Paul Bell Undecidability of the Membership Problem for

Paul Bell Undecidability of the Membership Problem for a Diagonal Matrix in a Matrix Semigroup* 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 3 D (Mortality problem)[1]. • Long standing open problems: • Reachability of identity matrix in any dimension > 2. • Membership problem in dimension 2. Dimension Zero Matrix Identity Matrix Membersh ip problem Scalar Matrix 1 D D 2 ? D ? ? 3 U ? 4 U ? [1] - “Unsolvability in 3 x 3 Matrices” – M. S. Paterson (1970)

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: 10 10 10 01 01 1 • 11 010 1 0 1 • Four dimensions are required in total. • This technique cannot be used for the reachability of the identity matrix.

PCP Encoding (2) • We use the following matrices for coding: • These form a free semigroup and can be used to encode the PCP words. 10 1 0 • 01 0 1

Index 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 Encoding • For a size n PCP we require 4 n+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.