Comparing the power of advice strings: a notion of complexity for infinite words

TL;DR

This paper introduces a framework for comparing infinite strings based on their ability to present objects within computational models, revealing a hierarchy of advice automatic structures and their logical and computational properties.

Contribution

It defines a new complexity measure for infinite words using advice strings and explores the hierarchy of advice automatic structures, linking them to MSO-transductions and transducers.

Findings

Different classifications of infinite words are studied in detail.

Logical and computational equivalences of measures are established.

Connections between advice automatic structures and transducer models are explored.

Abstract

This paper is the extended version of On the Complexity of Infinite Advice Strings (ICALP 2018). We investigate a notion of comparison between infinite strings. In a general way, if M is a computation model (e.g. Turing machines) and C a class of objects (e.g. languages), the complexity of an infinite word alpha can be measured with respect to the amount of objects from C that are presentable with machines from M using alpha as an oracle. In our case, the model M is finite automata and the objects C are either recognized languages or presentable structures, known respectively as advice regular languages and advice automatic structures. This leads to several different classifications of infinite words that are studied in detail; we also derive logical and computational equivalent measures. Our main results explore the connections between classes of advice automatic structures, MSO-transductions and two-way transducers. They suggest a closer study of the resulting hierarchy over infinite words.