CS 440: Introduction to AI
Homework 2  Part A Solution
Due: September 23 11:59PM, 2010
Your answers must be concise and clear.
Explain sufficiently that we can
easily determine what you understand.
We will give more points for a brief
interesting discussion with no answer than for a bluffing answer.
1
Validity and Satisfiability
(16 points) Determine whether each formula is valid, satisfiable (but not valid),
or unsatisfiable.
1.
P
(
x
)
Satisfiable since It holds for some denotational correspondence and some
world.
2.
∀
x P
(
x
)
Satisfiable.
3.
∃
x
∀
y P
(
x, y
)
Satisfiable.
4.
∀
x
(
P
(
x
)
∨ ¬
P
(
x
))
Valid. This is a tautology, which is always true.
5.
∃
x
∀
y
(
P
(
x, y
)
⇒ ∀
w
∃
z P
(
w, z
))
Satisfiable.
6.
∃
x
(
P
(
x
)
⇒ ¬
P
(
x
))
Satisfiable.
7.
∃
x
¬
(
¬
P
(
x
)
∨
P
(
x
))
Unsatisfiable.
¬
P
(
x
)
∨
P
(
x
) is a tautology, and it is negated, which makes
it always false.
8.
∀
x
∀
y
(
¬
P
(
x
)
∨
P
(
y
))
⇒
(
P
(
x
)
⇒
P
(
y
))
Valid. This is the same as Θ
⇒
Θ, which becomes
¬
Θ
∨
Θ, a tautology.
1
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2
Sentence Translation
(24 points) This problem has two parts.
In the first part, you are asked to
translate some first order predicate calculus (FOPC) sentences into English
sentences. The meanings of the predicates and functions in the FOPC sentences
are given. You should give as natural the English translations as possible. Few
points will be given for an English sentence like “For all
x
there exists a
y
such
that either
P
of
x
and
y
or
Q
of
y
.”
In the second part, you are asked to
translate English sentences into FOPC sentences. If the meaning of the English
sentence is ambiguous, give all possible readings. Note that sometimes
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 Spring '08
 Levinson
 Logic, Artificial Intelligence, Ron, FOPC, inference rule, FOPC sentences

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