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.