复杂度和具有最大简单复杂度和前缀复杂度的字符串的复杂度
Complexity of complexity and strings with maximal plain and prefix Kolmogorov complexity
摘要：
Peter Gacs证明（Gacs 1974），对于每一个n，存在一个长度为n的位串x，其简单复杂度C（x）相对于x几乎具有最大条件复杂度，即，C（C（x））|x）> logn-log ^（2）n-O（1）. (Here log ^（2）i = log log i.）在Elena Kalinina（Kalinina 2011）之后，我们提供了一个简单的基于博弈的证明;修改她的论点，我们得到了一个更好的（和紧的）界log n-O（1）。我们也展示了相同的无前缀复杂性的界限。Robert Solovay（Solovay 1975）证明了无限多个字符串x具有最大的普通复杂度，但不是最大的前缀复杂度（在相同长度的字符串中）：对于某个c，存在无限多个x使得|x|-C（x）<c和|x| + K（|x|）-K（x）> log ^（2）|x|-c log ^（3）|x|。事实上，Solovay和Gacs的结果是密切相关的。利用上述结果，我们对Solovay的结果给出了一个简短的证明.我们还推广了它，证明了对某些c和对所有n，存在长度为n的字符串x，使得n-C（x）<c和n + K（n）-K（x）> K（K（n））.| n）-3 K（K（K（n）|n）的范围|n）-c.我们还证明了一个封闭的上界K（K（n））|n）+O（1）.最后，我们提供了一个直接的游戏证明约瑟夫米勒的推广（米勒2006年）相同的Solovay的定理：如果一个可共混集（一个集与c.e.补数）对于每个长度都包含一个这个长度的字符串，那么它包含无限多个字符串x，使得|x| + K（|x|）-K（x）> log ^（2）|x| + O（log ^（3）|x|）。
英文摘要：
Peter Gacs showed (Gacs 1974) that for every n there exists a bit string x of length n whose plain complexity C(x) has almost maximal conditional complexity relative to x, i.e., C(C(x)|x) > log n - log^(2) n - O(1). (Here log^(2) i = log log i.) Following Elena Kalinina (Kalinina 2011), we provide a simple game-based proof of this result; modifying her argument, we get a better (and tight) bound log n - O(1). We also show the same bound for prefix-free complexity. Robert Solovay showed (Solovay 1975) that infinitely many strings x have maximal plain complexity but not maximal prefix complexity (among the strings of the same length): for some c there exist infinitely many x such that |x| - C(x) < c and |x| + K(|x|) - K(x) > log^(2) |x| - c log^(3) |x|. In fact, the results of Solovay and Gacs are closely related. Using the result above, we provide a short proof for Solovay's result. We also generalize it by showing that for some c and for all n there are strings x of length n with n - C (x) < c and n + K(n) - K(x) > K(K(n)|n) - 3 K(K(K(n)|n)|n) - c. We also prove a close upper bound K(K(n)|n) + O(1). Finally, we provide a direct game proof for Joseph Miller's generalization (Miller 2006) of the same Solovay's theorem: if a co-enumerable set (a set with c.e. complement) contains for every length a string of this length, then it contains infinitely many strings x such that |x| + K(|x|) - K(x) > log^(2) |x| + O(log^(3) |x|).
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