1. Logics for Data and Knowledge Representation
Exercises: DL

2. DL family of languages AL
SYNTAX :: SEMANTICS :: TBOX REASONING :: ABOX REASONING :: TABLEAU CALCULUS AL
<Atomic> ::= A | B | ... | P | Q | ... | ⊥ | ⊤
<wff> ::= <Atomic> | ¬<Atomic> | <wff> ⊓ <wff> | ∀R.C | ∃R.⊤
ALU <wff> ⊔ <wff>
ALE ∃R.C
ALN ≥nR | ≤nR
ALC ¬ <wff>
FL- is AL with the elimination of ⊤, ⊥ and 
FL0 is FL- with the elimination of ∃R.⊤

3. DL family of languages ∃R.C ≥2R Formula AL ALU ALE ALN ALC A  A⊔B
SYNTAX :: SEMANTICS :: TBOX REASONING :: ABOX REASONING :: TABLEAU CALCULUS
Examples of formulas in each of the languages:
Formula AL
ALU
ALE
ALN
ALC A 
A⊔B
∃R.C
≥2R
(A⊓B)
A⊔B 3

4. Formalization of a semantic network
SYNTAX :: SEMANTICS :: TBOX REASONING :: ABOX REASONING :: TABLEAU CALCULUS
instance-of
Person ⊑ ∃Drives.Car Person ⊑ ∃HasHobby.SportCar Person ⊑ ∃HasHobby.Opera Student ⊑ Person SportCar ⊑ Car Student(Ralf) Opera(DonCarlos)

5. Modeling a problem Formalize the following problem in DL:
SYNTAX :: SEMANTICS :: TBOX REASONING :: ABOX REASONING :: TABLEAU CALCULUS
Formalize the following problem in DL:
In a hospital patients, doctors and computers are equipped with proximity sensors able to detect whether doctors curated a patient or worked at their computer. The system detected that doctor Peter curated the patient Smith.
Doctor ⊑ cure.Patient ⊔ work.Computer cure ⊑ near work ⊑ near Doctor (Peter) Patient (Smith) cure(Peter, Smith)

6. Venn diagrams Provide the Venn diagram for: A ⊑ B ⊓ ¬C
SYNTAX :: SEMANTICS :: TBOX REASONING :: ABOX REASONING :: TABLEAU CALCULUS
Provide the Venn diagram for: A ⊑ B ⊓ ¬C
Note: Venn Diagrams can be used only for propositional DL B A C

7. Interpretation of Value Restriction
I(∀R.C) = {a ∈ ∆ | for all b, if (a,b)∈I(R) then b∈I(C)}
Those a that have only values b in C with role R. b
I(C) a
if (a,b) ∈I(R) b'
b''
if (a,b') ∈I(R)
if (a,b'') ∈I(R)

8. Interpretation of Existential Quantifier
I(∃R.C) = {a ∈ ∆ | exists b s.t. (a,b) ∈ I(R), b ∈ I(C)}
Those a that have some value b in C with role R. b
I(C) a
(a,b)∈I(R)

9. Proofs in DL semantics
SYNTAX :: SEMANTICS :: TBOX REASONING :: ABOX REASONING :: TABLEAU CALCULUS
Verify the following equivalences hold for all interpretations (∆,I). Use definitions or graphical representation.
I(¬(C ⊓ D)) = I(¬C ⊔ ¬D)
I(¬(C ⊔ D)) = I(¬C ⊓ ¬D)
I(¬∀R.C) = I(∃R.¬C)
I(¬∃R.C) = I(∀R.¬C)
Let us prove the last one:
I(¬∃R.C)= {a ∈ ∆ | not exists b s.t. (a,b) ∈ I(R), b ∈ I(C)}
= {a ∈ ∆ | not ∃b. R(a,b) and C(b)}
= {a ∈ ∆ | ∀b. not (R(a,b) and C(b))}
= {a ∈ ∆ | ∀b. if R(a,b) then ¬C(b)}
= I(∀R.¬C) 9

10. TBox and ABox reasoning
TBox reasoning requires in general the usage of the Tableaux algorithm (subject of the next lecture)
Now, we provide a list of cases that can be addressed WITHOUT it 10

11. Using DPLL for reasoning tasks
SYNTAX :: SEMANTICS :: TBOX REASONING :: ABOX REASONING :: TABLEAU CALCULUS
DPLL solves the CNFSAT-problem by searching a truth-assignment that satisfies all clauses θi in the input proposition P = θ1  …  θn
NOTE: DL sentences must to be translated in PL (via TBox and ABox elimination) and therefore it can only be used for propositional DL.
Model checking Does ν satisfy P? (ν ⊨ P?)
Check if ν(P) = true
Satisfiability Is there any ν such that ν ⊨ P?
Check that DPLL(P) succeeds and returns a ν
Unsatisfiability Is it true that there are no ν satisfying P?
Check that DPLL(P) fails
Validity Is P a tautology? (true for all ν)
Check that DPLL(P) fails 11

