Paul Bell
Undecidability of the Membership Problem
for a Diagonal Matrix in a Matrix Semigroup*
University of Liverpool
*Joint work with I.Potapov
Introduction
•
•
•
•
•
Definitions.
Motivation.
Description of the problem.
Outline of the proof.
Conclusion.
Some Definitions
• Reachability for a set of matrices asks if a
particular matrix can be produced by
multiplying elements of the set.
• Formally we call this set a generator, G, and
use this to create a semigroup, S, such that:
Known Results
•The reachability for the zero matrix is
undecidable in 3D (Mortality problem)[1].
• Long standing open problems:
• Reachability of identity matrix in any dimension >
2.
• Membership problem in dimension 2.
Dimension
Zero
Matrix
Identity
Matrix
Membership
problem
Scalar
Matrix
1
D
D
D
D
2
?
D
?
?
3
U
?
U
?
4
U
?
U
?
[1] - “Unsolvability in 3 x 3 Matrices” – M.S. Paterson (1970)
A Related Problem
• We consider a related problem to those on
the previous slide; the reachability of a
diagonal matrix.
• For a matrix semigroup:
• Theorem 1 : The reachability of the diagonal
matrix is undecidable in dimension 4.
• Theorem 2 : The reachability of the scalar matrix is
undecidable in dimension 4.
• We show undecidability by reduction of Post’s
correspondence problem.
The Scalar Matrix
• The scalar matrix can be thought of as the
product of the identity matrix and some k:
• The scalar matrix is often used to resize
an objects vertices whilst preserving the
object’s shape.
Post’s Correspondence Problem
• We are given a set of pairs of words.
• Try to find a sequence of these ‘tiles’ such that
the top and bottom words are equal.
• Some examples are much more difficult.
PCP Encoding
• We can think of the solution to the PCP as a
palindrome:
10 10 10 01 01 1 • 11 010 010 1 0 1
• Four dimensions are required in total.
• This technique cannot be used for the
reachability of the identity matrix.
PCP Encoding (2)
• We use the following matrices for coding:
 1 0
1  

2
1


1 2
0  

0
1


 1 0
11  


2
1


1  2
01  

0
1


• These form a free semigroup and can
be used to encode the PCP words.
10 1 0 • 01 0 1
 1 0  1 2  1 0  1 2   1  2  1 0  1  2  1 0 




  



  E
2
1
0
1
2
1
0
1
0
1

2
1
0
1

2
1




 




Index Coding
• We use an index coding which also forms a
palindrome:
1312  (1) 01000101001 (1) 00101000101
• We require two additional auxiliary matrices.
• We also used a prime factorization of
integers to limit the number of auxiliary
matrices.
Final PCP Encoding
• For a size n PCP we require 4n+2
matrices of the following form:
• W - Word part of matrix.
• I
- Index part.
•F
- Factorization part.
A Corollary
• By using this coding, a correct solution to the
PCP will be the matrix:
1

0
0

0
0 

1 0
0 
0 210 0 

0 0 210 
0
0
• We can now add a further auxiliary
matrix to reach the scalar matrix:
k

0
0


0

0
0
k
0
0
k
210
0
0
0 

0 
0 

k 

210 
• In fact we can reach any (non identity) diagonal
matrix where no element equals zero.
Conclusion
• We proved the reachability of any scaling
matrix (other than identity or zero) is
undecidable in any dimension >= 4.
• Future work could consider lower
dimensions.
• Prove a decidability result for the identity
matrix.