On the Joint Spectral Radius for Bounded Matrix Languages

Abstract

We show several problems concerning probabilistic finite automata with fixed numbers of letters and of fixed dimensions for bounded cut-point and strict cut-point languages are algorithmically undecidable by a reduction of Hilbert’s tenth problem using formal power series.

For a finite set of matrices $\{M_1, M_2, \ldots, M_k\} \subseteq \mathbb{Q}^{t \times t}$, we then consider the decidability of computing the joint spectral radius (which characterises the maximal asymptotic growth rate of a set of matrices) of the set $X = \{M_1^{j_1} M_2^{j_2} \cdots M_k^{j_k}| j_1, j_2, \ldots, j_k \geq 0\}$, which we term a bounded matrix language. Using an encoding of a probabilistic finite automaton shown in the paper, we prove the surprising result that determining whether the joint spectral radius of a bounded matrix language is less than or equal to one is undecidable, but determining if it is strictly less than one is in fact decidable (which is similar to a result recently shown for quantum automata).

This has an interpretation in terms of a control problem for a switched linear system with a fixed and finite number of switching operations; if we fix the maximum number of switching operations in advance, then determining convergence to the origin for all initial points is decidable whereas determining boundedness of all initial points is undecidable.