On the existence of hidden machines in computational time hierarchies

TL;DR

This paper explores the existence of hidden, unprovable total functions within infinite hierarchies of computational complexity classes, revealing fundamental incompleteness in formal complexity theories.

Contribution

It demonstrates that certain total functions in complexity hierarchies are unrecognizable by formal axiomatic systems, highlighting inherent limitations in formalizing computational complexity.

Findings

Existence of infinite hierarchy of hidden complexity classes

Some total functions are unprovable within formal systems

Incompleteness results in formal complexity theories

Abstract

Challenging the standard notion of totality in computable functions, one has that, given any sufficiently expressive formal axiomatic system, there are total functions that, although computable and "intuitively" understood as being total, cannot be proved to be total. In this article we show that this implies the existence of an infinite hierarchy of time complexity classes whose representative members are hidden from (or unknown by) the respective formal axiomatic systems. Although these classes contain total computable functions, there are some of these functions for which the formal axiomatic system cannot recognize as belonging to a time complexity class. This leads to incompleteness results regarding formalizations of computational complexity.