lecture 41 - cook%27s theorem

7/28/2019 Lecture 41 - Cook%27s Theorem

1/46

Cooks

Theorem

Robb T.Koether

HomeworkReview

Cooks

TheoremThe Computation

Table

Example

The Functioncell

The Functionstart

The Functionaccept

The Functionmove

3SAT is

NP-Complete

Assignment

Cooks TheoremLecture 41

Section 7.4

Robb T. Koether

Hampden-Sydney College

Fri, Dec 5, 2008
http://find/


2/46

Cooks

Theorem

Robb T.Koether

HomeworkReview

Cooks


Table

Example

The Functioncell

The Functionstart

The Functionaccept

The Functionmove

3SAT is

NP-Complete

Assignment

Outline

1 Homework Review

2 Cooks Theorem

The Computation Table

ExampleThe Function cellThe Function startThe Function acceptThe Function move

3 3SAT is NP-Complete

4 Assignment
http://goforward/http://find/http://goback/


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4/46

Cooks

Theorem

Robb T.Koether

HomeworkReview

Cooks


Table

Example

The Functioncell

The Functionstart

The Functionaccept

The Functionmove

3SAT is

NP-Complete

Assignment

Homework Review

Solution

Let G be a graph and let n be the number of vertices in

G.

The number of ways to choose three vertices isn

3

=

n(n 1)(n 2)

6,

which is a polynomial function of n.

At worst, we would check every possible set of three

vertices until we either found a triangle or exhausted all

possibilities.

Thus, TRIANGLE P.
http://find/


5/46

Cooks

Theorem

Robb T.Koether

HomeworkReview

Cooks


Table

Example

The Functioncell

The Functionstart

The Functionaccept

The Functionmove

3SAT is

NP-Complete

Assignment

Cooks Theorem

Theorem (Cooks Theorem)

SAT P if and only if P= NP.

Proof idea.

First show that SAT is in NP.

Then show that every problem in NP is polynomial-time

reducible to SAT.
http://find/http://goback/


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Cooks

Theorem

Robb T.Koether

HomeworkReview

Cooks


Table

Example

The Functioncell

The Functionstart

The Functionaccept

The Functionmove

3SAT is

NP-Complete

Assignment

Proof of Cooks Theorem

Proof ().

It is easy to see that SAT is in NP.Simply nondeterministically choose Boolean values for

each variable.

Then verify that the expression is true.
http://find/


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Cooks

Theorem

Robb T.Koether

HomeworkReview

Cooks


Table

Example

The Functioncell

The Functionstart

The Functionaccept

The Functionmove

3SAT is

NP-Complete

Assignment


Proof ().

Let A be a problem in NP.

Then there is a nondeterministic Turing machine M that

decides A in polynomial time.

Let the run time of M be at most nk.

When M is run, it can access at most the first nk cells.
http://find/


8/46

Cooks

Theorem

Robb T.Koether

HomeworkReview

Cooks


Table

Example

The Functioncell

The Functionstart

The Functionaccept

The Functionmove

3SAT is

NP-Complete

Assignment


Proof ().

Represent the execution of M by an nk nk array.

Each row contains a configuration of M.

The first row is the initial configuration, with input w.

The sequence of rows represents a computation of M

that reaches a halting state.
http://find/


9/46

Cooks

Theorem

Robb T.Koether

HomeworkReview

Cooks


Table

Example

The Functioncell

The Functionstart

The Functionaccept

The Functionmove

3SAT is

NP-Complete

Assignment


Proof ().

We will place the symbol # in the first and last cells of

each row, to help identify the ends of the tape.In each row, each cell between the #s will contain

A tape symbol x, orA state q, orA blank .


10/46

Cooks

Theorem

Robb T.Koether

HomeworkReview

Cooks


Table

Example

The Functioncell

The Functionstart

The Functionaccept

The Functionmove

3SAT is

NP-Complete

Assignment


Proof ().

