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A* shortest string decoding for non-idempotent semirings

Kyle Gorman, Cyril Allauzen

TL;DR

The paper tackles decoding weighted finite-state automata over non-idempotent monotonic negative semirings, where a shortest path may not exist, by introducing an A$^*$-based shortest-string decoding approach. It employs the backwards shortest distance $\beta$ as an admissible, consistent heuristic computed on-the-fly from a determinized DFA over a companion semiring, enabling correct shortest-string retrieval with on-the-fly determinization. Empirical evaluation on 700 word lattices from speech recognition demonstrates substantial efficiency gains over naïve full determinization, confirming practical scalability. The method broadens exact decoding capabilities to EM-based learning and interpolated language-model scoring, offering a principled and scalable alternative to Viterbi-like approximations.

Abstract

The single shortest path algorithm is undefined for weighted finite-state automata over non-idempotent semirings because such semirings do not guarantee the existence of a shortest path. However, in non-idempotent semirings admitting an order satisfying a monotonicity condition (such as the plus-times or log semirings), the notion of shortest string is well-defined. We describe an algorithm which finds the shortest string for a weighted non-deterministic automaton over such semirings using the backwards shortest distance of an equivalent deterministic automaton (DFA) as a heuristic for A* search performed over a companion idempotent semiring, which is proven to return the shortest string. While there may be exponentially more states in the DFA, this algorithm needs to visit only a small fraction of them if determinization is performed "on the fly".

A* shortest string decoding for non-idempotent semirings

TL;DR

The paper tackles decoding weighted finite-state automata over non-idempotent monotonic negative semirings, where a shortest path may not exist, by introducing an A-based shortest-string decoding approach. It employs the backwards shortest distance as an admissible, consistent heuristic computed on-the-fly from a determinized DFA over a companion semiring, enabling correct shortest-string retrieval with on-the-fly determinization. Empirical evaluation on 700 word lattices from speech recognition demonstrates substantial efficiency gains over naïve full determinization, confirming practical scalability. The method broadens exact decoding capabilities to EM-based learning and interpolated language-model scoring, offering a principled and scalable alternative to Viterbi-like approximations.

Abstract

The single shortest path algorithm is undefined for weighted finite-state automata over non-idempotent semirings because such semirings do not guarantee the existence of a shortest path. However, in non-idempotent semirings admitting an order satisfying a monotonicity condition (such as the plus-times or log semirings), the notion of shortest string is well-defined. We describe an algorithm which finds the shortest string for a weighted non-deterministic automaton over such semirings using the backwards shortest distance of an equivalent deterministic automaton (DFA) as a heuristic for A* search performed over a companion idempotent semiring, which is proven to return the shortest string. While there may be exponentially more states in the DFA, this algorithm needs to visit only a small fraction of them if determinization is performed "on the fly".
Paper Structure (19 sections, 4 theorems, 17 equations, 3 figures, 1 table)

This paper contains 19 sections, 4 theorems, 17 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Definitions
  3. Semirings
  4. Weighted finite-state acceptors
  5. Shortest distance
  6. Shortest path
  7. Determinization
  8. Shortest string
  9. A* search
  10. The algorithm
  11. Evaluation
  12. Data
  13. Implementation
  14. Methods
  15. Results
  16. ...and 4 more sections

Key Result

Lemma 2.1

In an idempotent semiring, a shortest path's string is also a shortest string.

Figures (3)

  • Figure 1: State diagrams showing a weighted NFA (left) and an equivalent DFA (right).
  • Figure 2: Comparison of word lattice decoding with the proposed algorithm vs. the naïve algorithm. The $x$-axis shows the number of states in the full DFA; the $y$-axis shows the number of DFA states visited by the proposed algorithm. Both axes are in logarithmic scale.
  • Figure 3: Comparison of word lattice decoding with the proposed algorithm to the size of the input NFA. The $x$-axis shows the number of states in the input NFA; the $y$-axis shows the number of states visited by the proposed algorithm. Both axes are in logarithmic scale.

Theorems & Definitions (51)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Remark 2.1
  • Definition 2.7
  • Remark 2.2
  • Definition 2.8
  • ...and 41 more