Deterministic regular functions of infinite words

TL;DR

This paper studies a class of deterministic regular functions of infinite words, showing they are computable, closed under composition, and characterized by various logical and automata-theoretic frameworks.

Contribution

It introduces a well-behaved subclass of regular functions of infinite words realized without look-ahead, establishing their computability and multiple characterizations.

Findings

They are computable.

They are closed under composition.

They are characterized by MSO-transductions and automata models.

Abstract

Regular functions of infinite words are (partial) functions realized by deterministic two-way transducers with infinite look-ahead. Equivalently, Alur et. al. have shown that they correspond to functions realized by deterministic Muller streaming string transducers, and to functions defined by MSO-transductions. Regular functions are however not computable in general (for a classical extension of Turing computability to infinite inputs), and we consider in this paper the class of deterministic regular functions of infinite words, realized by deterministic two-way transducers without look-ahead. We prove that it is a well-behaved class of functions: they are computable, closed under composition, characterized by the guarded fragment of MSO-transductions, by deterministic B\"uchi streaming string transducers, by deterministic two-way transducers with finite look-ahead, and by finite compositions of sequential functions and one fixed basic function called map-copy-reverse.