# CS代考计算机代写 algorithm CS 561a: Introduction to Artificial Intelligence

CS 561a: Introduction to Artificial Intelligence

CS 561, Sessions 9-10

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Knowledge and reasoning – second part

Knowledge representation

Logic and representation

Propositional (Boolean) logic

Normal forms

Inference in propositional logic

Wumpus world example

CS 561, Sessions 9-10

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Knowledge-Based Agent

Agent that uses prior or acquired knowledge to achieve its goals

Can make more efficient decisions

Can make informed decisions

Knowledge Base (KB): contains a set of representations of facts about the Agent’s environment

Each representation is called a sentence

Use some knowledge representation language, to TELL it what to know e.g., (temperature 72F)

ASK agent to query what to do

Agent can use inference to deduce new facts from TELLed facts

Knowledge Base

Inference engine

Domain independent algorithms

Domain specific content

TELL

ASK

CS 561, Sessions 9-10

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Generic knowledge-based agent

TELL KB what was perceived

Uses a KRL to insert new sentences, representations of facts, into KB

ASK KB what to do.

Uses logical reasoning to examine actions and select best.

CS 561, Sessions 9-10

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Wumpus world example

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Wumpus world characterization

Deterministic?

Accessible?

Static?

Discrete?

Episodic?

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Wumpus world characterization

Deterministic? Yes – outcome exactly specified.

Accessible? No – only local perception.

Static? Yes – Wumpus and pits do not move.

Discrete? Yes

Episodic? (Yes) – because static.

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Exploring a Wumpus world

A= Agent

B= Breeze

S= Smell

P= Pit

W= Wumpus

OK = Safe

V = Visited

G = Glitter

CS 561, Sessions 9-10

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Exploring a Wumpus world

A= Agent

B= Breeze

S= Smell

P= Pit

W= Wumpus

OK = Safe

V = Visited

G = Glitter

CS 561, Sessions 9-10

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Exploring a Wumpus world

A= Agent

B= Breeze

S= Smell

P= Pit

W= Wumpus

OK = Safe

V = Visited

G = Glitter

CS 561, Sessions 9-10

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Exploring a Wumpus world

A= Agent

B= Breeze

S= Smell

P= Pit

W= Wumpus

OK = Safe

V = Visited

G = Glitter

CS 561, Sessions 9-10

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Exploring a Wumpus world

A= Agent

B= Breeze

S= Smell

P= Pit

W= Wumpus

OK = Safe

V = Visited

G = Glitter

CS 561, Sessions 9-10

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Exploring a Wumpus world

A= Agent

B= Breeze

S= Smell

P= Pit

W= Wumpus

OK = Safe

V = Visited

G = Glitter

CS 561, Sessions 9-10

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Exploring a Wumpus world

A= Agent

B= Breeze

S= Smell

P= Pit

W= Wumpus

OK = Safe

V = Visited

G = Glitter

CS 561, Sessions 9-10

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Exploring a Wumpus world

A= Agent

B= Breeze

S= Smell

P= Pit

W= Wumpus

OK = Safe

V = Visited

G = Glitter

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Other tight spots

CS 561, Sessions 9-10

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Another example solution

No perception 1,2 and 2,1 OK

Move to 2,1

B in 2,1 2,2 or 3,1 P?

1,1 V no P in 1,1

Move to 1,2 (only option)

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Example solution

S and No S when in 2,1 1,3 or 1,2 has W

1,2 OK 1,3 W

No B in 1,2 2,2 OK & 3,1 P

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Logic in general

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Types of logic

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The Semantic Wall

Physical Symbol System World

+BLOCKA+

+BLOCKB+

+BLOCKC+

P1:(IS_ON +BLOCKA+ +BLOCKB+)

P2:((IS_RED +BLOCKA+)

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Syntax: says what is allowed on the LHS

Semantics: says how what is on the LHS relates to what is on the RHS

Inference: says how you can manipulate (formally, i.e., with no reference to the RHS) the symbols. [remember PSSH]

Want to be able to trust the results: want whatever the inference procedure does to “respect” what’s true or what follows in the world. So this is where we’re headed; good to keep in mind as we go through all the definitions now to follow. There is a method in this madness…

algebra example (put on board): but don’t use “entails” instead convey the idea

> n m: is this true or false? don’t know

if n=3, m=5, and > has its usual meaning, then (>n m) is false

(> n m) and (> m p) entail (n p)

CS 561, Sessions 9-10

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Truth depends on Interpretation

Representation 1 World

A

B

ON(A,B) T

ON(B,A) F

ON(A,B) F A

ON(B,A) T B

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block pictures

(on A B) and on(B C) entails (on A C)

(again don’t say “entails)

for this as well as for number example jus t did, need transitivity relationships

spend time on why it matters: if build a KB want to be able to verify its answers

introduce notion of “refer” “mean” idea of mapping into world; for this slide handwave about meaning of “on(a,b)”

CS 561, Sessions 9-10

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Entailment

Entailment is different than inference

CS 561, Sessions 9-10

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Logic as a representation of the World

Facts

World

Fact

follows

Refers to

(Semantics)

Representation: Sentences

Sentence

entails

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Models

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Inference

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Basic symbols

Expressions only evaluate to either “true” or “false.”

