# 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…

## 14 Citations

### On characterizations of randomized computation using plain Kolmogorov complexity

- MathematicsComput.
- 2014

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

- Mathematics, Computer ScienceElectron. Colloquium Comput. Complex.
- 2019

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

- Computer Science, MathematicsITCS
- 2020

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

- Computer Science, MathematicsElectron. 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

- 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

- 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

- Mathematics, Computer ScienceMFCS
- 2012

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

- Mathematics, Computer ScienceMFCS 2012
- 2012

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

- Computer Science, MathematicsAPPROX-RANDOM
- 2020

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

- Computer Science, Mathematics2020 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.

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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.

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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

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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…

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

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