logics for data and knowledge representation
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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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MODELING :: LOGICAL MODELING :: INTENTIONAL MODELS :: EXTENSIONAL MODELS
Example of informal Modeling
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World
MentalModel
SEMANTICGAP
ModelM
L: Informal description in NLD: {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
MODELING :: LOGICAL MODELING :: INTENTIONAL MODELS :: EXTENSIONAL MODELS
Logical Modeling
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Modeling
Realization
World
LanguageL
TheoryT
DomainD
ModelM
DataKnowledge
Meaning
MentalModel
SEMANTICGAP
InterpretationI
Entailmen
t⊨
NOTE: the key point is that in logical modeling we have formal semantics
MODELING :: LOGICAL MODELING :: INTENTIONAL MODELS :: EXTENSIONAL MODELS
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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MODELING :: LOGICAL MODELING :: INTENTIONAL MODELS :: EXTENSIONAL MODELS
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
MODELING :: LOGICAL MODELING :: INTENTIONAL MODELS :: EXTENSIONAL MODELS
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 :: LOGICAL MODELING :: INTENTIONAL MODELS :: EXTENSIONAL MODELS
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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MODELING :: LOGICAL MODELING :: INTENTIONAL MODELS :: EXTENSIONAL MODELS
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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MODELING :: LOGICAL MODELING :: INTENTIONAL MODELS :: EXTENSIONAL MODELS
Solution: Forest (II) A possible model:
I (Lion) = T I (Antelope) = TI (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) = FI (Catch) = F I (Survive) = T
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MODELING :: LOGICAL MODELING :: INTENTIONAL MODELS :: EXTENSIONAL MODELS
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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MODELING :: LOGICAL MODELING :: INTENTIONAL MODELS :: EXTENSIONAL MODELS
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.
Relevant objects are in red
L = {Person, Professor, Student, Master, PhD, Assistant}
D = {Fausto, Mary, Paul, Jane}
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MODELING :: LOGICAL MODELING :: INTENTIONAL MODELS :: EXTENSIONAL MODELS
Solution: Classroom (II) The corresponding Venn diagram
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PersonStudent
ProfessorPhD
Master
Assistant
MODELING :: LOGICAL MODELING :: INTENTIONAL MODELS :: EXTENSIONAL MODELS
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 :: LOGICAL MODELING :: INTENTIONAL MODELS :: EXTENSIONAL MODELS
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
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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MODELING :: LOGICAL MODELING :: INTENTIONAL MODELS :: EXTENSIONAL MODELS
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
Relevant objects are in red
L = {member, grandparent, parent, child, brother, sister, me}
D = {Bob, Fausto, Mary, Paul, Jane, Hugo, Robert}
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MODELING :: LOGICAL MODELING :: INTENTIONAL MODELS :: EXTENSIONAL MODELS
Solution: Family (II)
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Member
ParentGrandparent
Brother SisterChild
The corresponding Venn diagram
Me
MODELING :: LOGICAL MODELING :: INTENTIONAL MODELS :: EXTENSIONAL MODELS
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 :: LOGICAL MODELING :: INTENTIONAL MODELS :: EXTENSIONAL MODELS
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.
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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MODELING :: LOGICAL MODELING :: INTENTIONAL MODELS :: EXTENSIONAL MODELS
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.
Relevant objects are in red
L = {Friend, Forum, Close, PlayingChess}
D = {Bob, Fausto, Mary, Paul, Jane, Hugo, Robert}
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MODELING :: LOGICAL MODELING :: INTENTIONAL MODELS :: EXTENSIONAL MODELS
Solution: My friends (II)
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Friend
Forum
The corresponding Venn diagram
Close
PlayingChess
MODELING :: LOGICAL MODELING :: INTENTIONAL MODELS :: EXTENSIONAL MODELS
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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MODELING :: LOGICAL MODELING :: INTENTIONAL MODELS :: EXTENSIONAL MODELS
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 …
… …
ClassItalianBlackHair
PhD
MODELING :: LOGICAL MODELING :: INTENTIONAL MODELS :: EXTENSIONAL MODELS
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