On the joint spectral radius for bounded matrix languages

  • Authors:

  • Paul C. Bell;Vesa Halava;Mika Hirvensalo

  • Affiliations:

  • Department of Computer Science, University of Liverpool, Liverpool, UK;Turku Centre for Computer Science, Department of Mathematics, University of Turku, Turku, Finland;Turku Centre for Computer Science, Department of Mathematics, University of Turku, Turku, Finland

  • Venue:
  • RP'10 Proceedings of the 4th international conference on Reachability problems
  • Year:
  • 2010

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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 {M1, M2,...,Mk} ⊆ Qt×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 = {M1j1 M2j2 ... Mkjk|j1, j2, ..., jk ≥ 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.