Martin-Löf: universal effective tests for random sequences

What changed

Scientific scope

Martin-Löf constructs universal effective tests and defines infinite random binary sequences by avoidance of effectively specified null tests, connecting finite-string randomness to conditional description complexity. The universal quantifier is over recursively enumerable tests with stated significance bounds, not every conceivable arbitrary set or a practical finite battery. Fair-coin tests, tests for a fixed computable distribution, and conditional Bernoulli tests are distinct constructions. Finite critical levels depend on the test up to an additive constant; the infinite definition is invariant under universal-test choice. Finite complexity uses plain conditional program length given length/counts, not later prefix-free characterizations silently substituted. Almost-sure statements apply to the declared measure or Bernoulli class, not causal trust, scientific correctness or cryptographic unpredictability. Any finite prefix admits a computable nonrandom extension, so finite successful tests cannot certify infinite randomness. The final convergence proof sketches a recursive bound rather than fully displaying it. Full proof/glyph certification and experiment replay were not performed.

Evidence

Source-linked, scoped account of the original work. The published scope states the scientific assumptions and remaining verification limits. WRS has not independently certified the full proof, experimental result or system benchmark.

Contributors

Per Martin-Löf · SCIENCE: Original source-listed scientific author of this scoped work; byline does not determine effort fractions.

Per Martin-Löf, The Definition of Random Sequences (Information and Control 9,602-619, December 1966), original article scan

Provisional WRS assessment, edition 2026-10-16.1. Independent expert calibration is pending. This grade does not certify correctness or assign ownership.

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