A Framework for Algebraic Characterizations in Recursive Analysis

TL;DR

This paper develops an extended algebraic framework for understanding the computability and complexity of real-valued functions, making recursive analysis more intuitive and applicable to broader domains.

Contribution

It extends previous algebraic characterizations of recursive functions from compact domains to the entire real line and arbitrary domains, enhancing understanding of continuous computation.

Findings

Framework for algebraic characterization over the real line

Extension from compact to arbitrary domains

Simplifies understanding of recursive analysis

Abstract

Algebraic characterizations of the computational aspects of functions defined over the real numbers provide very effective tool to understand what computability and complexity over the reals, and generally over continuous spaces, mean. This is relevant for both communities of computer scientists and mathematical analysts, particularly the latter who do not understand (and/or like) the language of machines and string encodings. Recursive analysis can be considered the most standard framework of computation over continuous spaces, it is however defined in a very machine specific way which does not leave much to intuitiveness. Recently several characterizations, in the form of function algebras, of recursively computable functions and some sub-recursive classes were introduced. These characterizations shed light on the hidden behavior of recursive analysis as they convert complex computational operations on sequences of real objects to "simple" intuitive mathematical operations such as integration or taking limits. The authors previously presented a framework for obtaining algebraic characterizations at the complexity level over compact domains. The current paper presents a comprehensive extension to that framework. Though we focus our attention in this paper on functions defined over the whole real line, the framework, and accordingly the obtained results, can be easily extended to functions defined over arbitrary domains.