Machines as Programs: P $\neq$ NP

TL;DR

The paper generalizes the proofs-as-programs concept to introduce Ceiling Machines, demonstrating that certain nondeterministic solutions cannot be efficiently replicated by deterministic machines, thereby proving P ≠ NP.

Contribution

It introduces Ceiling Machines and the CNDS framework, providing a novel proof that P ≠ NP through machine-based formalization.

Findings

Nondeterministic solutions are not polynomial-time replicable by deterministic machines.

Introduction of Higher and Lower Ceiling Machines with distinct computational capabilities.

Formal proof that P ≠ NP based on the properties of Ceiling Machines and CNDS.

Abstract

The Curry-Howard correspondence is often called the proofs-as-programs result. I offer a generalization of this result, something which may be called machines as programs. Utilizing this insight, I introduce two new Turing Machines called "Ceiling Machines." The formal ingredients of these two machines are nearly identical. But there are crucial differences, splitting the two into a "Higher Ceiling Machine" and a "Lower Ceiling Machine." A potential graph of state transitions of the Higher Ceiling Machine is then offered. This graph is termed the "canonically nondeterministic solution" or CNDS, whose accompanying problem is its own replication, i.e., the problem, "Replicate CNDS" (whose accompanying algorithm is cast in Martin-L\"of type theory). I then show that while this graph can be replicated (solved) in polynomial time by a nondeterministic machine -- of which the Higher Ceiling Machine is a canonical example -- it cannot be solved in polynomial time by a deterministic machine, of which the Lower Ceiling Machine is also canonical. It is consequently proven that P $\neq$ NP.