P “P is true”

¬P “P is false” negation

P V Q “either P is true or Q is true or both” disjunction

P ^ Q “both P and Q are true” conjunction

P => Q “if P is true, then Q is true” implication

P Q “P and Q are either both true or both false” equivalence

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Propositional logic: syntax

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Propositional logic: semantics

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Truth tables

Truth value: whether a statement is true or false.

Truth table: complete list of truth values for a statement given all possible values of the individual atomic expressions.

Example:

P Q P V Q

T T T

T F T

F T T

F F F

CS 561, Sessions 9-10

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Truth tables for basic connectives

P Q ¬P ¬Q P V Q P ^ Q P=>Q PQ

T T F F T T T T

T F F T T F F F

F T T F T F T F

F F T T F F T T

CS 561, Sessions 9-10

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Propositional logic: basic manipulation rules

¬(¬A) = A Double negation

¬(A ^ B) = (¬A) V (¬B) Negated “and”

¬(A V B) = (¬A) ^ (¬B) Negated “or”

A ^ (B V C) = (A ^ B) V (A ^ C) Distributivity of ^ on V

A V (B ^ C) = (A V B) ^ (A V C) Distributivity of V on ^

A => B = (¬A) V B by definition

¬(A => B) = A ^ (¬B) using negated or

A B = (A => B) ^ (B => A) by definition

¬(A B) = (A ^ (¬B))V(B ^ (¬A)) using negated and & or

…

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Propositional inference: enumeration method

true

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Enumeration: Solution

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Propositional inference: normal forms

“sum of products of

simple variables or

negated simple variables”

“product of sums of

simple variables or

negated simple variables”

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Deriving expressions from functions

Given a boolean function in truth table form, find a propositional logic expression for it that uses only V, ^ and ¬.

Idea: We can easily do it by disjoining the “T” rows of the truth table.

Example: XOR function

P Q RESULT

T T F

T F T P ^ (¬Q)

F T T (¬P) ^ Q

F F F

RESULT = (P ^ (¬Q)) V ((¬P) ^ Q)

CS 561, Sessions 9-10

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Deriving expressions from functions

Given a boolean function in truth table form, find a propositional logic expression for it that uses only V, ^ and ¬.

Idea: We can easily do it by disjoining the “T” rows of the truth table.

Example: XOR function

P Q RESULT

T T F

T F T P ^ (¬Q)

F T T (¬P) ^ Q

F F F

RESULT = (P ^ (¬Q)) V ((¬P) ^ Q)

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A more formal approach

To construct a logical expression in disjunctive normal form from a truth table:

Build a “minterm” for each row of the table, where:

– For each variable whose value is T in that row, include

the variable in the minterm

– For each variable whose value is F in that row, include

the negation of the variable in the minterm

– Link variables in minterm by conjunctions

The expression consists of the disjunction of all minterms.

CS 561, Sessions 9-10

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Example: adder with carry

Takes 3 variables in: x, y and ci (carry-in); yields 2 results: sum (s) and carry-out (co). To get you used to other notations, here we assume T = 1, F = 0, V = OR, ^ = AND, ¬ = NOT.

co is:

s is:

CS 561, Sessions 9-10

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Tautologies

Logical expressions that are always true. Can be simplified out.

Examples:

T

T V A

A V (¬A)

¬(A ^ (¬A))

A A

((P V Q) P) V (¬P ^ Q)

(P Q) => (P => Q)

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Validity and satisfiability

Theorem

B

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Proof methods

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Inference Rules

Modus Tollens:

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Inference Rules

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Inference example

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Inference example

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Inference example

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Inference example

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Inference example

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Inference example

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Inference example

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Inference example

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Inference example

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Inference example

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Wumpus world: example

Facts: Percepts inject (TELL) facts into the KB

[stench at 1,1 and 2,1] S1,1 ; S2,1

Rules: if square has no stench then neither the square or adjacent squares contain the wumpus

R1: ¬S1,1 ¬W1,1 ¬W1,2 ¬W2,1

R2: ¬S2,1 ¬W1,1 ¬W2,1 ¬W2,2 ¬W3,1

…

Inference:

KB contains ¬S1,1 then using Modus Ponens we infer

¬W1,1 ¬W1,2 ¬W2,1

Using And-Elimination we get: ¬W1,1 ¬W1,2 ¬W2,1

…

CS 561, Sessions 9-10

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Limitations of Propositional Logic

1. It is too weak, i.e., has very limited expressiveness:

Each rule has to be represented for each situation:

e.g., “don’t go forward if the wumpus is in front of you” takes 64 rules

2. It cannot keep track of changes:

If one needs to track changes, e.g., where the agent has been before then we need a timed-version of each rule. To track 100 steps we’ll then need 6400 rules for the previous example.

Its hard to write and maintain such a huge rule-base

Inference becomes intractable

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Summary

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