Computing over the Reals: Foundations for Scientific Computing

TL;DR

This paper explores the foundational models of computability and complexity for real functions, emphasizing the bit-model's relevance to scientific computing and discussing the potential of physical systems to challenge the Church-Turing Thesis.

Contribution

It provides a comprehensive analysis of the bit-model for real computation and compares it with the Blum-Shub-Smale model, highlighting implications for scientific computing.

Findings

Bit-model effectively formalizes scientific computation problems

Comparison shows strengths and limitations of different models

Discussion on physical systems challenges to Church-Turing Thesis

Abstract

We give a detailed treatment of the ``bit-model'' of computability and complexity of real functions and subsets of R^n, and argue that this is a good way to formalize many problems of scientific computation. In the introduction we also discuss the alternative Blum-Shub-Smale model. In the final section we discuss the issue of whether physical systems could defeat the Church-Turing Thesis.