Mortality for 2 ×2 matrices is NP-hard

Abstract

We study the computational complexity of determining whether the zero matrix belongs to a finitely generated semigroup of two dimensional integer matrices (the mortality problem). We show that this problem is NP-hard to decide in the two-dimensional case by using a new encoding and properties of the projective special linear group. The decidability of the mortality problem in two dimensions remains a long standing open problem although in dimension three is known to be undecidable as was shown by Paterson in 1970. We also show a lower bound on the minimum length solution to the Mortality Problem, which is exponential in the number of matrices of the generator set and the maximal element of the matrices. © 2012 Springer-Verlag.