Computability and Complexity over the Product Topology of Real Numbers

TL;DR

This paper explores the computability and complexity of functions over the product topology of real numbers, revealing limitations in defining computable norms and differences in computational complexity based on query bounds.

Contribution

It extends Kawamura and Cook's framework to the product space of real numbers, analyzing computability and complexity under bounded and unbounded query scenarios.

Findings

No computable norm exists over the product topology.

Functions with bounded queries reduce to finite-dimensional problems.

Unbounded queries require non-uniform approaches, complicating computation.

Abstract

Kawamura and Cook have developed a framework for studying the computability and complexity theoretic problems over "large" topological spaces. This framework has been applied to study the complexity of the differential operator and the complexity of functionals over the space of continuous functions on the unit interval $C[0,1]$. In this paper we apply the ideas of Kawamura and Cook to the product space of the real numbers endowed with the product topology. We show that no computable norm can be defined over such topology. We investigate computability and complexity of total functions over the product space in two cases: (1) when the computing machine submits a uniformally bounded number of queries to the oracle and (2) when the number of queries submitted by the machine is not uniformally bounded. In the first case we show that the function over the product space can be reduced to a function over a finite-dimensional space. However, in general there exists functions whose computing machines must submit a non-uniform number of queries to the oracle indicating that computing over the product topology can not in general be reduced to computing over finite-dimensional spaces.