Symmetry of information and nonuniform lower bounds
CSR'07 Proceedings of the Second international conference on Computer Science: theory and applications
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Kolmogorov complexity is a measure that describes the compressibility of a string. Strings with low complexity contain a lot of redundancy, while strings with high Kolmogorov complexity seem to lack any kind of pattern. For instance, a string such as 5555 5555 5555 5555 5555 has low complexity, while a sequence such as 1732 7356 2748 7621 6552 would have high complexity. This thesis studies different notions of resource-bounded Kolmogorov complexity. In particular it studies Levin's Kt complexity and measures Kμ that are defined in a similar manner. Levin defined his measure as Kt( x) = min{|d| + logt | U(d) = x in t steps} where U is a universal Turing machine. It is shown that, contrary to common intuition, the measures Kμ behave differently from the resource-unbounded Kolmogorov complexity, even for generous resource bounds. In particular it is argued that a property called Symmetry of Information does not hold for some of these measures Kμ. One of the main results of this thesis addresses the question of the complexity of computing the measure Kt(x) for a given string x. It can be computed in exponential time, but no meaningful lower bound is known. However, it is shown that it is complete for exponential time under efficient, non-uniform reductions (i.e., reductions computable in P/poly) as well as nondeterministic polynomial time reductions (i.e., reductions computable in NP). Further completeness results of other complexity measures for different complexity classes are obtained as well. These results are of interest as the problem of computing the Kolmogorov complexity of a string is not a typical complete problem. Most problems that are complete for a complexity classes have a clear combinatorial structure representative of the complexity class they reside in. However, the problems studied seem to lack that property. This thesis also studies the relation between (a) the ability of sets in certain complexity classes to avoid simple strings and (b) the inclusion relation between different complexity classes. For instance, it is shown that every set in P contains simple strings, if and only if NEXP ⊆ P/poly.