False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-12

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.

FALSE: distributing an arbitrary formula into CNF always has polynomial size

Statement

Distributing an arbitrary Boolean formula into CNF always increases its size by at most a polynomial factor.

Facts & Assumptions

Given: The formula $φ_{n} : = ⋁_{i = 1}^{n} (a_{i} \land b_{i})$ .

[L1]

The page's CNF and satisfiability language are the ones from Boolean formulas, conjunctive normal form, and the satisfiability language SAT.

[L2]

The Tseitin transformation is kept precisely because it preserves satisfiability with only linear blowup, by The Tseitin transformation has linear size and preserves satisfiability.

Refutation

technique · direct

1.1L1givenalgebra

Fully distributing $φ_{n}$ into CNF chooses one literal from each conjunctive pair $(a_{i} \land b_{i})$ , so the resulting CNF has one clause for every function ${1, \dots, n} \to {a_{i}, b_{i}}$ . That is $2^{n}$ distinct clauses.

2.1L2step 1.1∎

Since the original formula has size $O (n)$ but the distributed CNF has $2^{n}$ clauses, the size blowup is exponential, not polynomial. Step 1.1 is exactly why [L2] keeps Tseitin variables available. Therefore the statement is false.

Depends on

Used by

Nothing in the library uses this result yet.

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

Sanjeev Arora and Boaz Barak, Computational Complexity: A Modern Approach (standard reference, not scraped)