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 problem to the MPC problem

4 Membership problem Input: recursive language string Question: Undecidable

5 Membership problem Input: unrestricted grammar string Question: Undecidable

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

8 Grammar For every production : special symbol string

9 Example: Grammar : String

10

11

12

13

14

15

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

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

18 construct MPC algorithm solution No-solution Membership machine

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

20 : input to the MPC problem

21 We construct a new sequences

22 We insert a special symbol between any two symbols

23

24 Special Cases

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

26 PC-solution MPC-solution

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

28 construct PC algorithm solution No-solution MPC algorithm