A note on time hierarchies for reasonable semantic classes without advice

TL;DR

This paper establishes time hierarchy results for reasonable semantic classes without advice by removing constant advice bits through a contrapositive argument applicable to various computational models.

Contribution

It introduces a method to eliminate constant advice bits in time hierarchy theorems for semantic classes, broadening understanding of computational complexity without advice.

Findings

Time hierarchies hold without advice for reasonable classes.

Advice bits can be effectively eliminated in hierarchy proofs.

The approach applies to various computational models.

Abstract

We show time hierarchies for reasonable semantic classes without advice by eliminating the constant bits of advice in previous results.The elimination is done by a contrapositive argument that for any reasonable computational model,let $\text{CTIME}(f(n))/{g(n)}$ denote the set of all languages decide by machines running in time $O(f(n))$ with advice of $g(n)$ bits in that model, if $\text{CTIME}(t(n))\subseteq \text{CTIME}(T(n))/{A(n)}$ then $\text{CTIME}(t(n))/a \subseteq \text{CTIME}(T(n))/{a+2^aA(n)}$ where $a$ is a constant integer.