Curiouser and Curiouser: The Link between Incompressibility and Complexity

@inproceedings{Allender2012CuriouserAC,
  title={Curiouser and Curiouser: The Link between Incompressibility and Complexity},
  author={Eric Allender},
  booktitle={CiE},
  year={2012}
}
This talk centers around some audacious conjectures that attempt to forge firm links between computational complexity classes and the study of Kolmogorov complexity. More specifically, let R denote the set of Kolmogorov-random strings. Let $\mbox{\rm BPP}$ denote the class of problems that can be solved with negligible error by probabilistic polynomial-time computations, and let $\mbox{\rm NEXP}$ denote the class of problems solvable in nondeterministic exponential time. Conjecture 1… 

On characterizations of randomized computation using plain Kolmogorov complexity

Allender, Friedman, and Gasarch recently proved an upper bound of pspace for the class DTTR K of decidable languages that are polynomial-time truth-table reducible to the set of prefix-free

On Nonadaptive Reductions to the Set of Random Strings and Its Dense Subsets

It is shown that a black-box nonadaptive randomized reduction to any distinguisher for (not only polynomial-time but also) exponential-time computable hitting set generators can be simulated in \(\mathsf {AM}\cap \mathsf{co} \mathSF {AM}\) even if there is no computational bound on a hitting set generator.

Unexpected Power of Random Strings

This paper shows that the exponential-time hierarchy EXPH can be solved in exponential time by nonadaptively asking the oracle whether a string is Kolmogorov-random or not, and provides an exact characterization of S 2 in terms of exponential- time-computable nonadaptive reductions to arbitrary dense subsets of random strings.

Unexpected hardness results for Kolmogorov complexity under uniform reductions

  • Shuichi Hirahara
  • Computer Science, Mathematics
    Electron. Colloquium Comput. Complex.
  • 2020
New proof techniques are developed for showing hardness of computing Kolmogorov complexity under surprisingly efficient reductions, which were previously conjectured to be impossible.

Reductions to the set of random strings Allender

It is shown that if a set A is reducible in polynomial time to the set of time-t-bounded Kolmogorov-random strings (for all large enough time bounds t), then A is in P/poly, and that if in addition such a reduction exists for any universal Turing machine one uses in the definition of Kolmogsorov complexity, then A are in PSPACE.

Reductions to the set of random strings

  • Allender
  • Mathematics, Computer Science
  • 2014
It is shown that if a set A is reducible in polynomial time to the set of time-t-bounded Kolmogorov-random strings (for all large enough time bounds t), then A is in P/poly, and that if in addition such a reduction exists for any universal Turing machine one uses in the definition of Kolmogsorov complexity, then A are in PSPACE.

Reductions to the Set of Random Strings: The Resource-Bounded Case

It is shown that if a set A is reducible in polynomial time to the set of time-t-bounded Kolmogorov random strings (for all large enough time bounds t), then A is in P/poly, and that if in addition such a reduction exists for any universal Turing machine one uses in the definition of Kolmogsorov complexity, then A are in PSPACE.

Reductions to the set of random strings: the resource-bounded case

It is shown that if a set A is reducible in polynomial time to the set of time-t-bounded Kolmogorov-random strings (for all large enough time bounds t), then A is in P/poly, and that if in addition such a reduction exists for any universal Turing machine one uses in the definition of Kolmogsorov complexity, then A are in PSPACE.

On Nonadaptive Security Reductions of Hitting Set Generators

It is argued that the recent worst-case to average-case reduction of Hirahara (FOCS 2018 [18]) is inherently non-black-box, without relying on any unproven assumptions, and the existence of a “non- black-box selector” for GapMCSP is exhibited.

Characterizing Average-Case Complexity of PH by Worst-Case Meta-Complexity

  • Shuichi Hirahara
  • Computer Science, Mathematics
    2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)
  • 2020
The equivalence provides fundamentally new proof techniques for analyzing average-case complexity through the lens of meta-complexity of time-bounded Kolmogorov complexity and resolves, as immediate corollaries, questions of equivalence among different notions of average- case complexity of PH.

References

SHOWING 1-10 OF 12 REFERENCES

Kolmogorov Complexity, Circuits, and the Strength of Formal Theories of Arithmetic

A collection of true statements in the language of arithmetic are presented and it is conjectured that C = BPP = P, and the possibility this might be an avenue for trying to prove the equality of BPP and P is discussed.

Derandomizing from Random Strings

It is shown that BPP is truth-table reducible to the set of Kolmogorov random strings R_K, and that strings of very high Kolmogsorov-complexity when used as advice are not much more useful than randomly chosen strings.

Power from random strings

We show that sets consisting of strings of high Kolmogorov complexity provide examples of sets that are complete for several complexity classes under probabilistic and non-uniform reductions. These

An Introduction to Kolmogorov Complexity and Its Applications

The book presents a thorough treatment of the central ideas and their applications of Kolmogorov complexity with a wide range of illustrative applications, and will be ideal for advanced undergraduate students, graduate students, and researchers in computer science, mathematics, cognitive sciences, philosophy, artificial intelligence, statistics, and physics.

Algorithmic Randomness and Complexity

This chapter discusses Randomness-Theoretic Weakness, Omega as an Operator, Complexity of C.E. Sets, and other Notions of Effective Randomness.

Computational Complexity: A Modern Approach

This beginning graduate textbook describes both recent achievements and classical results of computational complexity theory and can be used as a reference for self-study for anyone interested in complexity.

Completeness, the recursion theorem, and effectively simple sets