Computer Science > Computational Complexity
[Submitted on 20 Sep 2021]
Title:Machines as Programs: P $\neq$ NP
View PDFAbstract: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-Löf 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.
Current browse context:
cs.CC
References & Citations
Bibliographic and Citation Tools
Bibliographic Explorer (What is the Explorer?)
Litmaps (What is Litmaps?)
scite Smart Citations (What are Smart Citations?)
Code, Data and Media Associated with this Article
CatalyzeX Code Finder for Papers (What is CatalyzeX?)
DagsHub (What is DagsHub?)
Gotit.pub (What is GotitPub?)
Papers with Code (What is Papers with Code?)
ScienceCast (What is ScienceCast?)
Demos
Recommenders and Search Tools
Influence Flower (What are Influence Flowers?)
Connected Papers (What is Connected Papers?)
CORE Recommender (What is CORE?)
arXivLabs: experimental projects with community collaborators
arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.
Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.