The area of derandomization attempts to provide efficient deterministic simulations of randomized algorithms in various algorithmic settings. Goldreich and Wigderson introduced a notion of "typically-correct" deterministic simulations, which are allowed to err on few inputs. In this paper, we further the study of typically-correct derandomization in two ways. First, we develop a generic approach for constructing typically-correct derandomizations based on seed-extending pseudorandom generators, which are pseudorandom generators that reveal their seed. We use our approach to obtain both conditional and unconditional typically-correct derandomization results in various algorithmic settings. We show that our technique strictly generalizes an earlier approach by Shaltiel based on randomness extractors and simplifies the proofs of some known results. We also demonstrate that our approach is applicable in algorithmic settings where earlier work did not apply. For example, we present a typically-correct polynomial-time simulation for every language in BPP based on a hardness assumption that is (seemingly) weaker than the ones used in earlier work. Second, we investigate whether typically-correct derandomization of BPP implies circuit lower bounds. Extending the work of Kabanets and Impagliazzo for the zero-error case, we establish a positive answer for error rates in the range considered by Goldreich and Wigderson. In doing so, we provide a simpler proof of the zero-error result. Our proof scales better than the original one and does not rely on the result by Impagliazzo, Kabanets, and Wigderson that NEXP having polynomialsize circuits implies that NEXP coincides with EXP.
Bibliographical noteFunding Information:
A significant portion of this work was completed while the first author was a graduate student at the University of Wisconsin-Madison and while the second author was visiting the University of Haifa, the Weizmann Institute of Science, and Humboldt University in Berlin. Portions of this work were completed while the first author was supported by NSF award CCF-0728809, by a Cisco Systems Distinguished Graduate Fellowship, and Indiana State University Research Committee award #11-07, while the second author was supported by NSF awards CCF-0728809 and CCF-1017597 and by the Humboldt Foundation, and while the third author was supported by BSF grants 2004329 and 2010120 and ISF grants 686/07 and 864/11 and ERC grant 279559.
- Typically-correct derandomization
- circuit lower bounds
- pseudorandom generators
- randomized algorithms
ASJC Scopus subject areas
- Theoretical Computer Science
- Mathematics (all)
- Computational Theory and Mathematics
- Computational Mathematics