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

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

• Paul Bell

University of Liverpool Joint work with I.Potapov
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Introduction

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

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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

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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
  gt 2.
• Membership problem in dimension 2.

1 - Unsolvability in 3 x 3 Matrices M.S.
Paterson (1970)
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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 Posts
  correspondence problem.

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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 objects
  shape.

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Posts 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.

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PCP Encoding

• We can think of the solution to the PCP as a
  palindrome

10 10 10 01 01 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.

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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
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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.

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Final PCP Encoding

• For a size n PCP we require 4n2 matrices of the
  following form

• W - Word part of matrix.
• I - Index part.
• F - Factorization part.

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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.

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Conclusion

• We proved the reachability of any scaling matrix
  (other than identity or zero) is undecidable in
  any dimension gt 4.
• Future work could consider lower dimensions.
• Prove a decidability result for the identity
  matrix.