Undecidability of the Membership Undecidability of the Membership Problem for a Diagonal Matrix in a Problem for a Diagonal Matrix in a

Matrix Semigroup*Matrix Semigroup*

Paul BellPaul Bell

University of LiverpoolUniversity of Liverpool

*Joint work with I.Potapov*Joint work with I.Potapov

IntroductionIntroduction

• Definitions.• Motivation.• Description of the problem.• Outline of the proof.• Conclusion.

Some DefinitionsSome 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 ResultsKnown 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.

[1] - “Unsolvability in 3 x 3 Matrices” – M.S. Paterson (1970)

Dimension Zero

Matrix

Identity

Matrix

Membership problem

Scalar Matrix

1 D D D D

2 ? D ? ?

3 U ? U ?

4 U ? U ?

A Related ProblemA 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 MatrixThe 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 ProblemPost’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 EncodingPCP Encoding• We can think of the solution to the PCP as a

palindrome:

10 10 10 01 01 1 10 10 10 01 01 1 • • 11 010 010 1 0 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)PCP Encoding (2)• We use the following matrices for coding:

12

011

10

210

12

011 1

10

210 1

• These form a free semigroup and can be used to encode the PCP words.

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 CodingIndex 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 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.

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

• In fact we can reach any (non identity) diagonal matrix where no element equals zero.

ConclusionConclusion

• 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.