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

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False: Kolmogorov complexity is an absolute integer

Statement

The Kolmogorov complexity of a finite string is an absolute integer independent of the optimal machine.

Refutation

Given: an optimal machine $U$ .

1.1

Choose a finite word $z$ with $C_{U} (z) > 1$ ; such a word exists because there are infinitely many finite words but only three programs of length at most $1$ . Define $V (0 p) = U (p)$ and $V (1) = z$ , undefined otherwise. Then $V$ is optimal because it simulates $U$ with one extra bit.

given

2.1

But $C_{V} (z) = 1 < C_{U} (z)$ . The invariance theorem Invariance theorem for plain Kolmogorov complexity asserts only additive-constant agreement, so the absolute claim is false.

step 1.1given∎

Depends on

Invariance theorem for plain Kolmogorov complexity

Used by

Changing an optimal machine changes finite-string complexity Counterexample

Dependency tree · two levels

2 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

Shen, §3 (standard reference, not scraped)