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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 > 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?

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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 Post’s 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 object’s shape.

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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.

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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 4n+2 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 >= 4. Future work could consider lower dimensions. Prove a decidability result for the identity matrix.

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