# A theory of NP-completeness and ill-conditioning for approximate real computations

## TL;DR

This paper develops a complexity theory for approximate real computations, incorporating condition numbers and modeling practical numerical computations, and relates it to classical P vs NP questions.

## Contribution

It introduces a new complexity framework for approximate real computations that accounts for condition numbers and connects to classical complexity theory.

## Key findings

- Established a theory for exact computations with condition numbers.
- Developed a model for approximate computations reflecting floating point arithmetic.
- Linked the P vs NP question in the new theory to the classical P vs NP problem.

## Abstract

We develop a complexity theory for approximate real computations. We first produce a theory for exact computations but with condition numbers. The input size depends on a condition number, which is not assumed known by the machine. The theory admits deterministic and nondeterministic polynomial time recognizable problems. We prove that P is not NP in this theory if and only if P is not NP in the BSS theory over the reals. Then we develop a theory with weak and strong approximate computations. This theory is intended to model actual numerical computations that are usually performed in floating point arithmetic. It admits classes P and NP and also an NP-complete problem. We relate the P vs NP question in this new theory to the classical P vs NP problem.