Undecidability of the Membership Problem for a Diagonal Matrix in a Matrix Semigroup* Paul Bell 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. [1] - “Unsolvability in 3 x 3 Matrices” – M.S. Paterson (1970) DimensionZero Matrix Identity Matrix Membership problem Scalar Matrix 1DDDD 2?D?? 3U?U? 4U?U?

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

Index Coding We use an index coding which also forms a palindrome: 1312  (1) (1) 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: We can now add a further auxiliary matrix to reach the scalar matrix: 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.