On the Membership of Invertible Diagonal Matrices

Paul Bell and Igor Potapov (speaker)

The University of Liverpool, Computer Science Department

Developments in Language Theory - DLT 2005

Outline

• Membership problem– 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 problemin matrix semigroups

The motrality (membership for the zero matrix) is undecidable for 3x3 matrices. [M.Paterson]

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.

Membership problem

Dimension General Membership problem

Zero

Matrix

Identity

Matrix

Invertible Diagonal and Scalar Matices

1 D D D D

2 ? ? D ?

3 U U ? ?

4 U U ? ?

U

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

S1 S2 S3 S4 S1 S2

S5 S6 S7 S3 S4 S5 S6

S8 S7 S8

u1=

u2=

u3=

= v1

= v2

= v3

S1 S2 S3 S4

S1 S2

S4 S3 S2 S1 S1 S2S7 S6 S5 S3 S4 S5 S6

S3 S4 S5 S6

S5 S6 S7

S8 S7 S8

S8

S7 S8

Final PCP EncodingFinal 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:

12

011

10

210

12

011 1

10

210 1

10 1 0 10 1 0 • • 01 0 1 01 0 1

E

12

01

10

21

12

01

10

21

10

21

12

01

10

21

12

01

Index coding

• We use an index coding which also forms a palindrome:

(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.

1 3 1 2 2 1 3 1

Index coding

1 3 1 2 2 1 3 1

(1) 01000101001 (1) 00101000101

Final PCP EncodingFinal PCP EncodingFor a size n PCP we require 4n+2 matrices

• W - Word part of matrix.

• I - Index part.

• F - Factorization part.

A CorollaryA Corollary• By using this coding, a correct solution to the

PCP will be the matrix:

210000

021000

0010

0001

• We can now add a further auxiliary matrix to reach the scalar matrix:

210000

0210

00

000

000

k

kk

k

Membership of any scalar matrix (except identity) is undecidable for 4x4 rational matrix semigroup

Reduction to lower dimension

1

Theorem. The membership of a scalar matrix is undecidable for a semigroup generated by rational 3x3 matrices.

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

Matrix

Invertible Diagonal or Scalar Matrices

1 D D D D

2 ? ? D ?

3 U U ? U

4 U U ? U