Abstract
How powerful is the set of random strings? What can one say about a set A that is efficiently reducible to R, the set of Kolmogorov-random strings? We present the first upper bound on the class of computable sets in \(\mbox{\rm P}^R\) and \(\mbox{\rm NP}^R\).
The two most widely-studied notions of Kolmogorov complexity are the “plain” complexity C(x) and “prefix” complexity K(x); this gives two ways to define the set “R”: R C and R K . (Of course, each different choice of universal Turing machine U in the definition of C and K yields another variant \({R_{{C}_U}}\) or \({R_{{K}_U}}\).) Previous work on the power of “R” (for any of these variants [1,2,9]) has shown
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\(\mbox{\rm BPP} \subseteq \{A : A \mbox{$\leq^{\rm p}_{\it tt}$} R\}\).
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\(\mbox{\rm PSPACE} \subseteq \mbox{\rm P}^R\).
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\(\mbox{\rm NEXP} \subseteq \mbox{\rm NP}^R\).
Since these inclusions hold irrespective of low-level details of how “R” is defined, we have e.g.: \(\mbox{\rm NEXP} \subseteq \mbox{$\Delta^0_1$} \cap \bigcap_U \mbox{\rm NP}^{{R_{{K}_U}}}\). (\(\mbox{$\Delta^0_1$}\) is the class of computable sets.)
Our main contribution is to present the first upper bounds on the complexity of sets that are efficiently reducible to \({R_{{K}_U}}\). We show:
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\(\mbox{\rm BPP} \subseteq \mbox{$\Delta^0_1$} \cap \bigcap_U \{A : A \mbox{$\leq^{\rm p}_{\it tt}$} {R_{{K}_U}}\}\subseteq \mbox{\rm PSPACE}\).
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\(\mbox{\rm NEXP} \subseteq \mbox{$\Delta^0_1$} \cap \bigcap_U \mbox{\rm NP}^{{R_{{K}_U}}}\subseteq \mbox{\rm EXPSPACE}\).
Hence, in particular, \(\mbox{\rm PSPACE}\) is sandwiched between the class of sets Turing- and truth-table-reducible to R.
As a side-product, we obtain new insight into the limits of techniques for derandomization from uniform hardness assumptions.
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Allender, E., Friedman, L., Gasarch, W. (2011). Limits on the Computational Power of Random Strings. In: Aceto, L., Henzinger, M., Sgall, J. (eds) Automata, Languages and Programming. ICALP 2011. Lecture Notes in Computer Science, vol 6755. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22006-7_25
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