1. Logics for Data and Knowledge Representation
Exercises: ClassL
Fausto Giunchiglia, Rui Zhang and Vincenzo Maltese

2. SYNTAX

3. Symbols in ClassL Which of the following symbols are used in ClassL?
⊓  ⊤ ∨ ≡ ⊔ ⊑ → ↔ ⊥ ∧ ⊨
Which of the following symbols are in well formed formulas? 3

4. Symbols in ClassL (solution)
Which of the following symbols are used in ClassL?
⊓  ⊤ ∨ ≡ ⊔ ⊑ → ↔ ⊥ ∧ ⊨
Which of the following symbols are in well formed formulas? 4

5. Extended formation rules
The 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>) 5

6. Formation rules Which of the following is not a wff in ClassL?
 MonkeyLow ⊔ BananaHigh
  MonkeyLow ⊓ BananaHigh ⊑  GetBanana
MonkeyLow  ⊓ BananaHigh
MonkeyLow   GetBanana
NUM 2, 3, 4 ! 6

7. MODELING

8. Formalization of simple sentences
Propositional DL (ClassL) has pretty poor expressiveness. For instance, we cannot represent attributes and relations effectively.
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 8

9. Formalization of a problem in ClassL
Unicorns 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) 9

10. Formalization of a semantic network
This can be only partially expressed in ClassL: T = {BodyOfWater ⊑ Location, PopulatedPlace ⊑ Location, Lake ⊑ BodyOfWater, City ⊑ PopulatedPlace, Country ⊑ PopulatedPlace} A = {Person(GiorgioNapolitano), Lake(GardaLake), City(Trento), Country(Italy),} 10

11. Defining the TBox and ABox: the LDKR Class
Define a TBox and ABox for the following database:
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}
LDKR
Name
Nationality
Hair
Fausto
Italian
White
Enzo
Black
Rui
Chinese
Bisu
Indian
NOTE: ClassL is not expressive enough to represent database constrains such as keys involving two fields. 11

12. SEMANTICS

13. Semantics of ⊓ 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.
This might not be the case even when they are satisfiable by the same model. A A B B A B 13

14. TBOX REASONING

15. 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
ADD EXAMPLE 15

16. 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)
To prove satisfiability in one model it is enough to find one model; we can use Venn Diagrams. A B
To prove satisfiability in all models we need to prove validity; this can be proved analytically, or with DPLL as follows:
DPLL( (RewriteInPL(A⊑B)  RewriteInPL(B⊑A)  RewriteInPL((A⊓B) )))
DPLL( ((A B)  (B A))  (A  B)) 16

17. Satisfiability with respect to a TBox (II)
Given the TBox T={A⊑B, B⊑A}, is (A⊓B) satisfiable in ClassL?
This corresponds to the problem: T ⊨ (A⊓B)
To prove that it is not satisfiable in all models it is instead enough to find a counter example (i.e. one model of T which is not a model for the proposition). A B 17

18. Satisfiability with respect to a TBox (III)
Given the TBox T={C⊑A, C⊑B} is (A⊓B) satisfiable?
(in one model) A B C 18

19. Satisfiability with respect to a TBox (IV)
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 T
MonkeyLow ⊑  GetBanana
GetBanana ⊑ Survive
Is T satisfiable?
YES! Look at the
Venn diagram
Survive
GetBanana
MonkeyLow 19

20. Satisfiability with respect to a TBox (V)
Suppose we model the Monkey-Banana problem as follows:
TBox T
MonkeyLow ⊑  GetBanana
GetBanana ⊑ 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 ?
YES! Look at the
Venn diagram
Survive
GetBanana
MonkeyLow 20

21. Subsumption Suppose we describe the students/attendees in a course:
Are all assistants also undergraduates?
T ⊨ Assistant ⊑ Undergraduate
Assistant ≡ PhD ⊓ Teach ≡ Master ⊓ Research ⊓ Teach ≡
Student ⊓  Undergraduate ⊓ Research ⊓ Teach
Assistants are actually students who are not undergraduate.
Undergraduate ⊑  Teach
Bachelor ≡ Student ⊓ Undergraduate
Master ≡ Student ⊓  Undergraduate
PhD ≡ Master ⊓ Research
Assistant ≡ PhD ⊓ Teach
TBox T

22. Disjointness 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 ⊑  Teach
Bachelor ≡ Student ⊓ Undergraduate
Master ≡ Student ⊓  Undergraduate
PhD ≡ Master ⊓ Research
Assistant ≡ PhD ⊓ Teach
TBox T

23. Normalization of a TBox
Normalize the TBox below:
MonkeyLow ⊑  GetBanana
GetBanana ≡ Survive
Possible solution:
MonkeyLow ≡  GetBanana ⊓  ClimbBox 23

24. 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
Survive
GetBanana
MonkeyLow
ClimbBox
Notice that the fact that a monkey climbs the box does not necessarily mean that it survives. 24

25. ABOX REASONING

26. ABox: Consistency Check the consistency of A w.r.t. T via expansion. T
MonkeyLow ≡  GetBanana ⊓  ClimbBox
GetBanana ≡ Survive A
MonkeyLow(Cita)
Survive(Cita)
Expansion of A is consistent:
MonkeyLow(Cita)
 GetBanana(Cita)
ClimbBox(Cita)
 Survive(Cita) 26

27. 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? T
MonkeyLow ≡  GetBanana ⊓  ClimbBox
GetBanana ≡ Survive A
MonkeyLow(Cita)
Survive(Cita)
Expansion of A
MonkeyLow(Cita)
 GetBanana(Cita)
ClimbBox(Cita)
 Survive(Cita)
GetBanana(Cita) 27

28. Instance Retrieval. Consider the following expansion…
Undergraduate ⊑  Teach
Bachelor ≡ Student ⊓ Undergraduate
Master ≡ Student ⊓  Undergraduate
PhD ≡ Master ⊓ Research
Assistant ≡ PhD ⊓ Teach A
Master(Chen)
PhD(Enzo)
Assistant(Rui)
The expansion of A
Master(Chen)
Student(Chen)
Undergraduate(Chen)
PhD(Enzo)
Master(Enzo)
Research(Enzo)
Student(Enzo)
Undergraduate(Enzo)
Assistant(Rui)
PhD(Rui)
Teach(Rui)
Master(Rui)
Research(Rui)
Student(Rui)
Undergraduate(Rui) 28

29. Instance Retrieval. … find the instances of Master
Undergraduate ⊑  Teach
Bachelor ≡ Student ⊓ Undergraduate
Master ≡ Student ⊓  Undergraduate
PhD ≡ Master ⊓ Research
Assistant ≡ PhD ⊓ Teach A
Master(Chen)
PhD(Enzo)
Assistant(Rui)
The expansion of A
Master(Chen)
Student(Chen)
Undergraduate(Chen)
PhD(Enzo)
Master(Enzo)
Research(Enzo)
Student(Enzo)
Undergraduate(Enzo)
Assistant(Rui)
PhD(Rui)
Teach(Rui)
Master(Rui)
Research(Rui)
Student(Rui)
Undergraduate(Rui) 29

30. 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. T
MonkeyLow ≡  GetBanana ⊓  ClimbBox
GetBanana ≡ Survive A
MonkeyLow(Cita)
Survive(Cita) 30