Yes, I’m aware of this. It’s still true, I believe, that for any two finite strings s1 and s2 one can find description languages L1 and L2 (with complexity functions K1 and K2) such that
K1(s1) > K1(s2)
and
K2(s1) < K2(s2).
So there is no language-independent sense in which s1 is more complex than s2 (or vice versa). To make the claim more concrete, consider the fact that for any finite string, one could construct a Universal Turing Machine that outputs that string when given the input 0 (the string is essentially hard-coded into the structure of the machine). This corresponds to a description language in which that string has minimal K-complexity.
This is all compatible with the invariance theorem. As a simple illustration, let the constant c associated with L1 and L2 be 5, let K1(s1) be 10, K1(s2) be 9, K2(s1) be 6 and K2(s2) be 8. In this example, both the inequalities I’ve given above are true, and the invariance theorem isn’t violated.
Yes, I’m aware of this. It’s still true, I believe, that for any two finite strings s1 and s2 one can find description languages L1 and L2 (with complexity functions K1 and K2) such that
K1(s1) > K1(s2)
and
K2(s1) < K2(s2).
So there is no language-independent sense in which s1 is more complex than s2 (or vice versa). To make the claim more concrete, consider the fact that for any finite string, one could construct a Universal Turing Machine that outputs that string when given the input 0 (the string is essentially hard-coded into the structure of the machine). This corresponds to a description language in which that string has minimal K-complexity.
This is all compatible with the invariance theorem. As a simple illustration, let the constant c associated with L1 and L2 be 5, let K1(s1) be 10, K1(s2) be 9, K2(s1) be 6 and K2(s2) be 8. In this example, both the inequalities I’ve given above are true, and the invariance theorem isn’t violated.