First Order Logic

Oefening 7.2
Artificiële Intelligentie 1
First Order Logic
5 november 2002
Represent the following sentences in first-order logic, using a consistent vocabulary
(which you must define):
a. Not all students take both History and Biology.
b. Only one student failed History.
c. Only one student failed both History and Biology.
d. The best score in History was better than the best score in Biology.
e. Every person who dislikes all vegetarians is smart.
f. No person likes a smart vegetarian.
g. There is a woman who likes all men who are not vegetarians.
h. There is a barber who shaves all men in town who do not shave themselves.
i. No person likes a professor unless the professor is smart.
j. Politicians can fool some of the people all of the time, and they can fool all
of the people some of the time, but they can’t fool all of the people all of the time.
Oplossing
a. ∃x(Student(x) ∧ ¬(T ake(x, Biology) ∧ T ake(x, History)))
b. ∃x(Student(x) ∧ F ailed(x, History) ∧ ∀y(F ailed(y, History) ⇒ x = y))
c. ∃x(Student(x)∧F ailed(x, History)∧F ailed(x, Biology)∧(∀y((F ailed(y, History)∧
F ailed(y, Biology)) ⇒ x = y)))
d. ∃x(HistoryScore(x) ∧ ¬∃y(BiologyScore(y) ∧ Higher(y, x))
e. ∀x(P erson(x) ∧ ∀y(V egetarian(y) ⇒ Dislikes(x, y)) ⇒ Smart(x))
f. ¬∃x(P erson(x) ∧ ∃y(V egetarian(y) ∧ smart(y) ∧ Likes(x, y))
g. ∃x(W oman(x) ∧ ∀y((Man(y) ∧ ¬V egetarian(y)) ⇒ Likes(x, y)))
h. ∃xBarber(x) ∧ ∀y((Man(y) ∧ ¬Shaves(y, y)) ⇒ Shaves(x, y)))
i. ∀y(P rofessor(y) ∧ ¬Smart(y)) ⇒ ¬∃x(Likes(x, y))
j. ∀xP oliticain(x) ⇒ ((∃y∀t(T ime(t) ∧ (P eople(y)) ⇒ F ools(x, y, t))
∨ (∃t∀y(T ime(t) ∧ P eople(y)) ⇒ F ools(x, y, t))
∧ ¬∀x∀t(T ime(t) ∧ people(y) ⇒ F ools(x, y, t)))
Oefening 7.5
Represent the sentence “All Germans speak the same languages“ in predicate
calculus. Use Speaks(x, l), meaning that person x speaks language l.
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Oplossing
∀x, y, lGerman(x) ∧ German(y) ⇒ (Speaks(x, l) ⇔ Speaks(y, l))
Oefening 7.6
Write axioms describing the predicates GrandChild, GreatGrandparent, Brother,
Sister, Daughter, Son, Aunt, Uncle, BrotherInLaw and SisterInLaw.
Oplossing
a. GrandChild(x, y) ≡ ∃z(Child(x, z) ∧ Child(z, y))
b. GreatGrandP arent(x, y) ≡ ∃a, b(Child(y, a) ∧ Child(b, a) ∧ Child(b, x))
c. Brother(x, y) ≡ ∃z(Child(x, z) ∧ Child(y, z) ∧ Male(x))
d. Sister(x, y) ≡ ∃z(Child(x, z) ∧ Child(y, z) ∧ female(x))
e. Daughter(x, y) ≡ F emale(x) ∧ Child(x, y)
f. Son(x, y) ≡ F emale(x) ∧ Child(x, y)
g. Aunt(x, y) ≡ ∃z(Sister(x, z) ∧ Child(y, z))
h. Uncle(x, y) ≡ ∃z(Brother(x, z) ∧ Child(y, z))
i. BrotherInLaw(x, y) ≡ ∃z(Brother(x, z) ∧ (Spouse(z, y) ∨ Spouse(y, z)))
j. SisterInLaw(x, y) ≡ ∃z(Sister(x, z) ∧ (Spouse(z, y) ∨ Spouse(y, z)))
Oefening 7.9
This exercise can be done without the computer, although you may find it useful
to use a backward chainer to check your proof for the last part. The idea is to
formalize the blocks world domain using the situation calculus. The objects in
this domain are blocks, tables, and situations. The predicates are
On(x, y, s) ClearTop(x, s) Block(x) Table(x)
The only action is PutOn(x, y), where x must be a block whose top is clear
of any other blocks, and y can be either the table or a different block with a
clear top. The initial situation S0 has A on B on C on the table.
a. Write an axiom or axioms describing PutOn.
b. Describe the initial state, S0, in which there is a stack of three blocks, A on
B on C, where C is on the table, T.
c. Give the appropriate query that a theorem prover can solve to generate a
plan to build a stack where C is on top of B and B is on top of A. Write down
the solution that the theorem prover should return. (Hint: The solution will be
a situation described as the result of doing some actions to S0.)
d. Show formally that the solution fact follows from your description of the
situation and the axioms for PutOn.
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