Open
Unresolved assertion in the cited scope ↗
Avi Wigderson · Mathematics and Computation · §12.4, algebraic reductions and completeness →
Observed Oct 10, 2026
Conjecture · version 1
VP versus VNP
Earlier version 1. View current version 2 →
Statement
Over a field of characteristic zero, VP and VNP are distinct.
Assumptions
- Polynomial families with polynomially bounded degree; nonuniform algebraic circuits; arithmetic operation complexity over the field.
Scope
An algebraic analogue; no equivalence to Boolean P versus NP is asserted.
Evidence
Source reports
Connections
In this problem · publication v4 · Selected context
P versus NP →
Different statement · related variant · Open
This statement contributes context in its stated scope; it does not settle the canonical P versus NP question.
Analogue
analogous toP ≠ NP
VP versus VNP is an algebraic complexity analogue, with field operations and polynomial families.
No equivalence or implication to the classical Boolean question is asserted.
Details & sources
VP versus VNP →
Open · version 1
Over a field of characteristic zero, VP and VNP are distinct.
- Polynomial families with polynomially bounded degree; nonuniform algebraic circuits; arithmetic operation complexity over the field.
Scope: An algebraic analogue; no equivalence to Boolean P versus NP is asserted.
P ≠ NP →
Open · version 2
There exists a language in NP that is not decidable by any deterministic polynomial-time algorithm.
- Finite binary inputs; uniform classical computation; polynomial time in bit length.
Scope: The conjectural negative answer to the same canonical question.
Relationship not independently reviewed.
Connection in P versus NP →Relationship not independently reviewed. This concerns its use in this problem.
In this problem · publication v7
P versus NP →
Different statement · related variant · Open
This statement contributes context in its stated scope; it does not settle the canonical P versus NP question.
Analogue
analogous toP ≠ NP
VP versus VNP is an algebraic complexity analogue, with field operations and polynomial families.
No equivalence or implication to the classical Boolean question is asserted.
Details & sources
VP versus VNP →
Open · version 1
Over a field of characteristic zero, VP and VNP are distinct.
- Polynomial families with polynomially bounded degree; nonuniform algebraic circuits; arithmetic operation complexity over the field.
Scope: An algebraic analogue; no equivalence to Boolean P versus NP is asserted.
P ≠ NP →
Open · version 2
There exists a language in NP that is not decidable by any deterministic polynomial-time algorithm.
- Finite binary inputs; uniform classical computation; polynomial time in bit length.
Scope: The conjectural negative answer to the same canonical question.
Relationship not independently reviewed.
Connection in P versus NP →Relationship not independently reviewed. This concerns its use in this problem.
Sources
Avi Wigderson · Mathematics and Computation →
- §12.4, algebraic reductions and completeness → · Original passage ↗
Recorded wording
Over a field of characteristic zero, VP and VNP are distinct.
paraphrase · Only metadata and original OpenMath paraphrases are published; proof PDFs remain at their source.
Avi Wigderson · Published · Publication date 2019
Published author exposition or original paper.
Avi Wigderson · Mathematics and Computation — proof text ↗Manuscript / exposition · external
Revision & attribution
Source revision: author-draft-2019-08-06. Observed Oct 10, 2026.
AI involvement is not reported by this cited source; this is not a claim of its absence.
Avi Wigderson · Mathematics and Computation — proof text: author-draft-2019-08-06. External source text. Its mathematical claims and independent OpenMath review are assessed separately.
Stephen Cook · The P versus NP problem →
- Problem statement and discussion, pp. 2–3 → · Original passage ↗
Recorded wording
There exists a language in NP that is not decidable by any deterministic polynomial-time algorithm.
paraphrase · Only metadata and original OpenMath paraphrases are published; proof PDFs remain at their source.
Stephen Cook · Published
Published author exposition or original paper.
Revision & attribution
Source revision: official-description-sha256-018f3d473d16. Observed Oct 10, 2026.
AI involvement is not reported by this cited source; this is not a claim of its absence.
History
Statement versions
Statement versions and problem publication versions are separate. A new publication can update context or sources while retaining the same statement.
Assessment history · 1
Assessments retained in these publication records. Earlier statement versions retain their own source histories.
Open
Unresolved assertion in the cited scope ↗
Observed Oct 10, 2026 · Source passage →
Record details
- Claim ID
- claim-pnp-vp-vnp
- Statement version
- 1
- SHA-256
- 985cd77da3c4769da0fc81b1791aee2982f53a7e27262b3d427b2ad64254ad33