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