1 the wumpus game stenchbreeze stench gold breeze stenchbreeze start breeze

1

The Wumpus Game

Stench Breeze

BreezeStenchGold

Breeze

Stench Breeze

Start Breeze Breeze

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The Wumpus Game• Stench in the square containing the wumpus

and in the directly adjacent squares

• Breeze in the squares directly adjacent to a pit

• Glitter in the square where the gold is

• Bump when an agent walks into a wall

• Scream when the wumpus is killed

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The Wumpus Game• Agent’s percept: [Stench, Breeze, Glitter, Bump, Scream]

• Agent’s actions:– Go forward, turn right 90o, turn left 90o

– Grab to pick up an object in the same square– Fire 1 arrow in a straight line in the faced direction – Climb to leave the cave

• Death if entering a square of a pit or a live wumpus

• Agent’s goal: find and bring the gold back to the start

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The Wumpus Game

Stench Breeze

BreezeStenchGold

Breeze

StenchOK

Breeze

Start A

BreezeOK

Breeze

A = Agent

B = Breeze

G = Glitter/Gold

OK = Safe square

P = Pit

S = Stench

V = Visited

W = Wumpus

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The Wumpus Game

Stench Breeze

BreezeStenchGold

Breeze

StenchOK P?

Breeze

Start V

BreezeA P?

Breeze

A = Agent

B = Breeze

G = Glitter/Gold

OK = Safe square

P = Pit

S = Stench

V = Visited

W = Wumpus

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The Wumpus Game

Stench Breeze

BreezeStenchGold

Breeze

StenchA OK

Breeze

Start V

BreezeV

Breeze

A = Agent

B = Breeze

G = Glitter/Gold

OK = Safe square

P = Pit

S = Stench

V = Visited

W = Wumpus

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The Wumpus Game

Stench Breeze

BreezeStenchGold

Breeze

StenchV A

Breeze

Start V

BreezeV

Breeze

A = Agent

B = Breeze

G = Glitter/Gold

OK = Safe square

P = Pit

S = Stench

V = Visited

W = Wumpus

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The Wumpus Game

Stench Breeze

BreezeStenchGold A

Breeze

StenchV V

Breeze

Start V

BreezeV

Breeze

A = Agent

B = Breeze

G = Glitter/Gold

OK = Safe square

P = Pit

S = Stench

V = Visited

W = Wumpus

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Propositional Logic Syntax• Logical constants: true, false

• Propositional symbols: P, Q, …

• Logical connectives: , , , ,

• Sentences (formulas):– Logical constants– Proposition symbols– If is a sentence, then so are and ()– If and are sentences, then so are

, and

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Propositional Logic Semantics

• Interpretation: propositional symbol true/false

• The truth value of a sentence is defined by the truth table

P Q P P Q P Q P Q P Q

false false true false false true true

false true true false true true false

true false false false true false false

true true false true true true true

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Propositional Logic Semantics

• Satisfiable: true under an interpretation

• Valid: true under all interpretations

P Q P P (P Q) Q ((P Q) Q) P

false false false false true

false true false false true

true false false true true

true true false false true

satisfiableunsatisfiable

valid

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Propositional Logic Semantics

• Model: an interpretation under which the sentence is true

Q

P Q

P Q

P Q

P

Q

P Q

P Q

P Q

P

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Propositional Logic Semantics

• Entailment: KB iff every model of KB is a model of

is a logical consequence of KB

P Q P Q P Q, P

false false true false

false true true false

true false false false

true true true true

{P Q, P} Q

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Propositional Logic Semantics

• Equivalence: iff and

P Q P Q P Q

false false true true

false true true true

true false false false

true true true true

P Q P Q

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Propositional Logic Semantics

• Theorems: – iff is valid

KB can be proved by validity KB

– iff is unsatisfiable

KB can be proved by refutation of KB

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Inference Rules• Rule R:

Premises:1,2,, n

Conclusion:

• Soundness: R is sound iff {1,2,, n}

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Inference Rules• Modus Ponens:

,

-Elimination:12n

i

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Inference Rules -Introduction:

1, 2, … , n

1 2 … n

-Introduction:i

1 2 … n

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Inference Rules• Double-negation elimination:

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Resolution• Unit resolution:

P1 … Pi … Pn, QPi Q (all Pi and Q are literals)

P1 … Pi-1 Pi+1 … Pn

• Full resolution:

