Applied Computer Science IIChapter 5: Reducability

Prof. Dr. Luc De Raedt

Institut für InformatikAlbert-Ludwigs Universität Freiburg

Germany

Overview

• Examine several other undecidable problems

• Reducibility– Basic method to relate two problems to one

another in the light of “(un)solvability”– Reducibility is used for various types of

“unsolvability”, cf. complexity

• Mapping reducibility• The Post Correspondence Problem PCP

Reducability to compute

: map onto problem for

1. to map onto ( )

2. to compute ( )

3. return ( )

determine whether an NFA accepts

Assume you do not know an alg

w A

B

w f w

f w B

f w B

A w

Goal

Method

Example

Method

orithm that decides whether

an NFA accepts a string, but that you have an algorithm called that

decides whether a DFA accepts a string

You can obtain one using reduction

1. Transfor

P

m < , where is NFA into < ( ),

2. Run on < ( ),

3. Return answer of

A w A dfa A w

P dfa A w

P

Reductions

and proofs by contradiction

To show that is not solvable

Suppose you know that is not solvable.

Reduce to

Conclude that is not solvable

(because solvability would imply that can be s

A

B

B A

A

A B

Reductions

olved)

Undecidable problems from language theory

{ , | is a TM and halts on input }

is undecidable

173

TM

TM

HALT M w M M w

HALT

p

Theorem

Proof

Computable functions

• Cf. Loop-programs

• Examples : – f(<m,n>)=m+n– f(<M>) = M’ where M’ accepts the same

language as TM M except that it does not move its head against the left “wall”; if M is not a TM then return epsilon

* *A function : is if

there is a deterministic Turing Machine that on every

input halts with just

com

( ) on its tape.

putablef

T

w f w

If and is decidable, then is decidable

Let be a decider for

and the reduction from to

mA B B A

M B

f A B

Theorem

Proof

If and is undecidable, then is undecidablemA B A BCorollary

TM m TMA HALTTheorem

{ | is a TM and L( ) }

is undecidable ( )

174

TM

TM TM m TM

E M M M

E A E

p

Theorem

Proof

{ | is a TM and L( ) is a regular language}

is undecidable ( )

175

TM

TM TM m TM

REGULAR M M M

REGULAR A REGULAR

p

Theorem

Proof n

*

Recognize {0 1 | 0} if does not accept

and otherwise

n n M w

{ 1, 2 | 1 and 2 are TMs and ( 1) ( 2) }

is undecidable ( )

TM

TM TM m TM

EQ M M M M L M L M

EQ E EQ

Theorem

Reductions via computation histories

• Deterministic versus non-deterministic machines

• From now on we focus on deterministic machines

1 1

1

Let be a TM and a string

An for is

a sequence of configurations ,..., where is the starting configuration

of on ,

a

each corresponds to a legal transition,

a

ccepting computation history

n

m

i i

M w

M

C C C

M w C C

rejecting computation histor

d is an accepting configuration

A is similar except that it ends in

a rejecting configurat

y

ion.

mC

Linear bound automaton

Only limited memory available

A l is a restricted type of TM

where the tape head is not permitted to move off the portion

of the tape

inear bounde

containing t

d automato

he in

n

put.

LBA

• LBAs are quite powerful, e.g.

Deciders for , , , are all LBAsDFA CFG DFA CFGA A E E

{ , | is an LBA that accepts } is decidable

Let be an LBA with states, symbols in its tape alphabet,

There are exactly . . distinct configu

contra

ration

st

s of

this w

f

ithLB

n

T

A

MA

A M w M w

M q g

q n g M

Theorem

Lemma

or a tape of length n

• So, LBAs are fundamentally different than TMs !

{ | is an LBA and L( ) }

is undecidable ( )

179 180

LBA

LBA TM LBA

E M M M

E A E

p

Theorem

Proof

{ | is an accepting computation history of on }B v v M w

*

is undecidable

{ | is a CFG and ( ) }

CFG

CFG

ALL

ALL G G L G

Theorem

{ | is an instance of the Post correspondence problem

with a match that starts with the first domino}

MPCP P P

TM mA MPCPTheorem

0 1

#1. Put into P' as the first domino

# ... #

2. For every , and every , , where

if ( , ) ( , , ) put into P'

3. For every , , and every , , where

n

reject

reject

q w w

a b q r Q q q

qaq a r b R

br

a b c q r Q q q

if ( , ) ( , , ) put into P'

4. For every

put into P'

# #5. put and into P'

# #

6. For every

put and into P'

7

accept accept

accept accept

cqaq a r b L

rcb

a

a

a

a

aq q a

q q

. Finally, add

##

#acceptq

mMPCP PCPTheorem

Transitivity implies that is undecidablePCP

If and is Turing recognizable, then is Turing-recognizable

Let be a TM for

and the reduction from to

mA B B A

M B

f A B

Theorem

Proof

If and is not Turing recognizable,

then is not Turing recognizable

Usually, will be

Note also that if and only if

m

TM

m m

A B A

B

A A

A B A B

Corollary

Theorem 5.24 is neither Turing recognizable

nor co-Turing recognizable

1.

2.

TM

TM m TM

TM m TM

EQ

A EQ

A EQ

Theorem

Proof

Conclusions

• Examine several other undecidable problems

• Reducibility– Basic method to relate two problems to one

another in the light of “(un)solvability”– Reducibility is used for various types of

“unsolvability”, cf. complexity

• Mapping reducibility• The Post Correspondence Problem PCP