Decision Problems for Probabilistic Finite Automata on Bounded Languages

  • Authors:

  • Paul C. Bell;Vesa Halava;Mika Hirvensalo

  • Affiliations:

  • Department of Computer Science, Loughborough University, Loughborough, LE11 3TU, UK. p.bell@lboro.ac.uk;TUCS-Turku Centre for Computer Science, Department of Mathematics, University of Turku, FIN-20014, Turku, Finland, vehalava@utu.fi, mikhirve@utu.fi;TUCS-Turku Centre for Computer Science, Department of Mathematics, University of Turku, FIN-20014, Turku, Finland, vehalava@utu.fi, mikhirve@utu.fi

  • Venue:
  • Fundamenta Informaticae - MFCS & CSL 2010 Satellite Workshops: Selected Papers
  • Year:
  • 2013

Quantified Score

Hi-index 0.00

Visualization

Abstract

We show that several problems concerning probabilistic finite automata of a fixed dimension and a fixed number of letters for bounded cut-point and strict cut-point languages are algorithmically undecidable by a reduction of Hilbert's tenth problem. We then consider the set of so called “F-Problems” emptiness, infiniteness, containment, disjointness, universe and equivalence and show that they are also undecidable for bounded non-strict cut-point languages on probabilistic finite automata. 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 maximal spectral radius of any matrix in the set $X = \{M^{j_1}_1 M^{j_2}_2 \cdot M^{j_k}_k \vert j_1, j_2,\ldots, j_k \geq 0\}$, 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.