Lec 1 | Knowledge

Question 1

Agents that reason by operating on internal representations of knowledge

Answer 1

Knowledge Based Agents

Question 2

An assertion about the world in a knowledge representation language

Answer 2

Sentence

Question 3

It is based on propositions; statements about the world that can be either true or false

Answer 3

Propositional logic

Question 4

What are propositional symbols?

Answer 4

Propositional symbols are most often letters (P, Q, R) that are used to represent a proposition.

Question 5

What are the logical connectives?

Answer 5

not, and, or, implication, and biconditional

Question 6

Logical Connectives:

Inverses the truth value of the proposition.

Answer 6

Not

Question 7

Logical Connectives:

Connects two different propositions

Answer 7

And

Question 8

Logical Connectives:

It is true as long as either one of its arguments is true

Answer 8

Or

Question 9

Logical Connectives:

Represents a structure of “if P then Q.” Where P is the antecedent and Q is called the consequent

Answer 9

Implication

Question 10

Logical Connectives:

An implication that goes both directions. You can read it as “if and only if.”

Answer 10

Biconditional

Question 11

It is an assignment of a truth value to every propositional symbol (a “possible world”).

Answer 11

Model

Question 12

It is the truth-value assignment that provides information about the world.

Answer 12

Model

Question 13

A set of sentences known by a knowledge-based agent

Answer 13

Knowledge Base (KB)

Question 14

This is knowledge that the AI is provided about the world in the form of propositional logic sentences that can be used to make additional inferences about the world.

Answer 14

Knowledge Base (KB)

Question 15

In every model or world in which sentence α is true, sentence β is also true

Answer 15

Entailment

Question 16

The process of deriving new sentences
from old ones

Answer 16

Inference

Question 17

What is the Model Checking algorithm?

Answer 17

To determine if KB ⊨ α (in other words, answering the question: “can we conclude that α is true based on our knowledge base”)
* Enumerate all possible models.
* If in every model where KB is true, α is true as well, then KB entails α (KB ⊨ α).
* Otherwise KB does not entail α

Question 18

Is the model checking algorithm efficient?

Answer 18

Model Checking is not an efficient algorithm because it has to consider every possible model before giving the answer (a reminder: a query R is true if under all the models (truth assignments) where the KB is true, R is true as well).

Question 19

What is the difference between the Model Checking Algorithm and the Inference Rules?

Answer 19

Inference rules allow us to generate new information based on existing knowledge without considering every possible model.

Question 20

What are needed to run the Model Checking algorithm?

Answer 20

• Knowledge Base, which will be used to draw inferences
• A query, or the proposition that we are interested in whether it is entailed by the KB
• Symbols, a list of all the symbols (or atomic propositions) used (in our case, these are rain, hagrid, and dumbledore)
• Model, an assignment of truth and false values to symbols

Question 21

The process of figuring out how to represent propositions and logic in AI.

Answer 21

Knowledge Engineering

Question 22

What are the Inference Rules?

Answer 22

Modus Ponens, Elimination, Double Negation Elimination, Biconditional Elimination, De Morgan’s Law, and Distributive Property

Question 23

Theorem Proving

Answer 23

• Initial state: starting knowledge base
• Actions: inference rules
• Transition model: new knowledge base after inference
• Goal test: checking whether the statement that we are trying to prove is in the KB
• Path cost function: the number of steps in the proof

Question 24

A powerful inference rule that states that if one of two atomic propositions in an Or proposition is false, the other has to be true.

Answer 24

Resolution

Question 25

What does Resolution rely on?

Answer 25

Complementary Literals

Question 26

Two of the same atomic propositions where one is negated and the other is not, such as P and ¬P.

Answer 26

Complementary Literals

Question 27

It is a disjunction of literals (a propositional symbol or a negation of a propositional symbol, such as P, ¬P)

Answer 27

Clause

Question 28

Consists of propositions that are connected with an Or logical connective (P ∨ Q ∨ R)

Answer 28

Disjunction

Question 29

Logical sentence that is a conjunction of clauses. for example: (A ∨ B ∨ C) ∧ (D ∨ ¬E) ∧ (F ∨ G)

Answer 29

Conjunctive Normal Form (CNF)

Question 29

Consists of propositions that are connected with an And logical connective (P ∧ Q ∧ R)

Answer 29

Conjunctions

Question 30

What are the steps in Converting Propositions to CNF?

Answer 30

1. Eliminate biconditionals * Turn (α ↔ β) into (α → β) ∧ (β → α). 2. Eliminate implications * Turn (α → β) into ¬α ∨ β. 3. Move negation inwards until only literals are being negated (and not clauses), using De Morgan’s Laws. * Turn ¬(α ∧ β) into ¬α ∨ ¬β

Question 31

What process is used when a clause contains the same literal twice?

Answer 31

Factoring. The duplicate literal is removed.

