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Determinisation and Unambiguisation of Polynomially-Ambiguous Rational Weighted Automata

Ismaël Jecker, Filip Mazowiecki, David Purser

TL;DR

This work addresses determinisation and unambiguisation of weighted automata over the rational field. It develops a pumping-based approach that reduces questions to powers of transition matrices on patterns $uv^nw$, enabling a precise characterization via pumpability for polynomially-ambiguous automata. The main contributions show that both determinisation and unambiguisation are decidable in PSPACE in this setting, leveraging an exponential-size construction together with a polynomial-space zeroness test; the results also clarify non-constructive aspects of producing equivalent deterministic/unambiguous automata and leave open whether such automata can be efficiently constructed. The work builds on and extends the state of the art following Bell and Smertnig, connecting algebraic properties (linear hulls, Zariski closure) with automata-theoretic decisions and offering a clear path for complexity bounds in the bounded-ambiguity regime.

Abstract

We study the determinisation and unambiguisation problems of weighted automata over the rational field: Given a weighted automaton, can we determine whether there exists an equivalent deterministic, respectively unambiguous, weighted automaton? Recent results by Bell and Smertnig show that the problem is decidable, however they do not provide any complexity bounds. We show that both problems are in PSPACE for polynomially-ambiguous weighted automata.

Determinisation and Unambiguisation of Polynomially-Ambiguous Rational Weighted Automata

TL;DR

This work addresses determinisation and unambiguisation of weighted automata over the rational field. It develops a pumping-based approach that reduces questions to powers of transition matrices on patterns , enabling a precise characterization via pumpability for polynomially-ambiguous automata. The main contributions show that both determinisation and unambiguisation are decidable in PSPACE in this setting, leveraging an exponential-size construction together with a polynomial-space zeroness test; the results also clarify non-constructive aspects of producing equivalent deterministic/unambiguous automata and leave open whether such automata can be efficiently constructed. The work builds on and extends the state of the art following Bell and Smertnig, connecting algebraic properties (linear hulls, Zariski closure) with automata-theoretic decisions and offering a clear path for complexity bounds in the bounded-ambiguity regime.

Abstract

We study the determinisation and unambiguisation problems of weighted automata over the rational field: Given a weighted automaton, can we determine whether there exists an equivalent deterministic, respectively unambiguous, weighted automaton? Recent results by Bell and Smertnig show that the problem is decidable, however they do not provide any complexity bounds. We show that both problems are in PSPACE for polynomially-ambiguous weighted automata.
Paper Structure (10 sections, 9 theorems, 6 equations, 1 figure)

This paper contains 10 sections, 9 theorems, 6 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Weighted automata
  4. Paths, runs, counting runs, and the monoid of structures
  5. Determinisim, ambiguity and decision problems
  6. Closure properties
  7. Assumptions
  8. Constructing an equivalent automaton with non-negative transitions
  9. Overview of our main result
  10. Proof of Lemma \ref{['lemma:pumpMatrixS']}

Key Result

Lemma 1

The twin property of an unambiguous automaton $\mathcal{A}$ can be decided in polynomial time.

Figures (1)

  • Figure 1: Example of a unary weighted automaton, where the input label of all edge is the letter $a$ (omitted on the picture). Unlabelled edges are assumed to have weight $1$. Observe, if $n$ is even then $\mathcal{A}(a^n)=1+\sum_{i=0}^{n/2 -1} 2^i = 2^{n/2}$ and if $n$ is odd then $\mathcal{A}(a^n)=1$. The automaton on the left is polynomially-ambiguous, and unambiguisable as depicted on the right, but the function is not determinisable.

Theorems & Definitions (16)

  • Lemma 1
  • Proposition 2
  • Theorem 3
  • Definition 4
  • Remark 5
  • Remark 6
  • Definition 7
  • Lemma 7
  • Lemma 7
  • Claim 8
  • ...and 6 more