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 Bell Paul Bell University of Liverpool University of Liverpool *Joint work with I.Potapov *Joint work with I.Potapov
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Undecidability of the Membership Problem for a Diagonal Matrix in a Matrix Semigroup*
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. - PowerPoint PPT Presentation
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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