Logics for Data and Knowledge Representation

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Logics for Data and KnowledgeRepresentation

• Exercises Modeling

Fausto Giunchiglia, Rui Zhang and Vincenzo Maltese
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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
MODELING LOGICAL MODELING INTENTIONAL
MODELS EXTENSIONAL MODELS
Language L
Theory T
Data Knowledge
World
Mental Model
Domain D
Model M
Meaning
SEMANTIC GAP
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Example of informal Modeling
MODELING LOGICAL MODELING INTENTIONAL
MODELS EXTENSIONAL MODELS
Mental Model
Model M
Theory T
Language L
Domain D
World
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
SEMANTIC GAP
NOTE a database can be seen as an informal model
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Logical Modeling
MODELING LOGICAL MODELING INTENTIONAL
MODELS EXTENSIONAL MODELS
Language L
Theory T
Data Knowledge
Interpretation
Modeling
Entailment
World
Mental Model
?
I
Realization
Domain D
Model M
Meaning
SEMANTIC GAP
NOTE the key point is that in logical modeling
we have formal semantics
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What and How
MODELING LOGICAL MODELING INTENTIONAL
MODELS EXTENSIONAL MODELS

• 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
MODELING LOGICAL MODELING INTENTIONAL
MODELS EXTENSIONAL MODELS
Mental Model
Model M
Theory T
Language L
Domain D
World
L Monkey, Climbs, GetBanana, ?, ?, ? D T,
F T ? (Monkey ? Climbs) ? GetBanana A
possible model M I(Monkey) T I(Climbs)
T I(GetBanana) T
SEMANTIC GAP
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Example of formal (extensional) Modeling
MODELING LOGICAL MODELING INTENTIONAL
MODELS EXTENSIONAL MODELS
Mental Model
Model M
Theory T
Language L
Domain D
World
L Monkey, Climbs, GetBanana, ?, ?, ? D
Cita, ThatBanana T ? Climbs ? GetBanana
A possible model M I(Monkey)
Cita I(Climbs) Cita I(GetBanana) ThatBanana
SEMANTIC GAP
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Modeling Exercise Forest
MODELING LOGICAL MODELING INTENTIONAL
MODELS EXTENSIONAL MODELS

• 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)
MODELING LOGICAL MODELING INTENTIONAL
MODELS EXTENSIONAL MODELS

• 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)
MODELING LOGICAL MODELING INTENTIONAL
MODELS EXTENSIONAL MODELS

• 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
MODELING LOGICAL MODELING INTENTIONAL
MODELS EXTENSIONAL MODELS

• 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)
MODELING LOGICAL MODELING INTENTIONAL
MODELS EXTENSIONAL MODELS

• 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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Solution Classroom (II)
MODELING LOGICAL MODELING INTENTIONAL
MODELS EXTENSIONAL MODELS

• The corresponding Venn diagram

U
Person
Student
PhD
Assistant
Professor
Master
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Solution Classroom (III)
MODELING LOGICAL MODELING INTENTIONAL
MODELS EXTENSIONAL MODELS

• 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
MODELING LOGICAL MODELING INTENTIONAL
MODELS EXTENSIONAL MODELS

• 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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Solution Family (I)
MODELING LOGICAL MODELING INTENTIONAL
MODELS EXTENSIONAL MODELS

• 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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Solution Family (II)
MODELING LOGICAL MODELING INTENTIONAL
MODELS EXTENSIONAL MODELS

• The corresponding Venn diagram

U
Member
Grandparent
Parent
Brother
Sister
Me
Child
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Solution Family (III)
MODELING LOGICAL MODELING INTENTIONAL
MODELS EXTENSIONAL MODELS

• 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
MODELING LOGICAL MODELING INTENTIONAL
MODELS EXTENSIONAL MODELS

• 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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Solution My friends (I)
MODELING LOGICAL MODELING INTENTIONAL
MODELS EXTENSIONAL MODELS

• 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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Solution My friends (II)
MODELING LOGICAL MODELING INTENTIONAL
MODELS EXTENSIONAL MODELS

• The corresponding Venn diagram

U
Friend
Forum
Close
PlayingChess
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Solution My friends (III)
MODELING LOGICAL MODELING INTENTIONAL
MODELS EXTENSIONAL MODELS

• 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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A Database
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.

ID Name Nationality Hair Color Affiliation
1 Fausto Italian White Professor
2 Enzo Italian Black PhD
3 Rui Chinese Black Assistant
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Class
Italian
PhD
BlackHair