1. Creating Difficult Instances of the Post Correspondence Problem Presenter: Ling Zhao Department of Computing Science University of Alberta March 20, 2001 Authored by Richard J. Lorentz Revised on May 3, 2001

2. Motivation Post Correspondent Problem (PCP) poses unique difficulties because of the unbounded size of search space Post Correspondent Problem (PCP) poses unique difficulties because of the unbounded size of search space Theoretically, PCP is the classic undecidable problem Theoretically, PCP is the classic undecidable problem The bounded version of PCP is NP- complete The bounded version of PCP is NP- complete Even the instance with small size and small width may have very long solution Even the instance with small size and small width may have very long solution

3. Introduction Definition Definition Given an alphabet  and a finite set of pairs of strings ( g i, h i ) over the alphabet , does there exist a sequence i 1 i 2... i n of selections such that the strings g i 1 g i 2... g i n and h i 1 h i 2... h i n formed by concatenating are identical? width: the size of longest string in the pairs size: the number of pairs length of the solution: the size of the sequence of selection For simplicity, we restrict the alphabet  to be {0, 1}

4. Example 1 2 3 100 0 1 1 100 00 size = 3 width = 3

5. Example 1 2 3 100 0 1 1 100 00 100 1 1 1001 100 1001100 1001 3 1 1001100100 10011 1 10011001001 1001100 3 100110010010 1001100100 2 1001100100100 2 Solved! This instance has the shortest solution with length 7

6. Hard Instances 100 0 1 1 100 0 size = 3 width = 3 length = 75 2 shortest solutions 000 0 11 10 0 111 0 100 size = 4 width = 3 length = 204 1 shortest solution

7. Main Work in the Paper Solve PCP’s Solve PCP’s 1. How to identify that the instance has no solution? 2. How to find the solution quickly? 3. How to search the solution space efficiently and quickly? Generate the hard instances of PCP Generate the hard instances of PCP 1. How to find the instances with small size and small width but long shortest length of solution?

8. Identify instances with no solutions Prefix / postfix filters Prefix / postfix filters Length balance filter Length balance filter Element balance filter Element balance filter

9. Prefix / postfix filters 000 0 1 1 100 0 No starting point! 001 0 1 0 101 0 No ending point!

10. Length / element balance filters 000 100 1 0 10 0 The configuration will be always in the top! 101 1 01 10 01 0 The configuration will be always in the top and contain several 1’s while any pairs can not decrease the number of it

11. Algorithms Involved Hash function and cache scheme Hash function and cache scheme Iterative deepening Iterative deepening Tail recursion removal Tail recursion removal

12. Recognize hard instances Generate PCP instances Generate PCP instances Iterative deepening Iterative deepening Random algorithm Random algorithm Restart when the number of searched nodes exceeds a threshold Restart when the number of searched nodes exceeds a threshold Use some heuristic information Use some heuristic information e.g. discourage the repetitions of pairs

13. My Progress on this problem Identify the situation that the configuration can not be on the top or in the below Identify the situation that the configuration can not be on the top or in the below Identify that one postfix in the configuration can not lead to the solution Identify that one postfix in the configuration can not lead to the solution Bidirectional search Bidirectional search Other branch pruning methods Other branch pruning methods

14. Top mask and Bottom mask 100 0 10 0 100 100 If you want to make the configuration shrink its size to 0 or in the top, you can only use pair 1 At the beginning, the configuration is in the bottom Only two possible configurations satisfying the conditions: 101 No concatenations of the down strings can provide such postfixes! It has a bottom mask!

15. Useless Postfix DepthSelectionConfigurationProperty 1110Top 21 0110 Top 33101 41 01110 Top 53 1101 Top 61 101110 Top 71 01110110 Top 83 1101101 Top After depth 1, if we can prove that all configurations will contain the postfixes either 110 or 101, we can infer that this instance has no solution 110 0 1 1 111 01

16. Useless Postfix 110 0 1 1 111 01 DepthSelectionConfigurationProperty 1110Top 21 0110 Top 33101 41 01110 Top 53 1101 Top 61 101110 Top 71 01110110 Top 83 1101101 Top S 0 110 -> S 1 101 | S 2 110 S 3 110 -> S 4 101 | S 5 110 Notes: I made a mistake for the proof of this postfix. Though it probably has such postfix, I can not prove it. This instance can be proved of no solutions by using the exclusion method. Added by Ling Zhao on May 3

17. Conclusions Use the standard techniques and simple heuristics to generate interesting PCP instance Use the standard techniques and simple heuristics to generate interesting PCP instance Raise many instances with the length larger than 100 Raise many instances with the length larger than 100 Conjectures about the minimum length of solutions for some instances Conjectures about the minimum length of solutions for some instances Conjectures about the most difficult instances for the specific width and size. Conjectures about the most difficult instances for the specific width and size.