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Intelligent Agents

Problem 1

On the slide "A different view: What is AI?", one of the quadrants is labeled "Systems that think rationally". Explain what this means and what the obstacles are to this approach.

Propositional Logic

Problem 2

Translate the following propositions and statements into their equivalent representation in the form of an English sentences, where

  • $E$ is the statement "Liron is eating"
  • $H$ is the statement "Liron is hungry"

Part 1

$E \implies \neg H$

Answer:

If Liron is eating, then Liron is not hungry.

Part 2

$E \land \neg H$

Answer: Liron is eating and Liron is not hungry.

Part 3

$\neg (H \implies \neg E)$

Answer:

First, the statement can be simplified using De Morgan's law. Distributing the logical negation among the components within the parenthetical statement, we thereby transforming it into the following equivalent statement:

$(\neg \neg E \implies \neg H)$

Which can be further simplified to:

$(E \implies \neg H)$

"If Liron is eating, then Liron is not hungry."

However, in its original form, the proposition is more akin to something along the lines of:

"The proposition that if Liron is hungry, then he is not eating, is as a whole, false."

Problem 3

Translate the following English sentences to Propositional Logic.

Propositions:

  • (R)aining
  • Liron is (S)ick
  • Liron is (H)ungry
  • Liron is H(A)appy
  • Liron owns a (C)at
  • Liron owns a (D)og

Part 1

It is raining if and only if Liron is sick.

Answer:

$$ R \iff S $$

Part 2

If Liron is sick, then it is raining, and vice versa.

Answer:

$$ S \implies R \land R \implies S $$

Part 3

It is raining is equivalent to Liron is sick.

Answer:

$$ R \equiv S $$

Part 4

Liron is hungry but happy

Answer:

$$ H \land {A} $$

Part 5

Liron either owns a cat or a dog, but he does not own both.

Answer:

$$ C \oplus D $$

Problem 4

Which of the following propositions are tautologies? Which are contradictions? Why?

(a) Three is a prime number.

(b) It is raining or it is not raining.

(c) It is raining (P) and it is not raining (¬P).

Answer:

(a) is neither a tautology nor a contradiction.

(b) is a tautology.

(c) is a contradiction.

Problem 5

Which of the following propositions are tautologies? Why?

(a) P

(b) P ⇒ P

(c) (P ⇒ P) ⇒ P

(d) P ⇒ (P ⇒ P)

Answer:

(a) is not a tautology.

(b) is a tautology.

(c) is a tautology.

(d) is a tautology.

Problem 6

Which of the two following propositions are equivalent in the sense that one can always be substituted for the other one in any proposition without changing its truth value? Why?

(a)

  • first proposition: $P ⇒ Q$
  • second proposition: $¬P ∨ Q$

(b)

  • first proposition: $¬P$
  • second proposition: $P ⇒ False$

(c)

  • first proposition: $¬P$
  • second proposition: $False ⇒ P$

(d)

  • first proposition: $¬P$
  • second proposition: $¬P ∨ Q$