12. Modeling the problem Codify in DL the following problem:
SYNTAX :: SEMANTICS :: TBOX REASONING :: ABOX REASONING :: TABLEAU CALCULUS
Codify in DL the following problem:
“A federal agent can access top secret documents. An X file is a top secret document with no restricted access that can be read by policemen.”
FederalAgent ⊑ ∃Access.TopSecretDocument
XFile ⊑ TopSecretDocument ⊓ Restricted ⊓ ∀Read-1.Policeman

13. Satisfiability of a TBox (a set of formulas)
SYNTAX :: SEMANTICS :: TBOX REASONING :: ABOX REASONING :: TABLEAU CALCULUS
Say if the following TBox is satisfiable and provide a model in the case:
FederalAgent ⊑ ∃Access.TopSecretDocument
XFile ⊑ TopSecretDocument ⊓ Restricted ⊓ ∀Read-1.Policeman
I(FederalAgent) = {A} We have that I ⊨ T
I(TopSecretDocument) = {D1, D2}
I(Access) = {(A, D1), (A, D2)}
I(XFile) = {D1}
I(Restricted) = {D2}
I(Read) = {(B, D1)}
I(Policeman) = {B}
Otherwise you can apply the tableaux calculus and provide the ABox built

14. Satisfiability w.r.t. a TBox
SYNTAX :: SEMANTICS :: TBOX REASONING :: ABOX REASONING :: TABLEAU CALCULUS
Consider the TBox T:
FederalAgent ⊑ ∃Access.TopSecretDocument
XFile ⊑ TopSecretDocument ⊓ Restricted ⊓ ∀Read-1.Policeman
and the formula P: TopSecretDocument ⊓ Restricted
Is there any model such that T ⊨ P ?
Yes, if fact these is an interpretation I (e.g. the one of the previous exercise) such that I ⊨ T and I ⊨ P
I(TopSecretDocument) = {D1, D2}
I(Restricted) = {D1}
Notice that T does not affect P at all (i.e. we cannot further expand P w.r.t. T)

15. I(Access) = {(A, D1), (A, D2)}
Subsumption
SYNTAX :: SEMANTICS :: TBOX REASONING :: ABOX REASONING :: TABLEAU CALCULUS
Consider the TBox T:
FederalAgent ⊑ ∃Access.TopSecretDocument
XFile ⊑ TopSecretDocument ⊓ Restricted ⊓ ∀Read-1.Policeman
does T ⊨ TopSecretDocument ⊑ Restricted in all models?
By definition, it must be I(TopSecretDocument )  I(Restricted ) for every model I of T. Even if this is true for the I of the previous exercise, this is not true in general. It is enough to provide a counterexample:
I(FederalAgent) = {A}
I(TopSecretDocument) = {D2}
I(Access) = {(A, D1), (A, D2)}
I(XFile) = {D2}
I(Restricted) = {D1}
I(Read) = {(B, D2)}
I(Policeman) = {B}

16. Expansion of an ABox
SYNTAX :: SEMANTICS :: TBOX REASONING :: ABOX REASONING :: TABLEAU CALCULUS
Provide the expansion of A w.r.t. T (without normalization), where:
TBox T = {Student ⊑ Faculty, Professor ⊑ Faculty ⊓ Teach}
ABox A = {Professor(Bob), Faculty(Rui)}
Professor(Bob), Faculty(Bob), Teach(Bob)
Faculty(Rui)

17. ABox Reasoning services: Consistency
SYNTAX :: SEMANTICS :: TBOX REASONING :: ABOX REASONING :: TABLEAU CALCULUS
An ABox A is consistent with respect to a TBox T if there is an interpretation I which is a model of both A and T.
T = {Parent⊑≤1hasChild}
A = {hasChild(mary, bob), hasChild(mary, cate), Parent(mary)}
A is consistent ALONE but is not consistent with respect T.
In fact, from A mary has two children while T imposes maximum one 17

18. Drawing consequences (I)
SYNTAX :: SEMANTICS :: TBOX REASONING :: ABOX REASONING :: TABLEAU CALCULUS
Given the TBox and ABox below T:
Female⊑Human
Male⊑Human
Mother⊑Female
Father⊑Male
Child≡∃has.Mother⊓ ∃has.Father
Male⊓Female⊑⊥ A:
Mother(Anna)
Father(Bob)
has(Cate,Anna)
has(Cate,Bob)
Prove:
Human(Anna)
¬Female(Bob)
Child(Cate) 18

19. Drawing consequences (II)
SYNTAX :: SEMANTICS :: TBOX REASONING :: ABOX REASONING :: TABLEAU CALCULUS
Expand A w.r.t. T: A:
Mother(Anna)  Female(Anna)  Human(Anna)
Father(Bob)  Male(Bob)  Human(Bob) , ¬Female(Bob)
has(Cate,Anna)  Child(Cate)
has(Cate,Bob)  Child(Cate) 19