Logics for Data and KnowledgeRepresentation

Exercises: ClassL

Fausto Giunchiglia, Rui Zhang and Vincenzo Maltese

SYNTAX

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Symbols in ClassL

1. Which of the following symbols are used in ClassL?

⊓ ⊤ ∨ ≡ ⊔ ⊑ → ↔ ⊥ ∧ ⊨

2. Which of the following symbols are in well formed formulas?

⊓ ⊤ ∨ ≡ ⊔ ⊑ → ↔ ⊥ ∧ ⊨

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Symbols in ClassL (solution)

1. Which of the following symbols are used in ClassL?

⊓ ⊤ ∨ ≡ ⊔ ⊑ → ↔ ⊥ ∧ ⊨

2. Which of the following symbols are in well formed formulas?

⊓ ⊤ ∨ ≡ ⊔ ⊑ → ↔ ⊥ ∧ ⊨

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Extended formation rulesThe basic BNF grammar: <Atomic Formula> ::= A | B | ... | P | Q | ... | | ⊥ ⊤

<wff> ::= <Atomic Formula> | ¬<wff> | <wff> ⊓ <wff> | <wff> ⊔ <wff>

TBox: <definition> ::= <Atomic Formula> ≡ <wff>

<specialization> ::= <Atomic Formula> ⊑ <wff>

ABox: <individual> ::= a | b | ... |

<assertion> ::= <Atomic Formula> (<individual>)

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Formation rules Which of the following is not a wff in ClassL?

1. MonkeyLow ⊔ BananaHigh

2. MonkeyLow ⊓ BananaHigh ⊑ GetBanana

3. MonkeyLow ⊓ BananaHigh

4. MonkeyLow GetBanana

NUM 2, 3, 4 !

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MODELING

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Formalization of simple sentencesPropositional DL (ClassL) has pretty poor expressiveness. For instance, we cannot represent attributes and relations effectively.

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The set of games which are not legal Game ⊓ Legal

Lakes are locations Lake Location⊑

Lakes are locations made of water Lake Location MadeofWater⊑ ⊓

Persons can be distinguished into male and female

Male Person⊑Female Person⊑

Male and Female are disjoint Male Female ⊓ ⊑ ⊥

Persons have a birthplace Person hasBirthDate⊑

The set of documents about “programming in Java” are a subset of the documents about “programming languages” and “computer science”

JavaProgramming ⊑ProgrammingLanguage ⊓ComputerScience

Formalization of a problem in ClassLUnicorns are mythical horses having a horn. Pegasus is a unicorn while Mike is not.Unicorn mythical horse hasHorn⊑ ⊓ ⊓

Unicorn(Pegasus), Unicorn(Mike)

There are two kinds of students: master students and PhD students. All PhD students do research. Ronald is a master student that does research.MasterStudent Student⊑

PhDStudent Student doResearch⊑ ⊓

MasterStudent(Ronald)

doResearch(Ronald)

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Formalization of a semantic network

T = {BodyOfWater ⊑ Location, PopulatedPlace ⊑ Location, Lake ⊑ BodyOfWater, City ⊑ PopulatedPlace, Country ⊑ PopulatedPlace}

A = {Person(GiorgioNapolitano), Lake(GardaLake), City(Trento), Country(Italy), Part(GardaLake,Trento), Part(Trento, Italy), PresidentOf(GiorgioNapolitano, Italy)}

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Defining the TBox and ABox: the LDKR Class

Name Nationality Hair

Fausto Italian White

Enzo Italian Black

Rui Chinese Black

Bisu Indian Black

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ABox ={Italian(Fausto), Italian(Enzo), Chinese(Rui), Indian(Bisu), BlackHair(Enzo), BlackHair(Rui), BlackHair(Bisu),WhiteHair(Fausto)}

TBox ={Italian ⊑ LDKR, Indian ⊑ LDKR,Chinese ⊑ LDKR, BlackHair ⊑ LDKR,WhiteHair ⊑ LDKR}

Define a TBox and ABox for the following database:

LDKR

NOTE: ClassL is not expressive enough to represent database constrains such as keys involving two fields.

