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
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 NPcomplete
 Even the instance with small size and
small width may have very long solution

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}
Example
1
2
100
1
3
0
1
100 00
size = 3
width = 3
Example
1
100
1
2
1
100
1
3
1001
100
1
1001100
1001
1
1001100100
10011
3
10011001001
1001100
2
100110010010
1001100100
2
1001100100100
1001100100100
3
0
1
100 00
This instance has the shortest
solution with length 7
Solved!
Hard Instances
100
1
0
100
1
0
000 0 11 10
0 111 0 100
size = 3
width = 3
length = 75
2 shortest solutions
size = 4
width = 3
length = 204
1 shortest solution
Main Work in the Paper
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
1. How to find the instances with small size and
small width but long shortest length of solution?
Identify instances with no
solutions

Prefix / postfix filters

Length balance filter

Element balance filter
Prefix / postfix filters
000
1
0
100
1
0
No starting point!
001
0
0
101
1
0
No ending point!
Length / element balance filters
000 100
0
10
101
10
1
01
1
0
01
0
The configuration will be
always in the top!
The configuration will be always
in the top and contain several 1’s
while any pairs can not decrease
the number of it
Algorithms Involved
Hash function and cache scheme
 Iterative deepening
 Tail recursion removal

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

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

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!
10 1
It has a bottom mask!
Useless Postfix
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
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
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
110 0
1
1 111 01
S0110 -> S1101 | S2110
S3110 -> S4101 | S5110
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
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.
