Creating Difficult Instances of the Post Correspondence Problem

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Creating Difficult Instances of the Post
Correspondence Problem
Authored by Richard J. Lorentz

• Presenter Ling Zhao
• Department of Computing Science
• University of Alberta
• March 20, 2001

Revised on May 3, 2001
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Motivation

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

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Introduction

• Definition

Given an alphabet ? and a finite set of pairs of
strings ( gi , hi ) over the alphabet ?, does
there exist a sequence i1 i2 ... in of selections
such that the strings gi1gi2 ... gin and hi1hi2
... hin 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
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Example

• 1 2 3
• 100 0 1
• 1 100 00

size 3 width 3
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Example
100 1
1
1001 100
3
1001100 1001
1

• 1 2 3
• 100 0 1
• 1 100 00

1001100100 10011
1
10011001001 1001100
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This instance has the shortest solution with
length 7
100110010010 1001100100
2
1001100100100 1001100100100
2
Solved!
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Hard Instances
size 3 width 3 length 75 2 shortest
solutions

• 100 0 1
• 1 100 0

size 4 width 3 length 204 1 shortest
solution
000 0 11 10 0 111 0 100
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Main Work in the Paper

• Solve PCPs
• 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
• 1. How to find the instances with small size
  and
• small width but long shortest length of
  solution?

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Identify instances with no solutions

• Prefix / postfix filters
• Length balance filter
• Element balance filter

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Prefix / postfix filters

• 000 0 1
• 1 100 0

No starting point!
001 0 1 0 101 0
No ending point!
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Length / element balance filters

• 000 100 1
• 0 10 0

The configuration will be always in the top!
The configuration will be always in the top and
contain several 1s while any pairs can not
decrease the number of it
101 1 01 10 01 0
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Algorithms Involved

• Hash function and cache scheme
• Iterative deepening
• Tail recursion removal

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Recognize hard instances

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

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My Progress on this problem

• 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
• Bidirectional search
• Other branch pruning methods

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Top mask and Bottom mask

• 100 0 10
• 0 100 100

At the beginning, the configuration is in the
bottom
If you want to make the configuration shrink its
size to 0 or in the top, you can only use pair 1
Only two possible configurations satisfying the
conditions
No concatenations of the down strings can provide
such postfixes!
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1
It has a bottom mask!
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Useless Postfix
Depth Selection Configuration Property
1 1 10 Top
2 1 0110 Top
3 3 101 Top
4 1 01110 Top
5 3 1101 Top
6 1 101110 Top
7 1 01110110 Top
8 3 1101101 Top
110 0 1 1 111 01
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
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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
Useless Postfix
Depth Selection Configuration Property
1 1 10 Top
2 1 0110 Top
3 3 101 Top
4 1 01110 Top
5 3 1101 Top
6 1 101110 Top
7 1 01110110 Top
8 3 1101101 Top
110 0 1 1 111 01
S0110 -gt S1101 S2110
S3110 -gt S4101 S5110
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Conclusions

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