The single-use restriction for register automata and transducers over infinite alphabets

TL;DR

This thesis explores the single-use restriction in register automata and transducers over infinite alphabets, revealing their expressive equivalences and algebraic characterizations, and extending classical decomposition theorems to this setting.

Contribution

It introduces a unified framework for single-use models, establishes their equivalences, and extends algebraic and decomposition results to automata and transducers over infinite alphabets.

Findings

One-way and two-way register automata have equivalent expressive power.

Single-use Mealy machines and two-way transducers admit Krohn-Rhodes decompositions.

Single-use transducers are equivalent to streaming string transducers and regular list functions.

Abstract

This thesis studies the single-use restriction for register automata and transducers over infinite alphabets. The restriction requires that a read-access to a register should have the side effect of destroying its contents. This constraint results in robust classes of languages and transductions. For automata models, we show that one-way register automata, two-way register automata, and orbit-finite monoids have the same expressive power. For transducer models, we show that single-use Mealy machines and single-use two-way transducers admit versions of the Krohn-Rhodes decomposition theorem. Moreover, single-use Mealy machines are equivalent to an algebraic model called local algebraic semigroup transductions. Additionally, we show that single-use two-way transducers are equivalent to single-use streaming string transducers (SSTs) over infinite alphabets and to regular list functions with atoms. Compared with the previous work arXiv:1907.10504, this thesis offers a coherent narrative on the single-use restriction. We introduce an abstract notion of single-use functions and use them to define all the discussed single-use models. We also introduce and study the algebraic models of local semigroup transduction and local rational semigroup transduction.