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Have some fun

Checker:Checker: http://www.cs.caltech.edu/~vhuang/cs20/c/applet/more.html

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Local Beam Search

Run multiple searches to find theRun multiple searches to find the solution

The best K states are selected.

Like parallel hill climbing from different start points but:

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pAdvantage: They communicate to localize the searchDrawback: may miss the global optima

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Stochastic Beam Search

Instead of selecting the best KInstead of selecting the best K successors, use probability to select.

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Online SearchExplore the environment like MarsRoverExplore the environment like MarsRoverAgent knows:

ACTIONS(s): list of possible actionsStep cost c(s,a,s’): cost of going from s to s’ taking action aGOAL-TEST(s)

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Competitive ratio: path cost using exploration/optimal path cost Optimal path cost = off-line planning knowing the map

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Online Search (II)

Competitive ratio:Competitive ratio: Can be infinite if there are dead ends

Assumption:

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Assumption:Safely explorable: there are paths from each reachable state to goal.

Online Search (III)

DFS is acceptable methodDFS is acceptable method

Hill climbing is doable too

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LRTA*: Learning Real-Time A*

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Knowledge and reasoning

Knowledge representationKnowledge representationLogic and representationPropositional (Boolean) logicNormal formsI f i iti l l i

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Inference in propositional logicWumpus world example

Knowledge-Based Agent

Knowledge Base

Inference engine

Domain independent algorithms

TELL

ASK

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Domain specific content

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Generic knowledge-based agent

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1. TELL KB what was perceived

2. ASK KB what to do.

Wumpus world example

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Wumpus world characterization

Deterministic?Deterministic?

Accessible?

Static?

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Discrete?

Episodic?

Let’s play a game☺

http://www cogsci rpi edu/Otter/Wumphttp://www.cogsci.rpi.edu/Otter/Wumpus/

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Exploring a Wumpus world

A= AgentB= BreezeS= SmellP= PitW= WumpusOK = SafeV = VisitedG Gli

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G = Glitter

Other tight spots

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Another example solution

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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)

Let’s try blindedhttp://www.cogsci.rpi.edu/Otter/Wumpus/

Geeky games: Office environmenthttp://gpf-comics.com/games/wumpus/Dark environmentBlack holes suck you in (you feel slight breeze)Trudy with big mallet to smash you (you will feel her perfume)You have to find the check and your way back to the starting pointPay check can be in Trudy or black hole cubical.

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Trudy can be in the black hole cubicalComputer wants to find your check too

Nick is conservativeKi is taking chance once or twiceFooker is an aggressive one.

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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+

P (IS ON +BLOCKA+ +BLOCKB+)

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P1:(IS_ON +BLOCKA+ +BLOCKB+)P2:((IS_RED +BLOCKA+)

Truth depends on Interpretation

Representation 1 WorldRepresentation 1 WorldA

BON(A,B) T

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ON(A,B) F A

B

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Entailment

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Logic as a representation of the World

R i Sentails

Refers to (Semantics)

Representation: Sentences Sentence

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FactsWorld Factfollows

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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 y

P¬PP V QP ^ Q

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P => QP Q

Propositional logic: syntax

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Propositional logic: semantics

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Truth tables

Truth value: whether a statement is true or falseTruth 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 QT T

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T TT FF TF F

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Truth tables for basic connectives

P Q ¬P ¬Q P V Q P ^ Q P=>Q P QP Q ¬P ¬Q P V Q P ^ Q P=>Q P Q

T T F F TT F F T TF T T F T

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F F T T F

P Implies Q: (X>Y ^ Y>Z) => X>Z

P Q P=>QP Q P=>Q

T T TT F FF T T

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F F T

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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 > B ( A) V B by definition

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A => B = (¬A) V B by definition¬(A => B) = A ^ (¬B) using negated orA B = (A => B) ^ (B => A) by definition¬(A B) = (A ^ (¬B))V(B ^ (¬A))

using negated and & or

Propositional inference: enumeration method

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Propositional inference: normal forms

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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 Vpropositional 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 functionP Q RESULT

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T T FT F TF T TF F F

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A more formal approachTo 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 mintermFor each variable whose value is F in that row, include

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the negation of the variable in the minterm

Link variables in minterm by conjunctionsThe expression consists of the disjunction of all minterms.

Example: adder with carry

Takes 3 variables in: x y and ci (carry in);Takes 3 variables in: x, y and ci (carry-in);

results: sum (s) and carry-out (co).

To get you used to other notations, here we

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assume T = 1, F = 0, V = OR, ^ = AND, ¬ = NOT.

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Example: adder with carry

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co is:

s is:

TautologiesSimplify the logical expressions that are always true.

Examples:TT V AA V (¬A)

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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

Th

B

Theorem

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Model of a Formula

assignment a b a => b a^ b ~ (a b)A 0 0 1 0 1B 0 1 1 0 0C 1 0 0 0 0D 1 1 1 1 1

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a) The assignments A, B and D are models of the formula a => b.b) The assignment D is model of the formula a ^ b.c) The assignments A and D are models of the formula ~(a b).

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Satisfiability Exampleassignment a b a=>b a^b (a b) (a V b) a V ( a)assignment a b a=>b a^b (a b) ~(a V b) a V (~a)

A 0 0 1 0 1 1 1B 0 1 1 0 0 0 1C 1 0 0 0 0 0 1D 1 1 1 1 1 0 1

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Proof methods

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Inference Rules

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Inference Rules

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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

R l if h h h i h hRules: if square has no stench then neither the square or adjacent square contain the wumpus

R1: !S1,1 ⇒!W1,1 ∧ !W1,2 ∧ !W2,1R2: !S2,1 ⇒!W1,1 ∧!W2,2 ∧ !W2,2 ∧ !W3,1…

I f

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Inference: KB contains !S1,1 then using Modus Ponens we infer!W1,1 ∧ !W1,2 ∧ !W2,1Using And-Elimination we get: !W1,1 !W1,2 !W2,1…

Limitations of Propositional Logic

1 It is too weak (has very limited expressiveness):1. It is too weak (has very limited expressiveness):

2. 2. It cannot keep track of changes:

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Summary

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Next time

First order logic: [AIMA] Chapter 7First-order logic: [AIMA] Chapter 7

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