That is, each cell contains one symbol from the set

C = Q {#}.

We will create a Boolean expression from the table that

is satisfiable if and only if the table represents a valid

accepting computation of M.
http://find/


11/46

Cooks

Theorem

Robb T.Koether

HomeworkReview

Cooks

TheoremThe ComputationTable

Example

The Functioncell

The Functionstart

The Functionaccept

The Functionmove

3SAT is

NP-Complete

Assignment


Proof ().

Our Boolean expression will be of the form

= cell start accept move,

where

cell = true Every cell contains exactly one symbol.start = true Row 1 is a valid start configuration.accept = true One of the rows is a valid accept

configuration.move = true Each row yields the configuration in thenext row.
http://find/


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Cooks

TheoremRobb T.Koether

HomeworkReview

Cooks

TheoremThe ComputationTable

Example

The Functioncell

The Functionstart

The Functionaccept

The Functionmove

3SAT is

NP-Complete

Assignment


Proof ().

These expressions will all be true if and only if the table

represents a valid computation resulting in acceptance.


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Cooks


HomeworkReview

CooksTheorem

The Computation

Table

Example

The Functioncell

The Functionstart

The Functionaccept

The Functionmove

3SAT is

NP-Complete

Assignment


Proof ().

For each row number i from 1 to nk

, and for eachcolumn number j from 1 to nk, and for each symbol s in

C, let xi,j,s be a Boolean variable that is true if cell (i, j)contains the symbol s.
http://find/


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Cooks


HomeworkReview

CooksTheorem

The Computation

Table

Example

The Functioncell

The Functionstart

The Functionaccept

The Functionmove

3SAT is

NP-Complete

Assignment

Example

Consider the Turing machine below processing theinput 0110.

0 1, R

0 1, R

++, R

1 0, R 1 0, R

p

sr

q

(accept) (reject)

++, R


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Cooks


HomeworkReview

CooksTheorem

The Computation

Table

Example

The Functioncell

The Functionstart

The Functionaccept

The Functionmove

3SAT is

NP-Complete

Assignment

Example

The computation table.

# p 0 1 1 0 ## 1 q 1 1 0 #

# 1 0 q 1 0 #

# 1 0 0 q 0 #

# 1 0 0 1 p #

# 1 0 0 1 r #
http://find/


16/46

Cooks


HomeworkReview

CooksTheorem

The Computation

Table

Example

The Functioncell

The Functionstart

The Functionaccept

The Functionmove

3SAT is

NP-Complete

Assignment

Example

For example,

x1,1,# = truex1,1,0 = falsex1,2,p = truex1,3,0 = true
http://find/


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Cooks


HomeworkReview

CooksTheorem

The Computation

Table

Example

The Functioncell

The Functionstart

The Functionaccept

The Functionmove

3SAT is

NP-Complete

Assignment

The Expression cell

Proof ().

The expression cell consists of two parts.

The first part is true if and only if cell (i, j) contains atleastone symbol.

sC

xi,j,s
http://find/


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Cooks


HomeworkReview

CooksTheorem

The Computation

Table

Example

The Functioncell

The Functionstart

The Functionaccept

The Functionmove

3SAT is

NP-Complete

Assignment

The Expression cell

Proof ().

The second part is true if and only if cell (i, j) contains

at mostone symbol.sCtC

(xi,j,s xi,j,t)
http://find/


19/46

CooksTheorem

Robb T.Koether

HomeworkReview

CooksTheorem

The Computation

Table

Example

The Functioncell

The Functionstart

The Functionaccept

The Functionmove

3SAT is

NP-Complete

Assignment

The Expression cell

Proof ().

Therefore,

cell =

1ink

1jnk

sC

xi,j,s

sCtC

(xi,j,s xi,j,t)

.


