Notes on space complexity of integration of computable real functions in Ko-Friedman model

TL;DR

This paper investigates the space complexity of integrating computable real functions within the Ko-Friedman model, establishing that integration preserves linear-space computability for certain smooth functions and exploring the complexity class distinctions for integration.

Contribution

It proves that the integral of a linear-space computable $C^2$ real function is also linear-space computable, independent of open questions in complexity theory, and analyzes the complexity class of integration.

Findings

Integration of $C^2$ functions preserves linear-space computability.

Existence of a computable function with integrals in $FDSPACE(n^2)$ but not in $FP$ under complexity assumptions.

Complexity class distinctions for integration of computable functions are clarified.

Abstract

In the present paper it is shown that real function $g(x)=\int_{0}^{x}f(t)dt$ is a linear-space computable real function on interval $[0,1]$ if $f$ is a linear-space computable $C^2[0,1]$ real function on interval $[0,1]$, and this result does not depend on any open question in the computational complexity theory. The time complexity of computable real functions and integration of computable real functions is considered in the context of Ko-Friedman model which is based on the notion of Cauchy functions computable by Turing machines. In addition, a real computable function $f$ is given such that $\int_{0}^{1}f\in FDSPACE(n^2)_{C[a,b]}$ but $\int_{0}^{1}f\notin FP_{C[a,b]}$ if $FP\ne#P$.