Renormalization group approach to the P versus NP question

TL;DR

This paper explores using renormalization group techniques to differentiate between computational complexity classes, potentially offering new insights into the P versus NP problem by analyzing Boolean functions' behavior under transformation.

Contribution

It introduces a renormalization group transformation for Boolean functions and suggests its potential to distinguish between complexity classes, offering a novel approach to the P versus NP question.

Findings

The transformation behaves differently for generic and non-generic Boolean functions.

Non-generic functions can be identified through their transformation behavior.

This approach may lead to algorithms for proving functions are not in P.

Abstract

This paper argues that the ideas underlying the renormalization group technique used to characterize phase transitions in condensed matter systems could be useful for distinguishing computational complexity classes. The paper presents a renormalization group transformation that maps an arbitrary Boolean function of $N$ Boolean variables to one of $N-1$ variables. When this transformation is applied repeatedly, the behavior of the resulting sequence of functions is different for a generic Boolean function than for Boolean functions that can be written as a polynomial of degree $\xi$ with $\xi \ll N$ as well as for functions that depend on composite variables such as the arithmetic sum of the inputs. Being able to demonstrate that functions are non-generic is of interest because it suggests an avenue for constructing an algorithm capable of demonstrating that a given Boolean function cannot be computed using resources that are bounded by a polynomial of $N$.