Question 32

Proof by contradiction

Answer 32

``` To determine if KB ⊨ α: Check: is (KB ∧ ¬α) a contradiction? If so, then KB ⊨ α. Otherwise, no entailment. ```

Question 33

Another Resolution Algorithm

Answer 33

``` To determine if KB ⊨ α: - Convert (KB ∧ ¬α) to Conjunctive Normal Form. - Keep checking to see if we can use resolution to produce a new clause. - If we ever produce the empty clause (equivalent to False), a contradiction has been reached, thus proving that KB ⊨ α. - However, if contradiction is not achieved and no more clauses can be inferred, there is no entailment. ```

Question 34

Another type of logic that allows us to express more complex ideas more succinctly than propositional logic.

Answer 34

First Order Logic

Question 35

What are the two types of Symbols used in First Order Logic?

Answer 35

Constant Symbols and Predicate Symbols.

Question 36

What do constant symbols represent?

Answer 36

objects

Question 37

What are predicate symnols?

Answer 37

Predicate symbols are like relations or functions that take an argument and return a true or false value.

Question 38

It is a tool that can be used in first order logic to represent sentences without using a specific constant symbol.

Answer 38

Quantification

Question 39

What symbol is used in Universal Quantification?

Answer 39

Universal quantification uses the symbol ∀ to express “for all.”

Question 40

It is an idea parallel to universal quantification.

Answer 40

Existential Quantification

Question 41

It is used to create sentences that are true for at least one x.

Answer 41

Existential Quantification

Question 42

What symbol is used in Existential Quantification?

Answer 42

It is expressed using the symbol ∃.

Question 43

Can the Existential and Universal Quantification be used in the same sentence?

Answer 43

Yes. For example, the sentence ∀x. Person(x) → (∃y. House(y) ∧ BelongsTo(x, y)) expresses the idea that if x is a person, then there is at least one house, y, to which this person belongs. In other words, this sentence means that every person belongs to a house.

Question 44

# CS50 QUIZ Consider these logical sentences: 1. *If Hermione is in the library, then Harry is in the library.* 2. *Hermione is in the library.* 3. *Ron is in the library and Ron is not in the library.* 4. *Harry is in the library.* 5. *Harry is not in the library or Hermione is in the library.* 6. *Ron is in the library or Hermione is in the library.* Which of the following logical entailments is true? * Sentence 6 entails Sentence 2 * Sentence 1 entails Sentence 4 * Sentence 6 entails Sentence 3 * Sentence 2 entails Sentence 5 * Sentence 1 entails Sentence 2 * Sentence 5 entails Sentence 6

Answer 44

Sentence 2 entails Sentence 5

Question 45

# CS50 QUIZ There are other logical connectives that exist, other than the ones discussed in lecture. One of the most common is “Exclusive Or” (represented using the symbol ⊕). The expression A ⊕ B represents the sentence “A or B, but not both.” Which of the following is logically equivalent to A ⊕ B? * (A ∨ B) ∧ ¬ (A ∧ B) * (A ∧ B) ∨ ¬ (A ∨ B) * (A ∨ B) ∧ (A ∧ B) * (A ∨ B) ∧ ¬ (A ∨ B)

Answer 45

(A ∨ B) ∧ ¬ (A ∧ B)

Question 46

# CS50 QUIZ Let propositional variable R be that “It is raining,” the variable C be that “It is cloudy,” and the variable S be that “It is sunny.” Which of the following a propositional logic representation of the sentence “If it is raining, then it is cloudy and not sunny.”? * (R → C) ∧ ¬S * R → C → ¬S * R ∧ C ∧ ¬S * R → (C ∧ ¬S) * (C ∨ ¬S) → R

Answer 46

R → (C ∧ ¬S)

Question 47

# CS50 QUIZ Consider, in first-order logic, the following predicate symbols. Student(x) represents the predicate that “x is a student.” Course(x) represents the predicate that “x is a course.” Enrolled(x, y) represents the predicate that “x is enrolled in y.” Which of the following is a first-order logic translation of the sentence “There is a course that Harry and Hermione are both enrolled in.”? * ∃x. Course(x) ∧ Enrolled(Harry, x) ∧ Enrolled(Hermione, x) * ∀x. Course(x) ∧ Enrolled(Harry, x) ∧ Enrolled(Hermione, x) * ∃x. Enrolled(Harry, x) ∧ ∃y. Enrolled(Hermione, y) * ∀x. Enrolled(Harry, x) ∧ ∀y. Enrolled(Hermione, y) * ∃x. Enrolled(Harry, x) ∨ Enrolled(Hermione, x) * ∀x. Enrolled(Harry, x) ∨ Enrolled(Hermione, x)

Answer 47

∃x. Course(x) ∧ Enrolled(Harry, x) ∧ Enrolled(Hermione, x)