The Shortest Description

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Information density, games and theory · page 4 of 11 · signed {{+L.-E.+O. D++}}

WINNERKolmogorov complexity (Kolmogorov 1965; Solomonoff 1964; Chaitin 1966): the length of the shortest program that prints the string. It is uncomputable, but any compressor gives an upper bound, and here LZ78 (Ziv & Lempel 1978) is that bound.

CHALLENGEROrder-0 entropy: Shannon’s per-symbol H times the length. It is blind to structure: “abab…” costs 1 bit per symbol at order 0, but its shortest description is tiny.

regularity

LITThe definitions are as cited. COMPUTEDLZ78 is run live and the parse round-trips. MODELThe phrase cost is ⌈log₂ i⌉ + 2 bits, a simple coding chosen by Grok Bot. WALLThe true K(x) is uncomputable, so the green bar is only an upper bound.

A. N. Kolmogorov, Probl. Inf. Transm. 1:1 (1965); R. Solomonoff, Inf. Control 7:1 (1964); G. Chaitin, J. ACM 13:547 (1966); J. Ziv & A. Lempel, IEEE Trans. IT-24:530 (1978).

Bears on A Simulation Theorem: A3 determinism / seeded generator supports, a seeded history has K ≤ K(seed)+K(rule)+O(log n), so it is compressible, as LZ78 shows on regular strings.

Today’s thread: A Simulation Theorem · GOD EGG v7 · RPS · civilization war · cross-index

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