First Order Logic and Equality Concepts

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Questions and Answers

What does the statement ∀ x ¬Likes(x, Parsnips) imply?

At least someone likes parsnips.

No one likes parsnips. (correct)

There exists at least one person who dislikes parsnips.

Everyone likes parsnips.

What does ¬∃ x Likes(x, Parsnips) mean?

It is not the case that anyone likes parsnips. (correct)

At least one person likes parsnips.

Everyone dislikes parsnips.

Everyone likes parsnips.

How would you interpret ∃ y (∀ x Loves(x, y))?

There is at least one person who is loved by everyone. (correct)

Everyone loves someone.

Some people are loved by some others.

Every person loves many others.

Which of the following statements is equivalent to ¬∀ x Likes(x, IceCream)?

There exists someone who does not like ice cream. (A) Signup and view all the answers

What property does the statement ∀ x (∃ y Loves(x, y)) describe?

Everyone has someone they love. (D) Signup and view all the answers

What property of equality states that for any element x, x is equal to itself?

Reflexive property (C) Signup and view all the answers

Which rule allows for the replacement of a term in a clause with another equal term?

Demodulation (D) Signup and view all the answers

Which of the following statements is an example of a valid logical representation for the relation 'Horses, cows, and pigs are mammals'?

Horse(x) ⇒ Mammal(x) ∧ Cow(x) ⇒ Mammal(x) (B) Signup and view all the answers

In the context of equality, what does the transitive property imply?

If x = y and y = z, then x = z. (C) Signup and view all the answers

When performing existential instantiation, which of the following statements is NOT a legitimate result?

AsHighAs(Everest, Everest) (A) Signup and view all the answers

What is the purpose of substitution in first-order logic?

To replace variables with constants or other variables. (C) Signup and view all the answers

Which of the following logical expressions correctly defines the relationship between offspring and parent as inverse relations?

Both B and C are correct. (B) Signup and view all the answers

Which of the following predicates is NOT an example of a logical formulation related to equality?

x + y = z means y = x (D) Signup and view all the answers

What can be derived from the knowledge base given the statement ¬P1,2?

There is no pit in square [1,2]. (A) Signup and view all the answers

Which of the following statements accurately represents the knowledge that the agent is in square [1,1] and it is safe?

¬P1,1 and ¬W1,1 are true. (D) Signup and view all the answers

What can we conclude if the agent perceives a breeze in a room, indicated as Bx,y being true?

There is at least one pit in an adjacent room. (A) Signup and view all the answers

What is the implication of the statement (WumpusAhead ∧ WumpusAlive) ⇒ Shoot when WumpusAhead is false?

The action to shoot is now irrelevant. (A) Signup and view all the answers

When using And-Elimination, which statement can be inferred from (WumpusAhead ∧ WumpusAlive)?

Both WumpusAhead and WumpusAlive can be inferred as true. (D) Signup and view all the answers

What can be derived if we know ¬W2,3 is true, meaning there is no wumpus in square [2,3]?

There could be a pit in square [2,3]. (B) Signup and view all the answers

Which of the following accurately describes the role of negations in the agent's knowledge base?

Negations represent truths about the absence of threats. (B) Signup and view all the answers

What is the effect of knowing that there is a stench in square [x,y]?

The wumpus is guaranteed to be in an adjacent square. (D) Signup and view all the answers

Study Notes

Equality

Axiomatizing equality involves defining sentences about the equality relation in a knowledge base.
Equality is reflexive, symmetric, and transitive.
Equals can be substituted for equals in predicates and functions.
∀x x = x (Equality is reflexive)
∀x, y x = y ⇒ y = x (Equality is symmetric)
∀x, y, z (x = y ∧ y = z) ⇒ x = z (Equality is transitive)

Demodulation

Demodulation is a more efficient way to handle equality.
It uses inference rules instead of axioms.
A demodulation rule takes an equality clause (x = y) and a clause, and substitutes y for x.
The substitution happens if the term in the clause unifies with x.
The substitution is unidirectional; only x gets replaced by y.

Father/PaternalGrandfather

Birthdate(Father(Father(Bella)), 1926) ⇒ Birthdate(PaternalGrandfather(Bella), 1926) can be concluded via demodulation.

First Order Logic (FOL)

Horses, cows and pigs are mammals.
An offspring of a horse is a horse.
Bluebeard is a horse.
Bluebeard is Charlie's parent.
Offspring and parent relations are inverse.

Existential Instantiation

If a knowledge base has ∃ x AsHighAs(x, Everest), possible legitimate results of Existential Instantiation are:
AsHighAs(Everest, Everest)
AsHighAs(Kilimanjaro, Everest).
AsHighAs(Kilimanjaro, Everest) ^ AsHighAs(BenNevis, Everest). However, the last one introduces new variables and thus is unsound

Knowledge Base

A knowledge base (KB) is a set of sentences expressed in a knowledge representation language.
Sentences represent assertions about the world.
Axioms are sentences taken as given without derivation.

Operations (Knowledge Base)

TELL adds sentences to the KB
ASK queries the KB
Both may involve inference to derive new sentences from existing ones.

Wumpus World

A cave with rooms connected by passageways.
Contains a wumpus, pits and an agent.
The agent can perceive breeze and stench.

Inference and Proofs

Modus Ponens (modus ponens): If a → b and a are given, then b can be inferred.
And-Elimination: From a ^ b , a or b can me inferred.

Nested Quantifiers

∀ x ∀ y Brother(x, y) ⇒ Sibling(x, y) (Brothers are siblings)
∀ x, y Sibling(x, y) ⇒ Sibling (y, x).(siblinghood is symmetric)
"Everyone loves someone." – ∀x∃y Loves(x, y).
“There is someone who is loved by everyone.” – ∃y∀ x Loves(x, y).
"Everybody loves somebody" means that for every person there is someone they love.

De Morgan Rules

¬∀x P = ∃x ¬P
¬∃x P = ∀x ¬P

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Description

This quiz explores key concepts in First Order Logic, focusing on the axioms of equality, demodulation techniques, and implications regarding paternal relations. Test your understanding of reflexivity, symmetry, transitivity, and how these principles apply in logical reasoning.