Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Circuit families and P/poly
Definition
A circuit family is a sequence (C_n) with exactly n inputs at length n. It recognizes L if C_n(x)=1 exactly when x belongs to L for every n-bit x. P/poly contains exactly the languages recognized by families with polynomial size; no algorithm that constructs C_n from n is required.
Depends on
Used by
- Polynomial-size circuits imply a uniform polynomial-time generator False statement
- Every polynomial-time language has polynomial-size circuits Theorem
- P/poly equals polynomial time with polynomial advice Theorem
- Some undecidable languages have polynomial-size circuits Theorem
- The Karp--Lipton collapse Theorem
Dependency tree · one level
1 result within one dependency step of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Arora and Barak, Computational Complexity: A Modern Approach (standard reference, not scraped)
- Lance Fortnow, Counting Complexity (standard reference, not scraped)