Compression Complexity

The Kolmogorov complexity of x, denoted C(x), is the length of the shortest program that generates x. For such a simple definition, Kolmogorov complexity has a rich and deep theory, as well as applications to a wide variety of topics including learning theory, complexity lower bounds and SAT algorithms. Kolmogorov complexity typically focuses on decompression, going from the compressed program to the original string. This paper develops a dual notion of compression, the mapping from a string to its compressed version. Typical lossless compression algorithms such as Lempel-Ziv or Huffman Encoding always produce a string that will decompress to the original. We define a general compression concept based on this observation. For every m, we exhibit a single compression algorithm q of length about m which for n and strings x of length n ≥ m, the output of q will have length within n−m+O(1) bits of C(x). We also show this bound is tight in a strong way, for every n ≥ m there is an x of length n with C(x) ≈ m such that no compression program of size slightly less than m can compress x at all. We also consider a polynomial time-bounded version of compression complexity and show that similar results for this version would rule out cryptographic one-way functions.