The Post Correspondence Problem continued 1 1 We

The Post Correspondence Problem continued 1

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

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

Membership problem Input: recursive language string Question: Undecidable 4

Membership problem Input: unrestricted grammar string Question: Undecidable 5

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

Grammar : start variable : special symbol For every variable 7

Grammar string : special symbol For every production 8

Example: Grammar : String 9

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

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

Membership machine solution construct MPC algorithm No-solution 18

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

: input to the MPC problem 20

We construct a new sequences 21

We insert a special symbol between any two symbols 22

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Special Cases 24

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

PC-solution MPC-solution 26

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

MPC algorithm construct PC algorithm solution No-solution 28