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Level A · Machine-checkable Hard Logic & formalisation P-paper-disjoint-np-pairs

Complete disjoint NP pairs

There is no complete disjoint NP pair (Razborov [Ra94]; Pudlák's conjecture DisjNP [Pu17]). No disjoint NP pair is complete under polynomial-time many-one reductions of pairs.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

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Cite 
@misc{cairn-paper-disjoint-np-pairs,
  title        = {Complete disjoint NP pairs},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/paper-disjoint-np-pairs}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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The problem

The question

no_complete_disjoint_NP_pair. There is no complete disjoint NP pair (Razborov [Ra94]; Pudlák's conjecture DisjNP [Pu17]).

No disjoint NP pair is complete under polynomial-time many-one reductions of pairs. This is the strongest of the "incompleteness in the finite domain" conjectures: it implies that no length-optimal propositional proof system exists [Ra94, KMT03], hence NP ≠ coNP and P ≠ NP. Equivalently [Pu17], for every consistent polynomial-time axiomatized theory T ⊇ S¹₂ some disjoint NP pair is not provably disjoint in T.

no_complete_disjoint_coNP_pair. There is no complete disjoint coNP pair (Pudlák's conjecture DisjCoNP [Pu17]).

No disjoint coNP pair is complete under polynomial-time many-one reductions of pairs. This implies that TFNP has no complete problem and that SAT has no p-optimal proof system [Pu17, Kh22].

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Paper.DisjointNPPairs (2 statements).

theorem no_complete_disjoint_NP_pair : ¬∃ A B : DecisionProblem, IsCompleteNPPair A B
theorem no_complete_disjoint_coNP_pair : ¬∃ A B : DecisionProblem, IsCompleteCoNPPair A B

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References

  • [Ra94] Razborov, A. A. (1994). "On provably disjoint NP-pairs." BRICS Report Series RS-94-36; ECCC TR94-006.
  • [KMT03] Köbler, J., Messner, J., and Torán, J. (2003). "Optimal proof systems imply complete sets for promise classes." Inform. and Comput. 184, pp. 71--92.
  • [GSSZ04] Glaßer, C., Selman, A. L., Sengupta, S., and Zhang, L. (2004). "Disjoint NP-pairs." SIAM J. Comput. 33, pp. 1369--1416.
  • [Pu17] Pudlák, P. (2017). "Incompleteness in the finite domain." Bull. Symbolic Logic 23, pp. 405--441. arXiv:1601.01487
  • [Kh22] Khaniki, E. (2022). "New relations and separations of conjectures about incompleteness in the finite domain." J. Symbolic Logic 87, pp. 912--937. arXiv:1904.01362

Source and licence

Imported from Formal Conjectures (research papers), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.