On the mortality problem: From multiplicative matrix equations to linear recurrence sequences and beyond

Abstract

We consider the following variant of the Mortality Problem: given k × k matrices A1, A2, . . . , At, does there exist nonnegative integers m1, m2, . . . , mt such that the productis equal to the zero matrix? It is known that this problem is decidable when t ≤ 2 for matrices over algebraic numbers but becomes undecidable for sufficiently large t and k even for integral matrices.In this paper, we prove the first decidability results for t > 2. We show as one of our central results that for t = 3 this problem in any dimension is Turing equivalent to the well-known Skolem problem for linear recurrence sequences. Our proof relies on the Primary Decomposition Theorem for matrices that was not used to show decidability results in matrix semigroups before. As a corollary we obtain that the above problem is decidable for t = 3 and k ≤ 3 for matrices over algebraic numbers and for t = 3 and k = 4 for matrices over real algebraic numbers. Another consequence is that the set of triples (m1, m2, m3) for which the equation A m 1 1 A m 2 2 A m 3 3 equals the zero matrix is equal to a finite union of direct products of semilinear sets.For t = 4 we show that the solution set can be non-semilinear, and thus it seems unlikely that there is a direct connection to the Skolem problem. However we prove that the problem is still decidable for upper-triangular 2×2 rational matrices by employing powerful tools from transcendence theory such as Baker's theorem and S-unit equations. ACM Subject ClassificationTheory of computation → Computability; Mathematics of computing → Computations on matrices

Decidability of membership problems for flat rational subsets of $\mathrm{GL}(2,\mathbb{Q})$ and singular matrices

“…(also known as Knapsack problem). However, even in significantly restricted cases these problems become undecidable for high dimensional matrices over the integers, [3,20]; and a very few cases are known to be decidable, see [5,8]. The decidability of the membership problem remains open even for 2 × 2 matrices over integers [7,10,15,19,24] On the other hand, it is classical that membership in rational subsets of GL(2, Z) (the 2 × 2 integer matrices with determinant ±1) is decidable.…”