The single-use restriction for register automata and transducers over infinite alphabets
Rafał Stefański
TL;DR
The work addresses the challenge of extending regularity to words over infinite alphabets by enforcing a single-use restriction, which prevents reuse of read values and yields robust, algebraically tractable classes. It develops an abstract framework of single-use functions and uses it to redefine single-use register automata and transducers, proving Krohn-Rhodes–style decompositions and algebraic characterizations for single-use Mealy machines and two-way transducers. The results unify previously scattered insights by showing equivalences with orbit-finite monoids and by introducing local semigroup transduction theories, including local algebraic semigroup transductions, as well as infinite-alphabet SSTs and regular list functions. This coherent narrative advances a theory of regular languages and regular functions over infinite alphabets, with decidability and decomposition results that illuminate the algebraic structure behind computation on infinite data domains.
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.