1. Post's Correspondence Problem Word Problem in semi-Thue Systems
Hector Miguel Chavez
Western Michigan University
Jun 10, 2009

2. Post's Correspondence Problem
An instance of the Post's Correspondence Problem (PCP) consists of two lists of strings over some alphabet Σ;
A = w1, w2, . . ., wk
B = x1, x2, . . ., xk
The PCP has a solution if there is a sequence where:
wi, wi, . . ., wk = xi, xi, . . ., xk

3. Post's Correspondence Problem
Example 1:
List A
List B i wi xi 1
111 2
10111 10 3
This problem has a solution: 2, 1, 1, 3
w2w1w1w3 = x2x1x1x3 =

4. Post's Correspondence Problem
Example 2:
List A
List B i wi xi 1 10
101 2
011 11 3
w1 = 10
w3 = 101
x1 = 101
x3 = 011

5. Post's Correspondence Problem
The Modified “PCP”
The first pair in the solution must be the first pair in the lists.
w1, wi, . . ., wk = x1, xi, . . ., xk
List A
List B i wi xi 1
111 2
10111 10 3
No solution

6. Post's Correspondence Problem
Reducing a MPCP to PCP
List A
List B i wi xi
*1*
*1*1*1 1 1* 2
1*0*1*1*1*
*1*0 3
1*0* *0 4 $ *$
List A
List B i wi xi 1
111 2
10111 10 3

7. Post's Correspondence Problem
String Sequences
Solution?
MPCP
Decider A
YES B NO
Input
W ∈ L(G)
Membership
YES G w NO

8. Post's Correspondence Problem
Membership Problem A
Generate
A B
MPCP
Decider G
YES B w NO
MPCP can be reduced to PCP

9. Post's Correspondence Problem
Generating A & B A B G
FS → F
S: Start symbol
F: Special Symbol a
For every a V
For every V E
→ wE
String w
E: Special Symbol y x
For every production
X → Y →

10. Post's Correspondence ProblemA B
FS → F
Example:

11. Post's Correspondence ProblemA B
FS → F a b c
Example:

12. Post's Correspondence ProblemA B
FS → F a b c C S
Example:

13. Post's Correspondence ProblemA B
FS → F a b c C S E
→ aaacE
aABb
Bbb Bb
aac AC →
Example:

14. Post's Correspondence Problem
Membership Problem A
Generate
A B
MPCP
Decider G
YES B w NO
MPCP can be reduce to PCP

15. Word Problem for Semi-Thue Systems
A semi-Thue system S is a pair {Σ, P} where:
Σ is an alphabet
P is a set of rewrite rules or productions
In a rewriting x is called the antecedent and y the consequent.
x → y
A semi-Thue system is also known as a rewriting system.

16. Word Problem for Semi-Thue Systems
We say that a word v over Σ is immediately derivable from u if there is a rewrite rule x → y such that:
u = rxs and v = rys
If v is immediately derivable from u we write:
v ⇒ u

17. Word Problem for Semi-Thue Systems
Let P' be the set of all pairs (u, v) from Σ* x Σ* such that u ⇒ v. Then P ⊆ P' and if u ⇒ v , then
w u ⇒ w v and u w ⇒ v w for any word w
If a ⇒* b there is a sequence of derivations a = a1, a2, a3 = b.
If a ⇒* b and c ⇒* d imply ac ⇒* bd

18. Word Problem for Semi-Thue Systems
Example: Let S be a semi-Thue system where:
Σ = {a, b, c}
P = {ab → bc, bc → cb}.
The words ac3b, a2c2b and bc4 can be derived from a2bc2.
a2bc2 ⇒ a(bc)c2 ⇒ ac(bc)c ⇒ ac2(cb) = ac3b
a2bc2 ⇒ a2(cb)c ⇒ a2c(cb) = a2c2b
a2bc2 ⇒ a(bc)c2 ⇒ (bc)cc2 = bc4

19. Word Problem for Semi-Thue Systems
Given an arbitrary semi-Thue system S over Σ = {a, b} and two arbitrary words x, y, is y derivable from x in S?
The halting problem of the Turing Machines can be reduced to the Word Problem. Ex: If given an input X, the machine halts if Y can be produced.

20. References
Introduction to Automata Theory, Languages and Computation, John E. Hopcroft, Rajeev Motwani and Jeffrey D. Ullman, 2nd edition, Addison Wesley (ISBN: )
Mathematical Theory of Computation, Zohar Manna. Courier Dover Publications, 2003 (ISBN , )
Lecture Notes, The Post Correspondence Problem, Konstantin Busch. des/Post_Correspondence.ppt

21. Question Q: How can you reduce an MPCP to PCP List A List B i wi xi
*1*
*1*1*1 1 1* 2
1*0*1*1*1*
*1*0 3
1*0* *0 4 $ *$
List A
List B i wi xi 1
111 2
10111 10 3