logical modeling we have formal semantics


What and How World: the phenomenon that we are observing and want to model Domain (D) = the abstract relevant objects from the world Language (L) = a logical language with formal syntax and semantics:

The formal syntax is given by the set of rules to construct complex sentences (the grammar)

The formal semantics is given by the interpretation function I: L → D

Model (M) = the abstract (mathematical sense) representation of the intended truths via the interpretation I of the language L. M is called L-model of D M P, indicates that M satisfies P⊨

Theory (T) = the set of facts/constraints expressed in the language L. A fact defines a piece of knowledge (about D), something true in

the model.

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Example of formal (intentional) Modeling

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World

SEMANTICGAP

L = {Monkey, Climbs, GetBanana, , , }

D= {T, F}

T = { (Monkey Climbs) GetBanana}

A possible model M:I(Monkey) = TI(Climbs) = TI(GetBanana) = T

MentalModel

ModelM

TheoryT

LanguageL

DomainD


Example of formal (extensional) Modeling

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World

SEMANTICGAP

L = {Monkey, Climbs, GetBanana, , , }

D= {Cita, ThatBanana}

T = { Climbs GetBanana}

A possible model M:I(Monkey) = CitaI(Climbs) = CitaI(GetBanana) = ThatBanana

MentalModel

ModelM

TheoryT

LanguageL

DomainD


Modeling Exercise: Forest Description: There are two lions, Kimba and Simba,

in the forest. They are in competition for the food. There is a nice antelope they want to hunt. If they want to survive they have to catch it.

Problem: Model the problem by identify relevant objects, defining the domain, the language, the theory and providing a possible intentional model.

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Solution: Forest (I) Description: There are two lions, Kimba and Simba,

in the forest. They are in competition for the food. There is a nice antelope they want to hunt. If they want to survive they have to catch it.

Relevant objects are in red

D = {T, F}

L = {Lion, Antelope, Survive, Catch}

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Solution: Forest (II) A possible model:

I (Lion) = T I (Antelope) = T

I (Catch) = T I (Survive) = T

The theory T:

Antelope (Catch Survive)

Antelope Catch

I above is a model for T

I below is NOT a model for T

I (Lion) = T I (Antelope) = F

I (Catch) = F I (Survive) = T

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Modeling Exercise: Classroom Description: In a class there are several persons.

Usually there is one professor who teaches to some students. Students can be Master students or PhD students. Among PhD students there might be some Assistants of the professor.

Problem: Model the problem by identify relevant objects, defining the domain and the language, and providing a possible extensional model for it.

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Solution: Classroom (I) Description: In a class there are several persons.

Usually there is one professor who teaches to some students. Students can be Master students or PhD students. Among PhD students there might be some Assistants of the professor.


L = {Person, Professor, Student, Master, PhD, Assistant}

D = {Fausto, Mary, Paul, Jane}

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Solution: Classroom (II) The corresponding Venn diagram

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PersonStudent

ProfessorPhD

Master

Assistant


U

Solution: Classroom (III) A possible model:

I (Person) = {Fausto, Mary, Paul, Jane}

I (Professor) = {Fausto}

I (Student) = {Mary, Paul, Jane}

I (Master) = {Mary}

I (PhD) = {Paul, Jane}

I (Assistant) = {Paul}

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Modeling Exercise: Family Description: My family consists of several

members. There is a grandparent and my parents. Then there are some children, i.e. two sisters, one brother and me


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Solution: Family (I) Description: My family consists of several

members. There is a grandparent and my parents. Then there are some children, i.e. two sisters, one brother and me


L = {member, grandparent, parent, child, brother, sister, me}

D = {Bob, Fausto, Mary, Paul, Jane, Hugo, Robert}

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Solution: Family (II)

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Member

ParentGrandparent

Brother SisterChild

The corresponding Venn diagram

Me


U

Solution: Family (III) A possible model:

I (Member) = {Bob, Fausto, Mary, Paul, Jane, Hugo, Robert}

I (Grandparent) = {Bob}

I (Parent) = {Fausto, Mary, Bob}

I (Brother) = {Robert, Paul}

I (Sister) = {Jane}

I (Me) = {Hugo}

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Modeling Exercise: My friends Description: I have a lot of friends. I met some of

them on the forum of my website. However, only a few of them are close to me. In particular, I use to play chess with Paul.


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Solution: My friends (I) Description: I have a lot of friends. I met some of

them on the forum of my website. However, only a few of them are close to me. In particular, I use to play chess with Paul.


L = {Friend, Forum, Close, PlayingChess}

D = {Bob, Fausto, Mary, Paul, Jane, Hugo, Robert}

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Solution: My friends (II)

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FriendFriend

Forum

The corresponding Venn diagram

Close

PlayingChess


U

Solution: My friends (III) A possible model:

I (Friend) = {Bob, Paul, Jane, Robert, Richard, Samuel}

I (Forum) = {Bob, Paul, Jane}

I (Close) = {Bob, Paul, Samuel}

I (PlayingChess) = {Paul}

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Closed world assumption (CWA): The assumption that what is not currently

known to be true, is false.

I (Italian) = {Fausto, Enzo}

I (BlackHair) = {Enzo, Rui}

… Open world assumption

(OWA): anything which is not explicitly asserted is unknown.

Is Rui Italian? This is not asserted in the DB, therefore it is unknown.

A Database

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ID Name Nationality Hair Color Affiliation

1 Fausto Italian White Professor

2 Enzo Italian Black PhD

3 Rui Chinese Black Assistant

4 …

5 …

… …

ClassItalian

BlackHairPhD


Logics for Data and Knowledge Representation Exercises: Modeling Fausto Giunchiglia, Rui Zhang and...

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