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Tighter Undecidability Bounds for Matrix Mortality, Zero-in-the-Corner Problems, and More

We study the decidability of three well-known problems related to integer matrix multiplication: Mortality (M), Zero in the Left-Upper Corner (Z), and Zero in the Right-Upper Corner (R). Let d and k be positive integers. Define M(k, d x d) as the following special case of the Mortality problem: given a set X of d -by-d integer matrices such that the cardinality of X is not greater than k, decide whether the d-by-d zero matrix belongs to X^+, where X^+ denotes the closure of X under the usual matrix multiplication. In the same way, define the Z(k, d x d) problem as: given an instance X of M(k, d x d) (the instances of Z(k, d x d) are the same as those of M(k, d x d)), decide whether at least one matrix in X^+ has a zero in the left-upper corner. Define R(k, d x d) as the variant of Z(k, d x d) where "left-upper corner" is replaced with "right-upper corner". In the paper, we prove that M(6, 3 x 3), M(4, 5 x 5), M(3, 9 x 9), M(2, 15 x 15), Z(5, 3 x 3), Z(3, 5 x 5), Z(2, 9 x 9), R(6, 3 x 3), R(5, 4 x 4), and R(3, 6 x 6) are undecidable. The previous best comparable results were the undecidabilities of M(7, 3 x 3), M(3, 13 x 13), M(2, 21 x 21), Z(7, 3 x 3), Z(2, 13 x 13), R(7, 3 x 3), and R(2, 10 x 10).