P1 … Pi … Pn, Q1 … Qj … Qm

Pi Qj P1 … Pi-1 Pi+1 … Pn Q1 … Qj-1 Qj+1 … Qm

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Resolution• Conjunctive normal form (CNF): conjunction of

disjunctions of literals

(L11 … L1k) … (Ln1 … Lnm)

clause

• k-CNF: each clause contains at most k literals

• Every sentence can be written in CNF

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Inference Algorithms• To derive/assert a sentence given a knowledge

base as a set of sentences

• Inference rules + a searching algorithm

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Inference AlgorithmsAlgorithm PL-Resolution(KB, )

input: KB and are PL sentences

output: true (to assert KB ) or false (otherwise)

Clauses = set of clauses in the CNF representation of KB New = {}

loopfor each Ci, Cj in Clauses do

Resolvents = PL-Resolve(Ci, Cj)

if Resolvents contains an empty clause then return true

New = New Resolventsif New Clauses then return falseClauses = Clauses New

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Inference Algorithms• Soundness: every deduced answer is correct

if PL-Resolution returns true then KB

• Completeness: every correct answer is deducible

if KB then PL-Resolution returns true

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Inference Algorithms• PL-Resolution is sound and complete

• Modus Ponens with any searching algorithm is not complete

KB = {P Q, Q R}, = P R

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Inference Algorithms• Horn clause: disjunction of literals of which at

most one is positiveP1 P2 … Pn Q

orP1 P2 … Pn Q

• Forward or backward chaining can be used for inference on Horn clauses

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Satisfiability Problem • SAT problem: to determine if a sentence is

satisfiable

(to determine if the symbols in the sentence can be assigned true/false as to make it evaluate to true)

(D B C) (B A C) (C B E)

(E D B) (B E C)

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Satisfiability Problem • SAT problem in propositional logic is NP-

complete

This does not mean all instances of PL inference has the exponential complexity

Polynomial-time inference procedures for Horn clauses

• Many combinatorial problems in computer science can be reduced to SAT

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Satisfiability Problem • Backtracking algorithms

• Davis and Putnam (1960)

• Davis, Logemann, Loveland (1962)

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Satisfiability Problem • DPLL algorithm:

– Recursive, depth-first enumeration of possible models

– Early termination: a clause is true if any literal is true

(A B) (A C) is true if A is true regardless of B and C

– Pure symbol: has the same sign in all clauses

(A B) (B C) (C A)

A model would have the literals of pure symbols assigned true

Satisfied clause removal makes new pure symbols

– Unit clause: has just one literal

Unit clause satisfaction makes new unit clauses (unit propagation)

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Satisfiability Problem • Local search algorithms

Evaluation function counts the number of unsatisfied clauses

• Randomness to escape local minima

Flipping the truth value of one symbol at a time

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Statisfiability Problem Algorithm WalkSAT(Clauses, p, max_flips)

input: p is flipping probability, max_flips is number of flips allowed

output: a model or failure

Model = a random assignment of truth values to the symbols

for i = 1 to max_flips do

if Model satisfies Clauses then return Model

C = a randomly selected clause from Clauses that is false in Model

with probability p flip the value in Model of a symbol from C

else flip whichever symbol in C to maximizes no. of satisfied clauses

return failure

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Satisfiability Problem • Complexity depends on the clause/symbol ratio

Hard when the ratio is from than 4.3

• WalkSAT is much faster than DPLL

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Inference Monotonicity• Monotonicity: the set of entailed sentences

can only increase when information is added to the KB

if KB1 then (KB1KB2)

• Propositional logic is monotone

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An Agent for the Wumpus Game

• Knowledge Base: Xij = column i, row j

S11 B11 R1: S11 W11 W12 W21

S21 B21 R2: S21 W11 W21 W22 W31

S12 B12 R3: S12 W11 W12 W22 W13

R4: S12 W13 W12 W22 W11

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Agent for the Wumpus Game• Finding the wumpus:

– Modus Ponens (S11 + R1): W11 W12 W21

-Elimination: W11 W12 W21

– Modus Ponens (S21 + R2): W11 W21 W22 W31

-Elimination: W11 W21 W22 W31

– Modus Ponens (S12 + R4): W13 W12 W22 W11

– Unit resolution (W11 + W13 W12 W22 W11):

W13 W12 W22

– Unit resolution (W22 + W13 W12 W22): W13 W12

– Unit resolution (W12 + W13 W12): W13

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Agent for the Wumpus Game• Translating knowledge into action:

A11 EastA W21 Forward

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Agent for the Wumpus Game• Problems with the propositional agent:

– Too many propositions to handle:

"Don’t go forward if the wumpus is in front of you!" requires 16 x 4 = 64 propositions.

– Hard to deal with time and change

– Not expressive enough to represent or answer a question like "What action should the agent take?", …

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Homework• In Russell & Norvig’s AIMA (2nd ed.): Exercises

of Chapter 7.