Given a finite set $\mathcal{M}$ of $n \times n$ matrices over the integers, can you express the zero matrix as a product of matrices in $\mathcal{M}$? This is known as the mortality problem. Michael Paterson showed in 1970 that it is undecidable, even for $n=3$. Later it was shown that mortality remains undecidable for $n=3$ and $|\mathcal{M}| = 7$, and for $n=21$ and $|\mathcal{M}| = 2$. Decidability of mortality for $n=2$ is open. If all matrices in $\mathcal{M}$ are nonnegative, the mortality problem becomes decidable, because then it matters only whether matrix entries are $0$ or not. Specifically, view the given matrices as transition matrices of a nondeterministic finite automaton where all states are initial and accepting, and check whether there is a word that is not accepted. This problem is decidable and PSPACE-complete. One can construct cases where the shortest word that is not accepted by the automaton has exponential length. In this post we focus on the nonnegative
Last year I wrote a post on how Oxford selects undergraduate students for computer science. Oxford undergraduate admissions are politically charged: the British public suspects that Oxbridge is biased towards privileged applicants. This post is unpolitical though. I'm advocating a better way of solving a typical problem in academia: you have many candidates (think: papers, grant proposals, or Oxford applicants) and few assessors (think: members of program committees, assessment panels, or admission tutors), and the goal of the assessors is to select the best candidates. Since there are many candidates, each assessor can assess only some candidates. From a candidate's point of view: a paper may get 3 reviews, a grant proposal perhaps 5, and a shortlisted Oxford applicant may get 3 interviews. Each such review comes with a score that reflects the assessor's estimate of the candidate's quality. The problem is that different assessors will score the same candidate differ