Is Kolmogorov-random nonsensical for small numbers?

The Wikipedia definition of Kolmogorov-random defines a string (usually of bits) as being random if and only if it is shorter than any computer program that can produce that string.

Aren't nearly1 all strings shorter than a certain length2 going to be Kolmogorov-random?

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If a random Turing machine is chosen, then programs that output short strings will almost always be longer than the strings. If the programming language used for measurement is optimized for outputting a short string, then only the complexity of a subset of short strings can be made lower than their length. For example, if a Turing complete language defines $0$ as an operation that outputs 0 and halts, the program for outputting 1 and halting is forced to be longer than the length of its output.

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The exact length below which this occurs will depend on the programming language chosen. The length is 41 bits using 5 state, 2 symbol Turing machines.

Is Kolmogorov-random nonsensical for small numbers?

Although one could pick a specific description language to make Kolmogorov complexity into

a function from ​ {0,1}* ​ to ​ {0,1,2,3,...} , ​ most people don't pick as described ​ ( 0 , 1 , 2 , 3 ) ,

so basically:

Kolmogorov complexity is only defined up to an additive O(1), so one

needs an infinite set of strings to make sense of Kolmogorov randomness.

In particular, yes, Kolmogorov-randomness is nonsensical for small numbers.

StackExchange Computer Science Q#75220, answer score: 5