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 d (k) 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 d (k) problem as: given an instance X of M d (k) (the instances of Z d (k) are the same as those of M d (k)), decide whether at least one matrix in X + has a zero in the left-upper corner. Define R d (k) as the variant of Z d (k) where “left-upper corner” is replaced with “right-upper corner”. In the paper, we prove that M 3 (6), M 5 (4), M 9 (3), M 15 (2), Z 3 (5), Z 5 (3), Z 9 (2), R 3 (6), R 4 (5), and R 6 (3) are undecidable. The previous best comparable results were the undecidabilities of M 3 (7), M 13 (3), M 21 (2), Z 3 (7), Z 13 (2), R 3 (7), and R 10 (2).