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Probabilistic Finite Automaton Emptiness Is Undecidable for a Fixed Automaton

Author Günter Rote

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ACM Subject Classification
  • Theory of computation → Formal languages and automata theory
Keywords
  • Probabilistic finite automaton
  • Undecidability
  • Post Correspondence Problem

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Abstract

We construct a probabilistic finite automaton (PFA) with 7 states and an input alphabet of 5 symbols for which the PFA Emptiness Problem is undecidable. The only input for the decision problem is the starting distribution. For the proof, we use reductions from special instances of the Post Correspondence Problem.
We also consider some variations: The input alphabet of the PFA can be restricted to a binary alphabet at the expense of a larger number of states. If we allow a rational output value for each state instead of a yes-no acceptance decision, the number of states can even be reduced to 6.

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Günter Rote. Probabilistic Finite Automaton Emptiness Is Undecidable for a Fixed Automaton. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 86:1-86:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.MFCS.2025.86

Author Details

Günter Rote
  • Institut für Informatik, Freie Universität Berlin, Germany

References

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