SEMANTICS

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Proprieties of the and (I)∧ ⊓ Suppose that A and B are satisfiable.

Is A ∧ B always satisfiable in PL?

We can observe that the fact that A and B are satisfiable (alone) does not necessarily imply that A ∧ B is also satisfiable. It is instead the case when they are satisfiable by the same model.

Think for instance to the case B = A.

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A B A∧B

T T T

T F F

F T F

F F F

MODEL for A

MODEL for A

MODEL for B

MODEL for B

Proprieties of the and (II)∧ ⊓ Suppose that A and B are satisfiable.

Is A ⊓ B always satisfiable in ClassL?

We can easily observe that the fact that A and B are satisfiable does not imply that A ⊓ B is also satisfiable. Think to the case in which their extensions are disjoint. Differently from PL, this might not be the case even when they are satisfiable by the same model.

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A BA

B A B

TBOX REASONING

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Satisfiability with respect to a TBox T

RECALL:

Satisfiability in one model

A concept P is satisfiable w.r.t. a terminology T, if there exists an interpretation I with I ⊨ θ for all θ ∈ T, and such that I ⊨ P, namely I(P) is not empty

Satisfiability in all models (validity)

A concept P is satisfiable w.r.t. a terminology T, if for all interpretations I with I ⊨ θ for all θ ∈ T, and such that I ⊨ P, namely I(P) is not empty

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Satisfiability with respect to a TBox (I) Given the TBox T={A⊑B, B⊑A}, is (A⊓B) satisfiable in ClassL?

This corresponds to the problem: T ⊨ (A⊓B)

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AB

To prove satisfiability in all models we need to prove validity; this can be proved with DPLL:

DPLL(RewriteInPL(A⊑B) RewriteInPL(B⊑A) RewriteInPL((A⊓B) ))

DPLL(((A B) (B A)) (A B))

To prove satisfiability in one model it is enough to find one model; we can use Venn Diagrams.

Satisfiability with respect to a TBox (II) Given the TBox T={C⊑A, C⊑B} is (A⊓B) satisfiable in one

model?

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A BC

Satisfiability with respect to a TBox (III) Suppose we model the Monkey-Banana problem as follows:

“If the monkey is low in position then it cannot get the banana. If the monkey gets the banana it survives”.

TBox TMonkeyLow ⊑ GetBananaGetBanana ⊑ Survive

Is T satisfiable?

Survive

GetBanana MonkeyLow

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YES! Look at the Venn diagram

Satisfiability with respect to a TBox (IV) Suppose we model the Monkey-Banana problem as follows:

TBox TMonkeyLow ⊑ GetBananaGetBanana ⊑ Survive

Is it possible for a monkey to survive even if it does not get the banana?

We can restate the problem as follow: does T ⊨ GetBanana Survive⊓ ?

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Survive

GetBanana MonkeyLow

YES! Look at the Venn diagram

Subsumption

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Suppose we describe the students/attendees in a course:

Are assistants undergraduates?

T ⊨ Assistant ⊑ Undergraduate

Assistant ≡ PhD ⊓ Teach ≡ Master ⊓ Research ⊓ Teach ≡

Student ⊓ Undergraduate ⊓ Research ⊓ Teach

Assistants are actually students who are not undergraduate.

Undergraduate ⊑ TeachBachelor ≡ Student ⊓ UndergraduateMaster ≡ Student ⊓ UndergraduatePhD ≡ Master ⊓ ResearchAssistant ≡ PhD ⊓ Teach TBox T

Disjointness

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Suppose we describe the students/attendees in a course:

Are Bachelor and master disjoint?

