Post's Correspondence Problem Word Problem in semi-Thue Systems

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

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: This problem has a solution: 2, 1, 1, 3 w2w1w1w3 = x2x1x1x3 = 101111110

4. Post's Correspondence Problem Example 2: w1 = 10 w3 = 101 x1 = 101 x3 = 011 10101.. 101011...

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 No solution

6. Post's Correspondence Problem Reducing a MPCP to PCP

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

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

9. Post's Correspondence Problem Generating A & B

10. Post's Correspondence Problem Example:

11. Post's Correspondence Problem Example:

12. Post's Correspondence Problem Example:

13. Post's Correspondence Problem Example:

14. Post's Correspondence Problem Membership Problem A Generate A B MPCP Decider YES G 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. Given an arbitrary semi-Thue system S over Σ = {a, b} and two arbitrary words x, y, is y derivable from x in S? Word Problem for Semi-Thue Systems 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 2001 (ISBN: 0-201-44124-1) Mathematical Theory of Computation, Zohar Manna. Courier Dover Publications, 2003 (ISBN 0486432386, 9780486432380) Lecture Notes, The Post Correspondence Problem, Konstantin Busch. www.csc.lsu.edu/~busch/courses/theorycomp/fall2008/slides/Post_Correspondence.ppt

21. Question Q: How can you reduce an MPCP to PCP