1. 1 The Post Correspondence Problem continued

2. 2 1. We will prove that the MPC problem is undecidable 2. We will prove that the PC problem is undecidable

3. 3 1. We will prove that the MPC problem is undecidable We will reduce the membership problem to the MPC problem

4. 4 Membership problem Input: recursive language string Question: Undecidable

5. 5 Membership problem Input: unrestricted grammar string Question: Undecidable

6. 6 The reduction of the membership problem to the MPC problem: For unrestricted grammar and string we construct a pair such that has an MPC-solution if and only if

7. 7 : special symbol For every symbol Grammar : start variable For every variable

8. 8 Grammar For every production : special symbol string

9. 9 Example: Grammar : String

10. 10

11. 11

12. 12

13. 13

14. 14

15. 15

16. 16 Theorem: has an MPC-solution if and only if

17. 17 Algorithm for membership problem: Input: unrestricted grammar string Construct the pair If has an MPC-solution then else

18. 18 construct MPC algorithm solution No-solution Membership machine

19. 19 2. We will prove that the PC problem is undecidable We will reduce the MPC problem to the PC problem

20. 20 : input to the MPC problem

21. 21 We construct a new sequences

22. 22 We insert a special symbol between any two symbols

23. 23

24. 24 Special Cases

25. 25 Observation: There is a PC-solution for if and only if there is a MPC-solution for

26. 26 PC-solution MPC-solution

27. 27 MPC-algorithm Input: sequences Construct sequences Solve the PC problem for

28. 28 construct PC algorithm solution No-solution MPC algorithm