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Probabilistic Finite Automaton Emptiness is Undecidable for a Fixed Automaton

Günter Rote

TL;DR

This work proves that the PFA Emptiness problem is undecidable even for PFAs with a fixed automaton when the only input is the starting distribution or the end vector, by reducing instances of the Post Correspondence Problem (PCP) to PFA acceptance. The authors develop a 6-step matrix-construction pipeline that encodes PCP word pairs into matrix products, enforces column and row sums to yield stochastic matrices, and uses a small PCP instance (five word pairs) to obtain a fixed, small-state PFA (7 states) that captures PCP solvability. They further extend the results to binary alphabets and to versions where the end vector or the matrices are fixed, demonstrating undecidability for as few as two matrices in certain configurations and for 6×6 to 28×28 matrix sizes under various inputs. The results connect PFA emptiness to broad matrix-product undecidability questions, show how to amplify gaps between acceptance and rejection, and extend to rational-weighted automata, highlighting the robustness of undecidability in probabilistic and weighted automata. Overall, the paper advances understanding of the boundary between decidable and undecidable matrix-product problems and clarifies how small, fixed PFAs can encode complex computational questions.

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.

Probabilistic Finite Automaton Emptiness is Undecidable for a Fixed Automaton

TL;DR

This work proves that the PFA Emptiness problem is undecidable even for PFAs with a fixed automaton when the only input is the starting distribution or the end vector, by reducing instances of the Post Correspondence Problem (PCP) to PFA acceptance. The authors develop a 6-step matrix-construction pipeline that encodes PCP word pairs into matrix products, enforces column and row sums to yield stochastic matrices, and uses a small PCP instance (five word pairs) to obtain a fixed, small-state PFA (7 states) that captures PCP solvability. They further extend the results to binary alphabets and to versions where the end vector or the matrices are fixed, demonstrating undecidability for as few as two matrices in certain configurations and for 6×6 to 28×28 matrix sizes under various inputs. The results connect PFA emptiness to broad matrix-product undecidability questions, show how to amplify gaps between acceptance and rejection, and extend to rational-weighted automata, highlighting the robustness of undecidability in probabilistic and weighted automata. Overall, the paper advances understanding of the boundary between decidable and undecidable matrix-product problems and clarifies how small, fixed PFAs can encode complex computational questions.

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.

Paper Structure

This paper contains 36 sections, 9 theorems, 23 equations, 1 table.

Table of Contents

  1. Probabilistic finite automata (PFA)
  2. Formal problem definition
  3. Rational-Weigthed Automata.
  4. Statement of results
  5. Contributions of this paper and relations to other problems
  6. The Post Correspondence Problem (PCP)
  7. Proofs of \ref{['thm-matrices-forward']}\ref{['4-positive']} and \ref{['thm-matrices-forward']}\ref{['5-fixed']} (few states)
  8. Step 1: Modeling word pairs by matrices
  9. Equality detection
  10. Modeling the Post Correspondence Problem (PCP)
  11. History of ideas.
  12. Step 2: Making column sums 1
  13. Step 3: Making row sums 1 with an extra state
  14. Step 4: Making the matrices positive, and hence stochastic
  15. Step 5: Turning π into a probability distribution
  16. ...and 21 more sections

Key Result

Theorem 1

The PFA Emptiness Problem is undecidable for PFAs with a single accepting state and the following restrictions on the number of transition matrices (size of the input alphabet) and their size (number of states): All statements hold also for weak inequality $(\ge\lambda)$ as the acceptance criterion.

Theorems & Definitions (9)

  • Theorem 1
  • Theorem 2
  • Lemma 3: Multiplicative Law
  • Lemma 4: Equality Detection Lemma
  • Lemma 5
  • Lemma 6
  • Theorem 7: Gimbert and Oualhadj 2010 gimbert-oualhadj-2010:PFA
  • Theorem 7: Gimbert and Oualhadj 2010 gimbert-oualhadj-2010:PFA
  • Theorem 8