Transcendental Encoding conjecture

TL;DR

The paper introduces the Transcendental Encoding Conjecture, proposing a novel link between complexity classes and the algebraic or transcendental nature of their encodings, with implications for understanding NP-completeness.

Contribution

It formulates a new conjecture connecting complexity classes to the algebraic or transcendental properties of their encodings, providing initial examples and discussing open questions.

Findings

Languages in P encode to algebraic reals

NP-complete languages encode to transcendental reals

Some natural languages have provably rational or irrational encodings

Abstract

We propose the Transcendental Encoding Conjecture for decision problems, which asserts that every language in complexity class P encodes to an algebraic real (possibly rational or algebraic irrational) under its binary characteristic encoding or other relevant encodings, whereas every NP-complete language encodes to a transcendental real. In particular, we exhibit languages whose encodings are provably rational (hence algebraic), discuss the status of encodings for other "natural" languages such as PRIMES (its encoding is irrational but not known to be algebraic).