A* shortest string decoding for non-idempotent semirings

TL;DR

This paper introduces an A* search-based algorithm for finding the shortest string in weighted automata over non-idempotent semirings, leveraging determinization and heuristics to efficiently handle complex automata.

Contribution

It presents a novel A* algorithm that computes shortest strings in non-idempotent semirings using heuristics from deterministic automata, addressing limitations of existing shortest path methods.

Findings

Algorithm guarantees shortest string in non-idempotent semirings.

Heuristic reduces the number of states visited during search.

On-the-fly determinization improves efficiency.

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".