1. On the Membership of Invertible Diagonal Matrices
Developments in Language Theory - DLT 2005
On the Membership of Invertible Diagonal Matrices
Paul Bell and Igor Potapov (speaker)
The University of Liverpool, Computer Science Department

2. Outline Membership problem Known vs. new results
Case of diagonal and scalar matrices
Known vs. new results
Few tricks towards the goal
Technical details
Undecidability - PCP encoding
Artificial ordering
What to do next?

3. The membership problem in matrix semigroups
Given a finite set of matrices G={M1,M2,…,Mk } and a matrix M. Determine whether there exists a sequence of matrices Mi1,Mi2,…,Mil from G such that Mi1  Mi2  …  Mil = M
In other words, Let S be a given finitely generated semigroup of nn matrices from Znn (Qnn). Determine whether a matrix M belongs to S.
The motrality (membership for the zero matrix) is undecidable for 3x3 matrices. [M.Paterson]

4. Membership problem U Dimension General Membership problem Zero Matrix
Identity
Invertible Diagonal and Scalar Matices 1 D 2 ? 3 U 4 U

5. 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.

6. Main Results
Membership of invertible diagonal or scalar matrix is undecidable for:
3x3 rational matrix semigroup
4x4 integral matrix semigroup
Membership of any scalar matrix (except identity) is undecidable for
4x4 rational matrix semigroup

7. Some tricks towards the goal
Undecidability result
Post Corresponding Problem
Separate words and indexes coding
Mappings between words and matrices

8. Mappings between words and matrices
The group generated by matrices (0), (1) is free

9. Post Corresponding Problem
Given a finite alphabet X and a finite sequence of pairs of words in X*: (u1,v1),…, (uk,vk)
Is there a finite sequence of indexes{ij} :
PCP has a solution iff there a finite sequence of indexes{ij} iff

10. PCP example u1= S1 S2 S3 S4 S1 S2 = v1 S8 S7 S8 S7 S6 S5 S3 S4 S5 S6

11. 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.

12. Word coding
We use the following matrices for coding: •

13. Index coding We use an index coding which also forms a palindrome:
(1) (1)
We require two additional auxiliary matrices.
We also used a prime factorization of integers to limit the number of auxiliary matrices.

14. Index coding
(1) (1)

15. Final PCP Encoding For a size n PCP we require 4n+2 matrices
W - Word part of matrix.
I - Index part.
F - Factorization part.

16. A Corollary
By using this coding, a correct solution to the PCP will be the matrix:
We can now add a further auxiliary matrix to reach the scalar matrix:
Membership of any scalar matrix (except identity) is undecidable for 4x4 rational matrix semigroup

17. Reduction to lower dimension
Theorem. The membership of a scalar matrix is undecidable for a semigroup generated by rational 3x3 matrices. 1

18. Conclusion operations with rational vs integers
low dimensional systems
affine transformations
1D ax+b, cx+d
2D reachability is undecidable
Dimension
General Membership problem
Zero
Matrix
Identity
Invertible Diagonal or Scalar Matrices 1 D 2 ? 3 U 4