20/46

CooksTheorem

Robb T.Koether

HomeworkReview

CooksTheorem

The Computation

Table

Example

The Functioncell

The Functionstart

The Functionaccept

The Functionmove

3SAT is

NP-CompleteAssignment

The Expression start

Proof ().

The expression start is easy because we know what

the start configuration looks like.

#q0w1w2 . . . wn . . . #

We simply verify that the cells contain those symbols.
http://find/


21/46

CooksTheorem

Robb T.Koether

HomeworkReview

CooksTheorem

The Computation

Table

Example

The Functioncell

The Functionstart

The Functionaccept

The Functionmove

3SAT is


The Expression start

Proof ().

Therefore,

start = x1,1,# x1,2,q0 x1,3,w1 x1,4,w2

x1,n+2,wn x1,n+3, x1,nk1, x1,nk,#.
http://find/


22/46

CooksTheorem

Robb T.Koether

HomeworkReview

CooksTheorem

The Computation

Table

Example

The Functioncell

The Functionstart

The Functionaccept

The Functionmove

3SAT is


The Expression accept

Proof ().

The expression accept is also very simple.All we need to do is to guarantee that qaccept occurs

somewhere in the table.
http://find/


23/46

CooksTheorem

Robb T.Koether

HomeworkReview

CooksTheorem

The Computation

Table

Example

The Functioncell

The Functionstart

The Functionaccept

The Functionmove

3SAT is


The Expression accept

Proof ().

Therefore,accept =

1ink

1jnk

xi,j,qaccept .
http://find/


24/46

CooksTheorem

Robb T.Koether

HomeworkReview

CooksTheorem

The Computation

Table

Example

The Functioncell

The Functionstart

The Functionaccept

The Functionmove

3SAT isNP-Complete

Assignment

The Expression move

Proof ().

The expression move will take a bit more effort.

We will consider 2 3 windows in the table.

Theorem

The fullnk nk table represents a valid computation if andonly ifcell andstart are true and every2 3 window isvalid.

E l


25/46

CooksTheorem

Robb T.Koether

HomeworkReview

CooksTheorem

The Computation

Table

Example

The Functioncell

The Functionstart

The Functionaccept

The Functionmove

3SAT isNP-Complete

Assignment

Example

Proof ().

The computation table.

# p 0 1 1 0 ## 1 q 1 1 0 #

# 1 0 q 1 0 #

# 1 0 0 q 0 #

# 1 0 0 1 p #

# 1 0 0 1 r #

E l
http://find/


26/46

CooksTheorem

Robb T.Koether

HomeworkReview

CooksTheorem

The Computation

Table

Example

The Functioncell

The Functionstart

The Functionaccept

The Functionmove

3SAT isNP-Complete

Assignment

Example

Proof ().

Window (1, 2)

# p 0 1 1 0 ## 1 q 1 1 0 #

# 1 0 q 1 0 #

# 1 0 0 q 0 #

# 1 0 0 1 p #

# 1 0 0 1 r #

E l
http://find/


27/46

CooksTheorem

Robb T.Koether

HomeworkReview

CooksTheorem

The Computation

Table

Example

The Functioncell

The Functionstart

The Functionaccept

The Functionmove

3SAT isNP-Complete

Assignment

Example

Proof ().

Window (1, 3)

# p 0 1 1 0 ## 1 q 1 1 0 #

# 1 0 q 1 0 #

# 1 0 0 q 0 #

# 1 0 0 1 p #

# 1 0 0 1 r #


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Example


29/46

CooksTheorem

Robb T.Koether

HomeworkReview

CooksTheorem

The Computation

Table

Example

The Functioncell

The Functionstart

The Functionaccept

The Functionmove

3SAT isNP-Complete

Assignment

Example

Proof ().

Window (2, 4)

# p 0 1 1 0 ## 1 q 1 1 0 #

# 1 0 q 1 0 #

# 1 0 0 q 0 #

# 1 0 0 1 p #

# 1 0 0 1 r #

Example
http://find/


30/46

CooksTheorem

Robb T.Koether

HomeworkReview

CooksTheorem

The Computation

Table

Example

The Functioncell

The Functionstart

The Functionaccept

The Functionmove

3SAT isNP-Complete

Assignment

Example

Proof ().

