False statementConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07

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Every decision many-one reduction is parsimonious

Statement

Every polynomial-time many-one reduction between decision problems preserves the exact number of witnesses.

Facts & Assumptions

Given: a formula $θ$ with declared variables $x_{1}, \dots, x_{n}$ and a fresh variable $y$ .

[L1]

A decision many-one reduction need preserve only membership, by Polynomial-time many-one reductions.

[L2]

A parsimonious reduction preserves exact counts, by Parsimonious reductions between counting functions.

[L3]

NumberSAT counts assignments to every declared variable, including an unused one, by NumberSAT.

Refutation

technique · direct

1.1

Define $r (θ) = θ \land (y \lor \neg y)$ and declare $y$ in addition to the original variables. The map is polynomial time, and $θ$ is satisfiable iff $r (θ)$ is satisfiable, so it is a SAT-to-SAT many-one reduction by [L1].

L1givenconstruct

2.1

Each satisfying assignment of $θ$ has exactly two extensions, one for each value of $y$ . Hence [L3] gives $N u m b e r S A T (r (θ)) = 2 N u m b e r S A T (θ)$ . For the concrete input $θ = x_{1}$ , the counts are $1$ and $2$ , so the map is not parsimonious under [L2].

L2L3step 1.1algebra

3.1

This one polynomial-time decision reduction refutes the universal claim.

step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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

Arora and Barak, Computational Complexity: A Modern Approach (standard reference, not scraped)
Lance Fortnow, Counting Complexity (standard reference, not scraped)