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
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?
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 nn matrices from Znn (Qnn). Determine whether a matrix M belongs to S. The motrality (membership for the zero matrix) is undecidable for 3x3 matrices. [M.Paterson]
Membership problem U Dimension General Membership problem Zero Matrix Identity Invertible Diagonal and Scalar Matices 1 D 2 ? 3 U 4 U
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.
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
Some tricks towards the goal Undecidability result Post Corresponding Problem Separate words and indexes coding Mappings between words and matrices
Mappings between words and matrices The group generated by matrices (0), (1) is free
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
PCP example u1= S1 S2 S3 S4 S1 S2 = v1 S8 S7 S8 S7 S6 S5 S3 S4 S5 S6
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.
Word coding We use the following matrices for coding: 10 1 0 • 01 0 1
Index coding We use an index coding which also forms a palindrome: (1) 01000101001 (1) 00101000101 1 3 1 2 2 1 3 1 We require two additional auxiliary matrices. We also used a prime factorization of integers to limit the number of auxiliary matrices.
Index coding (1) 01000101001 (1) 00101000101 1 3 1 2 2 1 3 1
Final PCP Encoding For a size n PCP we require 4n+2 matrices 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: 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
Reduction to lower dimension Theorem. The membership of a scalar matrix is undecidable for a semigroup generated by rational 3x3 matrices. 1
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