Decision Problems for Probabilistic Finite Automata on Bounded Languages

: Bell PC, Halava V, Hirvensalo M

Publisher: IOS PRESS

: 2013

: Fundamenta Informaticae

: FUNDAMENTA INFORMATICAE

: FUND INFORM

: 1

: 123

: 1

: 14

: 0169-2968

DOI: https://doi.org/10.3233/FI-2013-797

For a finite set of matrices {M-1, M-2, ... , M-k} subset of Q(txt), we then consider the decidability of computing the maximal spectral radius of any matrix in the set X = {M-1(j1) M-2(j2) ... M-k(jk)vertical bar j(1), j(2), ... , j(k) >= 0}, which we call a bounded matrix language. Using an encoding of a probabilistic finite automaton shown in the paper, we prove the surprising result that determining if the maximal spectral radius of a bounded matrix language is less than or equal to one is undecidable, but determining whether it is strictly less than one is in fact decidable (which is similar to a result recently shown for quantum automata).