Semi-Countable Sets and their Application to Search Problems

TL;DR

The paper introduces semi-countable sets, a new class between countable and uncountable sets, and explores their implications for understanding computational complexity and the P vs NP problem.

Contribution

It defines semi-countable sets based on information efficiency and demonstrates their relevance to complexity class separation and search problems.

Findings

Semi-countable sets are between countable and uncountable sets.

These sets can be computed in exponential time but not in polynomial time.

The class $\phi_{ ext{Sigma}}$ encodes the Subset Sum problem, illuminating P vs NP.

Abstract

We present the concept of the \emph{information efficiency of functions} as a technique to understand the interaction between information and computation. Based on these results we identify a new class of objects that we call \emph{Semi-Countable Sets}. As the name suggests these sets form a separate class of objects between countable and uncountable sets. In principle these objects are countable, but the information in the descriptions of the elements of the class grows faster than the information in the natural numbers that index them. Any characterization of the class in terms of natural numbers is fundamentally incomplete. Semi-countable sets define one-to-one injections into the set of natural numbers that can be computed in exponential time, but not in polynomial time. A characteristic semi-countable object is $\phi_{\Sigma}$ the set of all additions for all finite sets of natural numbers. The class $\phi_{\Sigma}$ codes the Subset Sum problem. This gives a natural and transparant analysis of the separation between the classes $P$ and $NP$.