Window (3, 4)

# p 0 1 1 0 ## 1 q 1 1 0 #

# 1 0 q 1 0 #

# 1 0 0 q 0 #

# 1 0 0 1 p #

# 1 0 0 1 r #

The Expression
http://find/


31/46

CooksTheorem

Robb T.Koether

HomeworkReview

CooksTheorem

The Computation

Table

Example

The Functioncell

The Functionstart

The Functionaccept

The Functionmove

3SAT isNP-Complete

Assignment

The Expression move

Proof ().

For any symbol x and any state q, the legal moves are

Write x, move left, and change state to q.Write x, move right, and change state to q.

Recall that if we try to move left from the first cell, we

remain in the first cell.

The Expression
http://find/


32/46

CooksTheorem

Robb T.Koether

HomeworkReview

CooksTheorem

The Computation

Table

Example

The Functioncell

The Functionstart

The Functionaccept

The Functionmove

3SAT isNP-Complete

Assignment

The Expression move

Proof ().

Valid 2 3 windows for writing x, moving left, andchanging states from p to q (a, b, c are tape symbols).

p a bc x b

a p bq a x

a b pa q b

a b c

a b q

a b c

a b c

# p a

# q x

# a p

# q a

# a b

# a q

# a b

# a b

The Expression
http://find/


33/46

CooksTheorem

Robb T.Koether

HomeworkReview

CooksTheorem

The ComputationTable

Example

The Functioncell

The Functionstart

The Functionaccept

The Functionmove

3SAT isNP-Complete

Assignment

The Expression move

Proof ().

There is a similar set for a move to the right.

Let S be the set of all sextuples in C6 representing valid2 3 windows.

Then S includes (p, a, b, c, x, b), (a,p,b,q,a,x), etc., forall a,b,c,x and for all p, q Q.

The Expression


34/46

CooksTheorem

Robb T.Koether

HomeworkReview

CooksTheorem


Example

The Functioncell

The Functionstart

The Functionaccept

The Functionmove

3SAT isNP-Complete

Assignment

The Expression move

Proof ().

Then the expression

xi,j1,x1 xi,j,x2 xi,j+1,x3 xi+1,j1,x4

xi+1,j,x5 xi+1,j+1,x6

is true if and only if (x1, x2, x3, x4, x5, x6) S.

The Expression move
http://find/


35/46

CooksTheorem

Robb T.Koether

HomeworkReview

CooksTheorem


Example

The Functioncell

The Functionstart

The Functionaccept

The Functionmove

3SAT isNP-Complete

Assignment

The Expression move

Proof ().

So the expression

windowi,j =xS

(xi,j1,x1 xi,j,x2 xi,j+1,x3

xi+1,j1,x4 xi+1,j,x5 xi+1,j+1,x6)

is true if and only if window (i, j) is valid.

The Expression move
http://find/


36/46

CooksTheorem

Robb T.Koether

HomeworkReview

CooksTheorem


Example

The Functioncell

The Functionstart

The Functionaccept

The Functionmove

3SAT isNP-Complete

Assignment

The Expression move

Proof ().

Then the expression

move =

1i


37/46

CooksTheorem

Robb T.Koether

HomeworkReview

CooksTheorem


Example

The Functioncell

The Functionstart

The Functionaccept

The Functionmove

3SAT isNP-Complete

Assignment

Proof of Cook s Theorem

Proof ().

Then the Boolean expression

cell start accept move

is satisfiable if and only if w is accepted by M.

Therefore, SAT is NP-complete.

