Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2604.07406

On Formally Undecidable Propositions of Nondeterministic Complexity and Related Classes

Martin Kolář
View original source ↗Read source document ↗Export metadata ↓

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 Gödel'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 Gödel's First Incompleteness Theorem prohibits. The semantic definition of \NP\ is unsatisfiable, for the same reason that Hilbert's Program is.

Explore related subjects

cs.CC

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Martin Kolář. 2026-04-08. On Formally Undecidable Propositions of Nondeterministic Complexity and Related Classes. https://arxiv.org/abs/2604.07406

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Bit-counting complexity classes

We define bit-counting complexity classes whose membership depends on the binary profile of the number of accepting paths of non-deterministic polynomial time Turing machines. We study the relationship between this new family of complexity classes and the classical complexity classes. We prove that the complexity class ${\bf PP}$ is contained in our comparison based bit-counting complexity classes ${\bf B_{|0|=|1|}P}$, ${\bf B_{|0|<|1|}P}$ and ${\bf B_{|0|>|1|}P}$. We then show that the comparison based bit-counting complexity classes and the complexity class ${\bf PP}$ are Turing equivalent, that is ${\bf P}^{\bf PP} = {\bf P}^{{\bf B_{|0|=|1|}P}}={\bf P}^{{\bf B_{|0|>|1|}P}}={\bf P}^{{\bf B_{|0|<|1|}P}}$. We then prove that the complexity classes ${\bf NP}$ and ${\bf CoNP}$ are contained in both of our parity based bit-counting complexity classes ${\bf B_{|0| \oplus}P}$ and ${\bf B_{|1| \oplus}P}$. We also show that the Turing closures of the parity based bit-counting complexity classes coincide, that is ${\bf P}^{{\bf B_{|0|\oplus}P}}={\bf P}^{{\bf B_{|1|\oplus}P}}$. We do this by proving that when either parity based bit-counting complexity class is provided as an oracle for a polynomial time Turing machine, then it can simulate the other one, that is ${\bf B_{|1| \oplus}P}\subseteq {\bf P}^{{\bf B_{|0| \oplus}P}}$ and ${\bf B_{|0| \oplus}P}\subseteq {\bf P}^{{\bf B_{|1| \oplus}P}}$.

cs.CC↗
arXiv2026

Formalizing PARITY Circuit Lower Bounds in Lean

We formalize Hastad's PARITY lower bound in Lean using the switching lemma. For every fixed d >= 2, formulas and DAG circuits of computation depth at most d computing PARITY on n inputs require size exp(Omega_d(n^(1/(d-1)))) for all sufficiently large n. This matches the classical upper bound up to constants in the exponent and implies that PARITY is not in nonuniform AC0. We also construct a polynomial-size, logarithmic-depth bounded-fan-in formula family for PARITY, providing a witness to NC1 is not a subset of AC0 for the formalized models. The Lean source code is available at https://github.com/formalcs/circuit-complexity and is checked with Lean 4.33.1 and mathlib 4.33.1.

cs.CC↗
arXiv2026

Constant-Coin Complete-Information Debates for $\mathsf{P}$ with Arbitrarily Small Strong Error

We study complete-information debate systems in which a probabilistic finite-state verifier reads the alternating messages of a prover and a refuter. Demirci, Say, and Yakaryılmaz showed that every language in $\mathsf{P}$ has such debates checkable with a constant number of random bits and arbitrarily small weak error. Their strong-error construction, which also counts nontermination as failure, did not permit arbitrary error reduction. We close this gap: for every $L\in\mathsf{P}$ and every $\varepsilon>0$, there is a constant-space verifier using a constant number of private coin tosses that has perfect completeness and strong error at most $\varepsilon$. The verifier simulates a polynomial-time alternating multihead finite automaton, privately spot-checking one of its input heads. The key observation is that, on a nonmember, the refuter may concede any round in which the prover first misreports a head reading. This ensures termination against every prover when the refuter follows the specified strategy, and permits strong-error reduction by repetition.

cs.CC↗