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.
The class P
Definition
The class P is
Thus a language belongs to exactly when some deterministic Turing machine decides it in time bounded by a polynomial in the input length.
Remarks
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The exponent may depend on the language, but it is fixed for the chosen decider.
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On this page, the underlying machine model is the deterministic multitape model from The classes DTIME, NTIME, DSPACE, and NSPACE.
Depends on
Used by
- P is properly contained in EXP Corollary
- Polynomial hierarchy levels Definition
- Polynomial time with a Sharp-P oracle Definition
- Polynomial-time many-one reductions Definition
- Polynomial-time verifiers with polynomially bounded certificates Definition
- Relativized complexity class Definition
- The class NP via polynomial-time verifiers Definition
- Hierarchy theorems do not separate P from NP Proposition
- P ⊆ NP ∩ coNP Proposition
- Every polynomial-time language has polynomial-size circuits Theorem
- Max-3SAT has no PTAS unless P=NP Theorem
- Maximum independent set has no PTAS unless P=NP Theorem
- Polynomial-time many-one reductions transfer P-, NP-, and coNP-membership Theorem
Dependency tree · two levels
3 results within two dependency steps 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
- Michael Sipser, MIT 18.404J Theory of Computation, Lecture 14: P and NP, SAT, Poly-time Reducibility (standard reference, not scraped)
- Sanjeev Arora and Boaz Barak, Computational Complexity: A Modern Approach (standard reference, not scraped)