1

The Post Correspondence Problem

continued

2

1. We will prove that the MPC problem is undecidable

2. We will prove that the PC problem is undecidable

3

1. We will prove that the MPC problem is undecidable

We will reduce the membership problemto the MPC problem

4

Membership problem

Input: recursive language string

Lw

Question: ?Lw

Undecidable

5

Membership problem

Input: unrestricted grammar string

Gw

Question: ?)(GLw

Undecidable

6

The reduction of the membership problemto the MPC problem:

For unrestricted grammar and string G w

we construct a pair such thatBA,

BA, has an MPC-solution if and only if )(GLw

7

A

FS

B

FF : special symbol

aa For every symbol

Grammar G

S : start variable

a

V V For every variable V

8

A

y

B

x

wEE

Grammar G

For every production

yx

E : special symbol

string w

9

Example:

Grammar : G

aacAC

CBb

BbbaABbS

|

String aaacw

10

A B

FS F:1w :1v

a a

b b

c c

A

B

CS

A

B

CS

:2w

:8w

:2v

:8v

11

A B

E aaacE:9w :9v

aABb S

Bbb S

C Bbaac

AC

:14w

:14v

12

)(GLaaac

aaacaACaABbS

aaacw

13

bBAaSF

A

B

1w

aABbS

10w

1v 10v

14

CAabBAaSF

A

B

1w

aACaABbS

10w

1v 10v

14w 2w 5w 12w

14v 2v 5v 12v

15

EcaaaCAabBAaSF

A

B

1w

aaacaACaABbS

10w

1v 10v

14w 2w 5w 12w

14v 2v 5v 12v 14v 2v 13v 9v

14w 2w 13w 9w

16

Theorem:

),( BA has an MPC-solution if and only if )(GLw

17

Algorithm for membership problem:

Input: unrestricted grammar string

Gw

Construct the pair BA,

If has an MPC-solution then BA, )(GLw

else )(GLw

18

G

w

constructBA,

BA, MPCalgorithm

solution

No-solution

)(GLw

)(GLw

Membership machine

19

2. We will prove that the PC problem is undecidable

We will reduce the MPC problemto the PC problem

20

: input to the MPC problemBA,

nwwwA ,,, 21

nvvvB ,,, 21

21

We construct a new sequences DC,

nwwwA ,,, 21

nvvvB ,,, 21

110 ,,,, nn wwwwC

110 ,,,, nn vvvvD

22

abcadwi

***** dacbawi

A

C

We insert a special symbol between any two symbols

23

abcadvi

dacbavi *****

B

D

24

1nw

C

Special Cases

10 *ww

*1nv

D

10 vv

25

Observation:

There is a PC-solution forif and only ifthere is a MPC-solution for

DC,

BA,

26

PC-solution 1010 nknk vwvwww

MPC-solution kk vvww 11

27

MPC-algorithm

Input: sequences BA,

Construct sequences DC,

Solve the PC problem for DC,

28

BA,construct

DC,DC, PC

algorithm

solution

No-solution

MPC algorithm