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Probabilistic Finite Automaton Emptiness is undecidable

Günter Rote

TL;DR

This work shows that the emptiness problem for probabilistic finite automata is undecidable and strengthens this result through multiple independent proofs, including self-contained renditions of the classic Nasu–Honda, Claus, and Condon–Lipton approaches. It demonstrates undecidability under varied restrictions: fixed starting distributions or accepting states, small transition matrices (as small as 9×9 or 11×11 with a handful of matrices), and even binary alphabets with fixed matrices, by reductions from PCP and 2-counter machine Halting problems. A key theme is translating matrix product questions into PFA acceptance probabilities and using amplification techniques to create decisive probability gaps, yielding robust, parameter-rich undecidability results. The paper also surveys alternative universal machines and strategies for encoding computations, highlighting the deep connections between automata theory, PCP, and matrix-product problems in the landscape of undecidability. Overall, these results illuminate the subtle frontier where seemingly simple probabilistic models exhibit intractable decision problems, with implications for joint spectral radius, bilinear systems, and complexity of matrix products.

Abstract

It is undecidable whether the language recognized by a probabilistic finite automaton is empty. Several other undecidability results, in particular regarding problems about matrix products, are based on this important theorem. We present three proofs of this theorem from the literature in a self-contained way, and we derive some strengthenings. For example, we show that the problem remains undecidable for a fixed probabilistic finite automaton with 11 states, where only the starting distribution is given as input.

Probabilistic Finite Automaton Emptiness is undecidable

TL;DR

This work shows that the emptiness problem for probabilistic finite automata is undecidable and strengthens this result through multiple independent proofs, including self-contained renditions of the classic Nasu–Honda, Claus, and Condon–Lipton approaches. It demonstrates undecidability under varied restrictions: fixed starting distributions or accepting states, small transition matrices (as small as 9×9 or 11×11 with a handful of matrices), and even binary alphabets with fixed matrices, by reductions from PCP and 2-counter machine Halting problems. A key theme is translating matrix product questions into PFA acceptance probabilities and using amplification techniques to create decisive probability gaps, yielding robust, parameter-rich undecidability results. The paper also surveys alternative universal machines and strategies for encoding computations, highlighting the deep connections between automata theory, PCP, and matrix-product problems in the landscape of undecidability. Overall, these results illuminate the subtle frontier where seemingly simple probabilistic models exhibit intractable decision problems, with implications for joint spectral radius, bilinear systems, and complexity of matrix products.

Abstract

It is undecidable whether the language recognized by a probabilistic finite automaton is empty. Several other undecidability results, in particular regarding problems about matrix products, are based on this important theorem. We present three proofs of this theorem from the literature in a self-contained way, and we derive some strengthenings. For example, we show that the problem remains undecidable for a fixed probabilistic finite automaton with 11 states, where only the starting distribution is given as input.
Paper Structure (84 sections, 25 theorems, 80 equations, 14 figures, 2 tables)

This paper contains 84 sections, 25 theorems, 80 equations, 14 figures, 2 tables.

Table of Contents

  1. Probabilistic finite automata (PFA)
  2. Formal problem definition
  3. Statement of results
  4. More general acceptance.
  5. Uniqueness of solutions.
  6. Smaller transition matrices.
  7. Preface: history and matrix products
  8. Three proofs
  9. Interlude: Other problems on matrix products
  10. … back to the proofs of PFA Emptiness:
  11. Overview
  12. Comparison of the proofs
  13. The Condon--Lipton proof via 2-counter machines
  14. The Equality Checker
  15. Correctness Test: checking a 2CM computation
  16. ...and 69 more sections

Key Result

theorem 1

For any fixed $\lambda$ with $0< \lambda< 1$, the PFA Emptiness Problem eq:accept with cutpoint $\lambda$ is undecidable, even when restricted to instances where $\mathcal{M}$ consists of only two transition matrices, all of whose entries are from the set $\{0,\frac{1}{2},1\}$, and $\pi$ and $f$ are

Figures (14)

  • Figure 1: The coin flips for the input $\texttt{a}^{10}\texttt{b}^{22}\texttt{\#}$
  • Figure 2: The behavior of the Equality Checker, assuming $i \equiv j \pmod{12}$
  • Figure 3: The Correctness Test for a computation $A$, and a hypothetical position where equality does not hold.
  • Figure 4: The binary automaton with acceptance probability $\phi$
  • Figure 5: Acceptance probability $\phi\psi$
  • ...and 9 more figures

Theorems & Definitions (41)

  • theorem 1
  • Definition 1
  • theorem 2
  • theorem 3
  • theorem 4
  • theorem 5: Claus claus81
  • theorem 6: hirvensalo07
  • lemma 1
  • proof
  • lemma 2
  • ...and 31 more