Space-Efficient Bimachine Construction Based on the Equalizer Accumulation Principle

TL;DR

This paper introduces a space-efficient bimachine construction method based on the equalizer accumulation principle, which significantly reduces automaton size compared to standard methods, especially for certain classes of transducers.

Contribution

The paper presents a novel bimachine construction technique that minimizes automaton size by considering parallel transducer paths and maximizing output at each step.

Findings

Standard bimachine construction can produce factorial-sized automata.

The new method reduces automaton size to exponential in the number of states.

Applicable to a broad class of monoids, including free monoids and groups.

Abstract

Algorithms for building bimachines from functional transducers found in the literature in a run of the bimachine imitate one successful path of the input transducer. Each single bimachine output exactly corresponds to the output of a single transducer transition. Here we introduce an alternative construction principle where bimachine steps take alternative parallel transducer paths into account, maximizing the possible output at each step using a joint view. The size of both the deterministic left and right automaton of the bimachine is restricted by $2^{\vert Q\vert}$ where $\vert Q\vert$ is the number of transducer states. Other bimachine constructions lead to larger subautomata. As a concrete example we present a class of real-time functional transducers with $n+2$ states for which the standard bimachine construction generates a bimachine with at least $\Theta(n!)$ states whereas the construction based on the equalizer accumulation principle leads to $2^n + n +3$ states. Our construction can be applied to rational functions from free monoids to "mge monoids", a large class of monoids including free monoids, groups, and others that is closed under Cartesian products.