False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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.

A pseudopolynomial algorithm is polynomial in the binary input length

Statement

A pseudopolynomial algorithm is polynomial in the binary input length.

Facts & Assumptions

Given: The standard dynamic program for SUBSET SUM that fills an $n \times T$ table for an instance with $n$ input numbers and target $T$ .

[F1]

A SUBSET SUM instance writes its target integer in binary, by Subset sum and partition decision problems.

[L1]

Worst-case running time is measured as a function of the input length, not of the numeric value of a parameter written inside that input, by Worst-case time and space complexity of a machine.

Refutation

technique · direct

1.1

Consider the one-number SUBSET SUM instances $([2^{m}], 2^{m})$ for $m \geq 1$ . By [F1], the binary input length is $O (m)$ .

F1given

2.1

The standard table-filling algorithm uses $Θ (n T) = Θ (2^{m})$ time on this family because here $n = 1$ and $T = 2^{m}$ .

step 1.1givenalgebra

3.1

Since $2^{m}$ is exponential in the binary length $m$ , the running time in step 2.1 is not polynomial in the input length. By [L1], polynomial-time complexity is measured against that binary length. Therefore pseudopolynomial dependence on $T$ does not imply polynomial dependence on the bit-length of $T$ .

L1step 1.1step 2.1

4.1

The statement is false.

step 3.1∎

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

Fabrizio Iozzi and Luca Trevisan, Handout NP2 (standard reference, not scraped)
Sanjeev Arora and Boaz Barak, Computational Complexity: A Modern Approach (standard reference, not scraped)