This question is about *sophistication*, a way of measuring the amount of "interesting, non-random information" in a binary string, which was proposed by Kolmogorov and others in the 1980s. I'll define all the needed concepts below, but for further reading, I recommend this paper by Antunes and Fortnow, this PhD thesis by Antunes, or this paper by Gacs, Tromp, and Vitanyi.

Given an n-bit string x, recall that K(x), the *Kolmogorov complexity* of x, is the length in bits of the shortest program p (in some fixed universal programming language) such that p()=x: that is, p halts and outputs x when given a blank input. Given a set S ⊆ {0,1}^{n}, one can also define K(S) to be the length in bits of the shortest program that outputs the 2^{n}-bit characteristic sequence of S. Finally, one can define K(x|S) to be the length in bits of the shortest program p such that p(S)=x: that is, p halts and outputs x when fed the characteristic sequence of S as input.

The "problem" with Kolmogorov complexity is that it's maximized by random strings, which are intuitively not very "complex" at all. This motivates the following alternatives to K(x):

Given an n-bit string x and a constant c>0, the oxymoronically-named *naïve sophistication* of x, or NSoph_{c}(x), is the smallest possible value of K(S), over all sets S ⊆ {0,1}^{n} such that x∈S and K(x|S) ≥ log_{2}|S| - c. Intuitively, NSoph measures the minimum number of bits needed to specify a set of which x is an incompressible or Kolmogorov-random element. I call it "naïve" because it's the first measure I would think of that's sort of like Kolmogorov complexity but small for random strings (small because for random strings, one can take S={0,1}^{n}, whence NSoph_{c}(x)=O(1)).

Meanwhile, the *coarse sophistication* of x or CSoph(x), defined by Antunes, is the smallest possible value of 2K(S)+log_{2}|S|-K(x), over all sets S ⊆ {0,1}^{n} such that x∈S. Intuitively, CSoph measures the minimum number of bits needed to specify x via a "two-part code," where the first part specifies a set S containing x, the second part gives the index of x in S, and a penalty gets applied both for K(S) (the length of the first part of the code) and for K(S)+log_{2}|S|-K(x) (the amount by which the total code length exceeds K(x)). Despite the unwieldy definition, Antunes amasses evidence that CSoph is in various ways the "right" measure of the non-random information in a string.

My question is now the following:

Let c=O(1). Is NSoph_{c}(x), my "unsophisticated kind of sophistication," always close to CSoph(x), Antunes' "sophisticated kind of sophistication"? Or can there be a large gap between the two? If so, how large?

Here's what I know about this question:

- CSoph(x) ≤ 2NSoph
_{c}(x)+c. To see this: let the set S minimize K(S) subject to x∈S and K(x|S) ≥ log_{2}|S| - c. Then CSoph(x) ≤ 2K(S)+log_{2}|S|-K(x) ≤ 2NSoph_{c}(x)+log_{2}|S|-K(x) ≤ 2NSoph_{c}(x)+log_{2}|S|-K(x|S) ≤ 2NSoph_{c}(x)+c. - NSoph
_{c}(x)*can*be about twice as large as CSoph(x). To see this: first, as observed by Antunes, if x is an n-bit string, then CSoph(x) never exceeds n/2+o(n). (For we can always achieve that bound by setting S={x} if K(x)≤n/2, or S={0,1}^{n}if K(x)>n/2.) Second, as discussed by Gacs, Tromp, Vitanyi, it's possible to construct what Kolmogorov called "absolutely non-random objects," meaning n-bit strings x such that K(x|S) ≤ log_{2}|S| - O(1) whenever K(S) ≤ n - clog(n). For these strings, we clearly have NSoph_{c}(x) ≥ n-O(log n) if c=O(1). Combining now yields the result.

As a final note, NSoph and CSoph are *both* different from the "ordinary sophistication" Soph, which is defined as follows: Soph_{c}(x) is the smallest possible value of K(S), over all sets S ⊆ {0,1}^{n} such that x∈S and K(S) + log_{2}|S| ≤ K(x)+c. Intuitively, Soph_{c}(x) measures the minimum number of bits needed for the *first* part of a near-minimal two-part code specifying the string x. One can observe the following (I'll give details on request):

- NSoph
_{c}(x) ≤ Soph_{c}(x) - CSoph(x) ≤ Soph
_{c}(x)+c - There exist strings x for which Soph
_{c}(x) is very large but NSoph_{c}(x) and CSoph(x) are both very small.

I'll also observe that NSoph_{c}(x), CSoph(x), and Soph_{c}(x) are all upper-bounded by the Kolmogorov complexity K(x) (or rather, by K(x)+c).

**Update:** Sorry, just minutes after writing this post, I think I see the answer to *one* direction of my problem! Consider the "absolutely non-random objects" x discussed above. These objects satisfy K(x) ≥ NSoph_{c}(x) ≥ n-O(log n). But precisely *because* their Kolmogorov complexity is so large, they should also satisfy CSoph(x)=O(log n), achieved by setting S={0,1}^{n}. On the other hand, I still don't know whether CSoph(x) can ever be larger than NSoph_{c}(x) (only that, if so, it's never more than a factor of 2 larger). And I'd still be extremely interested if anyone could answer that question.