From Two-Way to One-Way Finite State Transducers

TL;DR

This paper investigates whether functions defined by two-way finite state transducers can be equivalently represented by one-way transducers, providing a decidable procedure to construct such one-way transducers when possible.

Contribution

It extends Rabin and Scott's classical automaton equivalence result to finite state transductions, establishing the decidability of converting two-way transducers to one-way transducers.

Findings

Decidability of the definability problem for two-way to one-way transducers

A procedure to construct equivalent one-way transducers when they exist

Extension of classical automaton equivalence results to transductions

Abstract

Any two-way finite state automaton is equivalent to some one-way finite state automaton. This well-known result, shown by Rabin and Scott and independently by Shepherdson, states that two-way finite state automata (even non-deterministic) characterize the class of regular languages. It is also known that this result does not extend to finite string transductions: (deterministic) two-way finite state transducers strictly extend the expressive power of (functional) one-way transducers. In particular deterministic two-way transducers capture exactly the class of MSO-transductions of finite strings. In this paper, we address the following definability problem: given a function defined by a two-way finite state transducer, is it definable by a one-way finite state transducer? By extending Rabin and Scott's proof to transductions, we show that this problem is decidable. Our procedure builds a one-way transducer, which is equivalent to the two-way transducer, whenever one exists.