Foundations of Online Structure Theory II: The Operator Approach

TL;DR

This paper develops a unified operator-based framework for online structure theory, connecting online algorithms with computable analysis and finite combinatorics, and integrating ideas from various computational fields.

Contribution

It introduces a novel operator approach that unifies online algorithms, computable analysis, reverse mathematics, and other areas, providing new tools for analyzing finite and infinite structures.

Findings

Links online algorithms with computable analysis.

Uses modifications of computable analysis notions for combinatorics.

Finitizes reverse mathematics for finite combinatorial problems.

Abstract

We introduce a framework for online structure theory. Our approach generalises notions arising independently in several areas of computability theory and complexity theory. We suggest a unifying approach using operators where we allow the input to be a countable object of an arbitrary complexity. We give a new framework which (i) ties online algorithms with computable analysis, (ii) shows how to use modifications of notions from computable analysis, such as Weihrauch reducibility, to analyse finite but uniform combinatorics, (iii) show how to finitize reverse mathematics to suggest a fine structure of finite analogs of infinite combinatorial problems, and (iv) see how similar ideas can be amalgamated from areas such as EX-learning, computable analysis, distributed computing and the like. One of the key ideas is that online algorithms can be viewed as a sub-area of computable analysis. Conversely, we also get an enrichment of computable analysis from classical online algorithms.