Computable real function F such that F is not polynomial time computable on [0,1]

TL;DR

This paper constructs a computable real function on [0,1] that can be evaluated in exponential time but not in polynomial time, highlighting a separation in computational complexity for real functions.

Contribution

It provides an explicit example of a computable real function that is not polynomial time computable, demonstrating a fundamental complexity distinction.

Findings

Existence of a computable function not polynomial time evaluable

Function is evaluable in exponential time

Separation holds for all rational points in (0,1)

Abstract

A computable real function F on [0,1] is constructed such that there exists an exponential time algorithm for the evaluation of the function on [0,1] on Turing machine but there does not exist any polynomial time algorithm for the evaluation of the function on [0,1] on Turing machine (moreover, it holds for any rational point on (0,1))