Building from Cooks Theorem


38/46

CooksTheorem

Robb T.Koether

HomeworkReview

CooksTheorem


Example

The Functioncell

The Functionstart

The Functionaccept

The Functionmove

3SAT isNP-Complete

Assignment

Building from Cook s Theorem

Now we may show that other problems areNP-complete by showing that SAT is polynomial-time

reducible to them.

3SAT is NP-Complete
http://find/


39/46

CooksTheorem

Robb T.Koether

HomeworkReview

CooksTheorem


Example

The Functioncell

The Functionstart

The Functionaccept

The Functionmove

3SAT isNP-Complete

Assignment

3SAT is NP Complete

Theorem

3SAT is NP-complete.

Proof.

We need only show that SAT is polynomial-time

reducible to 3SAT.

Any boolean expression may be put into 3-conjunctive

normal form by the following technique.

3SAT is NP-Complete
http://find/


40/46

CooksTheorem

Robb T.Koether

HomeworkReview

CooksTheorem


Example

The Functioncell

The Functionstart

The Functionaccept

The Functionmove

3SAT isNP-Complete

Assignment

3S s Co p ete

Proof.

Rewrite the each clause

x1 x2 x3 xn

as

(x1 x2 y1) (y1 x3 y2) (yn3 xn1 xn).

3SAT is NP-Complete


41/46

CooksTheorem

Robb T.Koether

HomeworkReview

CooksTheorem


Example

The Functioncell

The Functionstart

The Functionaccept

The Functionmove

3SAT isNP-Complete

Assignment

p

Proof.

The expressions cell, start, and accept are already in

conjunctive normal form.Apply the above technique to put them in 3-conjunctive

normal form.

The expression move is not in conjunctive normal form.

3SAT is NP-Complete


42/46

CooksTheorem

Robb T.Koether

HomeworkReview

CooksTheorem


Example

The Functioncell

The Functionstart

The Functionaccept

The Functionmove

3SAT isNP-Complete

Assignment

p

Proof.

However, it may be rewritten in conjunctive normal form

by repeatedly applying the distributive laws

x (y z) = (x y) (x z)

and

x (y z) = (x y) (x z).

3SAT is NP-Complete


43/46

CooksTheorem

Robb T.Koether

HomeworkReview

CooksTheorem


Example

The Functioncell

The Functionstart

The Functionaccept

The Functionmove

3SAT isNP-Complete

Assignment

p

Proof.

For example,

(x y) (z w) = ((x y) z) ((x y) w)= ((x z) (y z))

((x w) (y w))

= (x z) (y z) (x w)

(y w).

3SAT is NP-Complete
http://find/


44/46

CooksTheorem

Robb T.Koether

HomeworkReview

CooksTheorem


Example

The Functioncell

The Functionstart

The Functionaccept

The Functionmove

3SAT isNP-Complete

Assignment

p

Proof.

Therefore, the expression

cell start accept move

can be rewritten in 3-conjunctive normal form.

Therefore, 3SAT is NP-complete.

Other NP-Complete Problems
http://find/


45/46

CooksTheorem

Robb T.Koether

HomeworkReview

CooksTheorem


Example

The Functioncell

The Functionstart

The Functionaccept

The Functionmove

3SAT isNP-Complete

Assignment

There are many other NP-complete problems.

See NP-Complete

See List of NP-Complete Problems

Assignment
http://en.wikipedia.org/wiki/NP-completehttp://en.wikipedia.org/wiki/List_of_NP-complete_problemshttp://en.wikipedia.org/wiki/List_of_NP-complete_problemshttp://en.wikipedia.org/wiki/NP-completehttp://find/


46/46

CooksTheorem

Robb T.Koether

HomeworkReview

CooksTheorem


Example

The Functioncell

The Functionstart

The Functionaccept

The Functionmove

3SAT isNP-Complete

Assignment

Homework

Read Section 7.4, pages 276 - 283.

Problems 26, 27, 29, 30, 31, 36, 37, 38, page 297 -

298.
http://find/

lecture 41 - cook%27s theorem

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