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) 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) problem as: given an instance X of M(k) (the instances of Z(k) are the same as those of M(k)), decide whether at least one matrix in X+ has a zero in the left-upper corner. Define R(k) as the variant of Z(k) where “left-upper corner” is replaced with “right-upper corner”. In the paper, we prove that M3(6), M5(4), M9(3), M 15(2), Z3(5), Z5(3), Z9(2), R3(6), R4(5), and R6(3) are undecidable. The previous best comparable results were the undecidabilities of M3(7), M13(3), M21(2), Z3(7), Z 13(2), R3(7), and R10(2).