Statement copy in the gpt 6 astra pro collection (CS-1)

P versus NP Problem

Definitions and mathematical statement

A language is a set $L\subseteq\{0,1\}^*$. Write $\mathsf P$ for languages decided by a deterministic Turing machine in time bounded by a polynomial in the input length. A language belongs to $\mathsf{NP}$ if there are a deterministic polynomial-time verifier $V$ and a polynomial $p$ such that \[ x\in L\quad\Longleftrightarrow\quad \exists y\in\{0,1\}^{\le p(|x|)}\;V(x,y)=1. \] The polynomial and machine may depend on $L$, but not on its individual inputs. The question is whether $\mathsf P=\mathsf{NP}$. A negative answer requires a language in $\mathsf{NP}$ outside $\mathsf P$, not merely a slow algorithm for one problem.

Formulation and concept references: [S1], [E1].

Short English statement

Does an efficient way to check a proposed yes-answer always imply an efficient way to decide whether the answer is yes?

Sources

• S1 Clay Mathematics Institute, The Millennium Prize Problems. https://www.claymath.org/library/monographs/MPPc.pdf. Catalogue formulation source.
• S2 P vs NP. https://www.claymath.org/millennium/p-vs-np/. Catalogue research/status reference; not a proof endorsement.
• S3 Constructive solvability and the P versus NP problem — Arne Hole. https://arxiv.org/abs/2406.16843. Catalogue research/status reference; not a proof endorsement.
• S4 PNP Labs — P versus NP formal reconstruction. https://pnplabs.com.au/. Catalogue research/status reference; not a proof endorsement.
• E1 Stephen Cook, The P versus NP Problem, Clay Mathematics Institute, Section 1 and Appendix. https://www.claymath.org/wp-content/uploads/2022/06/pvsnp.pdf. Added primary source: Definitions and formulation. Consulted 24 September 2026.

• E2 Theodore Baker, John Gill and Robert Solovay, Relativizations of the $\mathcal P=?\mathcal{NP}$ Question, SIAM Journal on Computing 4$4$ $1975$, 431--442, oracle equality and separation results. https://doi.org/10.1137/0204037. Consulted 24 September 2026: publisher abstract; used only for the stated oracle obstruction, not a new proof audit.
• E3 Alexander A. Razborov and Steven Rudich, Natural Proofs, Journal of Computer and System Sciences 55$1$ $1997$, 24--35; author manuscript dated 30 November 1996, abstract and the natural-property/hardness framework. https://www.cs.umd.edu/~gasarch/BLOGPAPERS/natural.pdf. Consulted 24 September 2026. The pseudorandomness assumption is retained; no unconditional barrier is asserted.
• E4 Scott Aaronson and Avi Wigderson, Algebrization: A New Barrier in Complexity Theory, ACM Transactions on Computation Theory 1$1$ $2009$, Article 2; author manuscript, Sections 1--2 and 5, especially Theorems 5.1 and 5.3. https://www.scottaaronson.com/papers/alg.pdf. Consulted 24 September 2026: definitions and barrier statements.
• E5 Ryan Williams, Nonuniform ACC Circuit Lower Bounds, Journal of the ACM 61$1$ $2014$, Article 2, 32 pages; author manuscript, Theorem 1.1 and Section 1.1. https://people.csail.mit.edu/rrw/acc-lbs.pdf. https://doi.org/10.1145/2559903. Consulted 24 September 2026.
• E6 Lijie Chen, Avishay Tal and Yichuan Wang, Super-quadratic Lower Bounds for Depth-2 Linear Threshold Circuits, ECCC TR26-039, 15 March 2026, abstract and reported $\mathsf E^{\mathsf{NP}}$ threshold-circuit bound. https://eccc.weizmann.ac.il/report/2026/039/. Consulted 24 September 2026: report page and abstract. A recent manuscript result, not an independent verification of its full proof.
• E7 Adnan Darwiche and Pierre Marquis, A Knowledge Compilation Map, Journal of Artificial Intelligence Research 17 $2002$, 229--264, Sections 1--2 and the separation of succinctness, queries and transformations. https://arxiv.org/abs/1106.1819. https://doi.org/10.1613/jair.989. Consulted 24 September 2026: article introduction and language definitions. Used as methodological context, not as the source of Lemma 1.1.