On Formally Undecidable Propositions of Nondeterministic Complexity and Related Classes

TL;DR

This paper explores the limitations of the class NP by connecting its definition to G"odel's Incompleteness Theorem, showing that certain semantic properties of NP are inherently unsatisfiable.

Contribution

It demonstrates that the semantic definition of NP leads to contradictions similar to those in Hilbert's Program, revealing fundamental limitations.

Findings

NP contains languages with properties G"odel's theorem prohibits.

The polynomial-time checking relation can include G"odel's proof-checking relation.

The semantic definition of NP is inherently unsatisfiable.

Abstract

The definition of \NP\ requires, for each member language~$L$, a polynomial-time checking relation~$R$ and a constant~$k$ such that $w \in L \iff \exists y\,(|y| \leq |w|^k \wedge R(w,y))$. We show that this biconditional instantiates, for each member language, Hilbert's triple: a sound, complete, decidable proof system in which truth-in-$L$ and bounded provability coincide by fiat. We show further that the polynomial-time restriction on~$R$ does not exclude G\"odel's proof-checking relation, which is itself polynomial-time and fits the definition as a literal instance. Hence \NP, taken as a totality over all polynomial-time~$R$, contains languages for which the biconditional asserts a property that G\"odel's First Incompleteness Theorem prohibits. The semantic definition of \NP\ is unsatisfiable, for the same reason that Hilbert's Program is.