c1 propositional logic

8/8/2019 C1 Propositional Logic

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The Wumpus Game

Stench in the square containing the wumpus andin 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

Agents percept : [Stench, Breeze, Glitter, Bump, Scream]

Agents 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 Agents goal: find and bring the gold back to the

start



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The Wumpus Game

Stench Breeze

Breeze

Stench

Gold

Breeze

Stench

OK

Breeze

Start

A

Breeze

OK

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

Breeze

Stench

Gold

Breeze

Stench

OK P?

Breeze

Start

V

Breeze

A 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

BreezeStench

Gold

Breeze

Stench

A OK

Breeze

Start

V

Breeze

V

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

BreezeStench

Gold

Breeze

Stench

V A

Breeze

Start

V

Breeze

V

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

BreezeStench

Gold A

Breeze

Stench

V V

Breeze

Start

V

Breeze

V

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 E is a sentence, then so are E and (E)

If E and F are sentences, then so are E FE F

E F, and E F



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Propositional Logic Semantics

Interpretation: propositional symbol p true/false

The truth value of a sentence is defined by thetruth 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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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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Model: an interpretation under which the sentenceis true

Q

P Q

P Q

P Q

P

Q

P Q

P Q

P Q

P



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Entailment : KB `! E iff every model of KB is a modelof E

E 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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Equivalence: E| F iff E `! F and F `! E

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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Theorems:

E `! F iff E F is valid

KB `! E can be proved by validity KB E

E `! F iff E F is unsatisfiable

KB `! E can be proved by refutation of KB E



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Inference Rules

Rule R:Premises: E1, E2, , En

Conclusion: F

Soundness: R is sound iff {E1, E2, , En} `! F



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Inference Rules

Modus Ponens:E F, E

F

-Elimination:E1 E2 En

Ei



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Inference Rules

-Introduction:E1, E2, , En

E1 E2 En

-Introduction:Ei

E1 E2 En



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Inference Rules

Double-negation elimination:E

E



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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 Q j Qm

Pi | Q j P1 Pi-1 Pi+1 Pn Q1 Q j-1 Q j+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 baseas a set of sentences

Inference rules + a searching algorithm



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Algorithm PL-Resolution(KB, E)

input: KB and E are PL sentences

output: true (to assert KB `! E) or false (otherwise)

Clauses = set of clauses in the CNF representation of KB E

New = {}

loop

for each Ci, C j in Clauses do

Resolvents = PL-Resolve(Ci, C j)

if Resolvents contains an empty clause then return true

New = New Resolvents

if New Clauses then return false

Clauses = Clauses New



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Soundness: every deduced answer is correct

if PL-Resolution returns true then KB `! E

Completeness: every correct answer is deducible

if KB `! Ethen PL-Resolution returns true



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PL-Resolution is sound and complete

Modus Ponens with any searching algorithm is not complete

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



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Horn clause: disjunction of literals of which at most one is positive

P1 P2 Pn Q

orP1 P2 Pn Q

Forward or backward chaining can be used forinference 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 assignedtrue/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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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 sciencecan be reduced to SAT



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Backtracking algorithms

Davis and Putnam (1960)

Davis, Logemann, Loveland (1962)



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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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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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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 canonly increase when information is added to theKB

if KB1 `! E then (KB1KB2) `! E

Propositional logic is monotone



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An Agent for the Wumpus Game

Knowledge Base: X ij = 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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Translating knowledge into action:

A11 East A W21 Forward



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Problems with the propositional agent:

Too many propositions to handle:

"Dont 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 aquestion like "What action should the agent take?",



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Homework

In Russell & Norvigs AIMA (2nd ed.): Exercises of Chapter 7.

c1 propositional logic

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