DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-02

How 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.

Polynomial-time many-one reductions

Definition

Let $A \subseteq Σ^{*}$ and $B \subseteq Γ^{*}$ be languages over finite alphabets. A polynomial-time many-one reduction from $A$ to $B$ is a total function $f : Σ^{*} \to Γ^{*}$ such that:

$f$ is computable by a deterministic Turing machine in polynomial time, and
for every $x \in Σ^{*}$ , $x \in A ⟺ f (x) \in B .$

When such an $f$ exists, write $A \leq_{p} B .$

Remarks

"Many-one" means that the source instance is transformed once into a single target instance.
This is not the same notion as an oracle or Turing reduction, where later queries may depend on earlier answers.

Depends on

Used by

A forward-only mapping that is not a correct many-one reduction Counterexample
Gap promise problems and gap-preserving reductions Definition
NP-hard and NP-complete languages Definition
Reduction gadgets and interface invariants Definition
A gadget reduction is correct as soon as every yes-instance maps to a yes-instance False statement
Every decision many-one reduction is parsimonious False statement
Polynomial-time many-one reductions compose Lemma
3-SAT is NP-complete Theorem
3SAT polynomial-time many-one reduces to 3-COLORABILITY Theorem
3SAT polynomial-time many-one reduces to CLIQUE Theorem
3SAT polynomial-time many-one reduces to directed Hamiltonian cycle Theorem
3SAT polynomial-time many-one reduces to subset sum Theorem
Bounded alternation qbf is level complete Theorem
CircuitSAT is NP-complete Theorem
CLIQUE, INDEPENDENT SET, and VERTEX COVER are polynomially interreducible Theorem
Constant-gap binary CSP is NP-hard Theorem
Directed Hamiltonian cycle polynomial-time many-one reduces to undirected Hamiltonian cycle Theorem
Max-3SAT has no PTAS unless P=NP Theorem
Polynomial-time many-one reductions transfer P-, NP-, and coNP-membership Theorem
SAT is NP-complete Theorem
Subset sum polynomial-time many-one reduces to partition Theorem
TQBF is PSPACE-complete Theorem
Vertex cover polynomial-time many-one reduces to set cover Theorem

Dependency tree · two levels

6 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

Stephen A. Cook, The Complexity of Theorem-Proving Procedures (standard reference, not scraped)
Sanjeev Arora and Boaz Barak, Computational Complexity: A Modern Approach (standard reference, not scraped)