T ⊨ Bachelor ⊓ Master ⊑ ⊥

(Student ⊓ Undergraduate) ⊓ (Student ⊓ Undergraduate)

Student ⊓ (Undergraduate ⊓ Undergraduate) ≡ ⊥

Undergraduate ⊑ TeachBachelor ≡ Student ⊓ UndergraduateMaster ≡ Student ⊓ UndergraduatePhD ≡ Master ⊓ ResearchAssistant ≡ PhD ⊓ Teach TBox T

Normalization of a TBox

Normalize the TBox below:MonkeyLow ⊑ GetBanana

GetBanana ≡ Survive

Possible solution:MonkeyLow ≡ GetBanana ⊓ ClimbBox

GetBanana ≡ Survive

Note that, with this theory, the monkey necessarily needs to get the banana to survive.

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Expansion of a TBox

Expand the TBox below:MonkeyLow ≡ GetBanana ⊓ ClimbBox

GetBanana ≡ Survive

T’, expansion of T (The Venn diagram gives a possible model):MonkeyLow ≡ Survive ⊓ ClimbBox

GetBanana ≡ Survive

Survive

GetBanana MonkeyLow

ClimbBoxNotice that the fact that a monkey climbs the box does not necessarily mean that it survives.

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ABOX REASONING

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ABox: Consistency Check the consistency of A w.r.t. T via expansion.

TMonkeyLow ≡ GetBanana ⊓ ClimbBoxGetBanana ≡ Survive

AMonkeyLow(Cita)Survive(Cita)

Expansion of A is consistent:MonkeyLow(Cita) GetBanana(Cita) ClimbBox(Cita)

Survive(Cita) GetBanana(Cita)

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ABox: Instance checking Given T and A below

Is Cita an instance of MonkeyLow?

YES Is Cita an instance of ClimbBox?

NO Is Cita an instance of GetBanana?

NO

TMonkeyLow ≡ GetBanana ⊓ ClimbBoxGetBanana ≡ Survive

AMonkeyLow(Cita)Survive(Cita)

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Expansion of A MonkeyLow(Cita) GetBanana(Cita)ClimbBox(Cita)

Survive(Cita)GetBanana(Cita)

Instance Retrieval. Consider the following expansion…

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TUndergraduate ⊑ TeachBachelor ≡ Student ⊓ UndergraduateMaster ≡ Student ⊓ UndergraduatePhD ≡ Master ⊓ ResearchAssistant ≡ PhD ⊓ Teach

Assistant(Rui) PhD(Rui)Teach(Rui) Master(Rui) Research(Rui)Student(Rui)Undergraduate(Rui)

AMaster(Chen)PhD(Enzo)Assistant(Rui)

Master(Chen) Student(Chen) Undergraduate(Chen)

PhD(Enzo)Master(Enzo)Research(Enzo)Student(Enzo)Undergraduate(Enzo)

The expansion of A

Instance Retrieval. … find the instances of Master

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TUndergraduate ⊑ TeachBachelor ≡ Student ⊓ UndergraduateMaster ≡ Student ⊓ UndergraduatePhD ≡ Master ⊓ ResearchAssistant ≡ PhD ⊓ Teach

Assistant(Rui) PhD(Rui)Teach(Rui) Master(Rui) Research(Rui)Student(Rui)Undergraduate(Rui)

AMaster(Chen)PhD(Enzo)Assistant(Rui)

Master(Chen) Student(Chen) Undergraduate(Chen)

PhD(Enzo)Master(Enzo)Research(Enzo)Student(Enzo)Undergraduate(Enzo)

The expansion of A

ABox: Concept realization Find the most specific concept C such that A ⊨ C(Cita)

Notice that MonkeyLow directly uses GetBanana and ClimbBox, and it uses Survive. The most specific concept is therefore MonkeyLow.

TMonkeyLow ≡ GetBanana ⊓ ClimbBoxGetBanana ≡ Survive

AMonkeyLow(Cita)Survive(Cita)

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