For Friday

• Read chapter 8

• Homework:– Chapter 7, exercise 1

Program 1

• Any questions?

Alpha-Beta Effectiveness

• Depends on the order in which siblings are considered

• Optimal ordering would reduce nodes considered from O(bd) to O(bd/2)--but that requires perfect knowledge

• Simple ordering heuristics can help quite a bit

Chance

• What if we don’t know what the options are?

• Expectiminimax uses the expected value for any node where chance is involved.

• Pruning with chance is more difficult. Why?

Imperfect Knowledge

• What issues arise when we don’t know everything (as in standard card games)?

State of the Art

• Chess – Deep Blue and Fritz

• Checkers – Chinook

• Othello – Logistello

• Backgammon – TD-Gammon (learning)

• Go – Computers are very bad

• Bridge

What about the games we play?

Knowledge

• Knowledge Base– Inference mechanism (domain-independent)– Information (domain-dependent)

• Knowledge Representation Language– Sentences (which are not quite like English sentence)– The KRL determine what the agent can “know”– It also affects what kind of reasoning is possible

• Tell and Ask

Getting Knowledge

• We can TELL the agent everything it needs to know

• We can create an agent that can “learn” new information to store in its knowledge base

The Wumpus World

• Simple computer game

• Good testbed for an agent

• A world in which an agent with knowledge should be able to perform well

• World has a single wumpus which cannot move, pits, and gold

Wumpus Percepts• The wumpus’s square and squares adjacent to it smell

bad.

• Squares adjacent to a pit are breezy.

• When standing in a square with gold, the agent will perceive a glitter.

• The agent can hear a scream when the wumpus dies from anywhere

• The agent will perceive a bump if it walks into a wall.

• The agent doesn’t know where it is.

Wumpus Actions

• Go forward• Turn left• Turn right• Grab (picks up gold in that square)• Shoot (fires an arrow forward--only once)

– If the wumpus is in front of the agent, it dies.

• Climb (leave the cavern--only good at the start square)

Consequences

• Entering a square containing a live wumpus is deadly

• Entering a square containing a pit is deadly

• Getting out of the cave with the gold is worth 1,000 points.

• Getting killed costs 10,000 points

• Each action costs 1 point

Possible Wumpus Environment

S te n c h

S te n c h

S te n c hS te n c h

B re e ze

B re e zeB re e ze

B re e ze

B re e ze B re e ze

G o ld

P it

P it

P it

W u m p u s

A g e n t

Knowledge Representation

• Two sets of rules:– Syntax: determines what atomic symbols exist

in the language and how to combine them into sentences

– Semantics: Relationship between the sentences and “the world”--needed to determine truth or falsehood of the sentences

Reasoning

• Entailment

• Inference– May produce new sentences entailed by KB– May be used to determine which a particular

sentence is entailed by the KB

• We want inference procedures that are sound, or truth-preserving.

What Is a Logic?

• A set of language rules– Syntax– Semantics

• A proof theory– A set of rules for deducing the entailments of a

set of sentences

Distinguishing LogicsLanguage Ontological

Commitment (whatexists in the world)

EpistemologicalCommitment (Whatan agent believesabout facts)

PropositionalLogic

facts true/false/unknown

First-order logic facts, objects,relations

true/false/unknown

Temporal logic facts, objects,relations, times

true/false/unknown

Probability theory facts degree of belief 0…1

Fuzzy logic degree of truth degree of belief 0…1

Propositional Logic

• Simple logic

• Deals only in facts

• Provides a stepping stone into first order logic

Syntax• Logical Constants: true and false• Propositional symbols P, Q ... are sentences• If S is a sentence then (S) is a sentence.• If S is a sentence then ¬S is a sentence.

• If S1 and S2 are sentences, then so are:

– S1 S2

– S1 S2

– S1 S2

– S1 S2

Semantics• true and false mean truth or falsehood in the world

• P is true if its proposition is true of the world

• ¬S is the negation of S

• The remainder follow standard truth tables– S1 S2 : AND

– S1 S2 : inclusive OR

– S1 S2 : True unless S1 is true and S2 is false

– S1 S2 : bi-conditional, or if and only if

Inference

• An interpretation is an assignment of true or false to each atomic proposition

• A sentence true under any interpretation is valid (a tautology or analytic sentence)

• Validity can be checked by exhaustive checking of truth tables

Rules of Inference

• Alternative to truth-table checking

• A sequence of inference rule applications leading to a desired conclusion is a logical proof

• We can check inference rules using truth tables, and then use to create sound proofs

• We can treat finding a proof as a search problem

Propositional Inference Rules

• Modus Ponens or Implication Elimination

• And-Elimination

• Unit Resolution

• Resolution

Simpler Inference

• Horn clauses

• Forward-chaining

• Backward-chaining

Building an Agent with Propositional Logic

• Propositional logic has some nice properties– Easily understood– Easily computed

• Can we build a viable wumpus world agent with propositional logic???

The Problem

• Propositional Logic only deals with facts.

• We cannot easily represent general rules that apply to any square.

• We cannot express information about squares and relate (we can’t easily keep track of which squares we have visited)

More Precisely

• In propositional logic, each possible atomic fact requires a separate unique propositional symbol.

• If there are n people and m locations, representing the fact that some person moved from one location to another requires nm2 separate symbols.