Question 1
Three options were wrong: Modalities, Trust, and Term vectors.
Modalities are only considered in modal logics. Trust could be expressed with the help of ontologies, but are not inherent part. Term vectors are used in document retrieval.
Question 2
All propositional formulas were satisfied by the given interpretation except for ( ¬a ∨ ¬b ) → ( a ∨ ¬c ).
Inserting the interpretations in this formula gives us:
( ¬false ∨ ¬true ) → ( false ∨ ¬true )
( true ∨ false ) → ( false ∨ false )
true → false
false
Question 3
The formulas (∃X) (∀Y) p(X,Y) ≡ (∀Y) (∃X) p(X,Y)
and (∀X) (∀Y) p(X) ∨ ¬q(Y) ≡ (∀X) (∀Y) p(Y) → q(X)
are each not equivalent, all the others are.
The first formula can easily be translated to a natural language sentence, e.g. the first part to Someone loves everyone., while the second part would be Everyone is loved by someone. Th esecond part is fulfilled if everybody loves himself, while the first part is not. Both parts are not equivalent. You could falsify this with an interpretation:
Individuals: {a, b}
p: {(a,a), (b,b)}
When applying this to the formula, we see: false ≡ true
Question 4
We want to show whether F can be deduced from the knowledge base K, which can be written as K → F.
In the Tableaux algorithm we proof this by refutation, so we need to show that the opposite ¬(K → F) is not satisfiable.
¬(K → F) would result in one very big formula that needs to be resolved before applying the tableaux, but it is equivalent to (K ∧ ¬F), so we simply can add ¬F to our knowledge base K and no more resolution is needed.
All other options not even were equivalent formulas.
Question 5
To proof that a formula F can be deduced from a knowledge base K, the implication (K → F) has to be negated: (¬(A ∧ B → C)) or (A ∧ B ∧ ¬C). This results in the following:
- A
- B
- ¬C
At this point you are already finished with the tableaux (without closing it) because you cannot deduce new knowledge which could lead to closing the opened branch. Therefore, the correct answer is: No, the tableaux algorithm on ¬(A ∧ B → C) cannot be closed.
Question 6
The correct translation is:
Department ⊑ ∃partOf.Company Employee ≡ ∃employedBy.Company ⊔ ∃employedBy.∃partOf.Company
Department is modeled as a subclass of that things that are part of something of class Company.
Employee is modeled as the union of the class of individuals that are employed by something of class Company and the class of individuals that are employed by something that is part of something of class Company.
Question 7
Only one answer was correct:
Bicycle ⊑ ∃hasPart.Frontwheel Bicycle ⊑ ∃hasPart.Rearwheel Bicycle ⊑ ∃hasPart.Saddle
It constructs the class Bicycle as a subclass of three other classes, those having a part of class Frontwheel, Rearwheel, and Saddle respectively.
Regrading
We had a typo in the option
Bicycle ⊑ ∃hasPart.Frontwheel ⊔ ∃hasPart.Rearwheel ⊔ ∃hasPart.Saddle
Our intention was to use intersection ⊓, instead the union operator ⊔ was used. Therefore, this option is not correct. A regrading will be done asap.
Question 8
The Open World Assumption assumes incomplete knowledge and therefore the the existence of any individual, as long as it is not explicitly excluded.
While the Closed World Assumption demands all possible individuals to be contained in the knowledge base.
Regrading
We found another typo in the option "In OWA the existence of any individual is possible, unless it is not explicitly excluded." The option should rather be "In OWA the existence of any individual is possible, unless it is explicitly excluded." This will be corrected asap.