In a World of P=BPP

• Published in: Studies in Complexity and Cryptography. Miscellanea on the Interplay between Randomness and Computation pp. 191 - 232
• Main Author: Goldreich, Oded
• Format: Book Chapter
• Language: English
• Published: Berlin, Heidelberg Springer Berlin Heidelberg 2011
• Series: Lecture Notes in Computer Science
• Subjects: BPP; derandomization; FPTAS; promise problems; pseudorandom generators; randomized constructions; search problems
• ISBN: 3642226698; 9783642226694
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-642-22670-0_20

We show that proving results such as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{BPP}=\mathcal{P}$\end{document} essentially necessitate the construction of suitable pseudorandom generators (i.e., generators that suffice for such derandomization results). In particular, the main incarnation of this equivalence refers to the standard notion of uniform derandomization and to the corresponding pseudorandom generators (i.e., the standard uniform notion of “canonical derandomizers”). This equivalence bypasses the question of which hardness assumptions are required for establishing such derandomization results, which has received considerable attention in the last decade or so (starting with Impagliazzo and Wigderson [JCSS, 2001]). We also identify a natural class of search problems that can be solved by deterministic polynomial-time reductions to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{BPP}$\end{document}. This result is instrumental to the construction of the aforementioned pseudorandom generators (based on the assumption \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{BPP}=\mathcal{P}$\end{document}), which is actually a reduction of the “construction problem” to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{BPP}$\end{document}. Caveat: Throughout the text, we abuse standard notation by letting \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{BPP},\mathcal{P}$\end{document} etc denote classes of promise problems. We are aware of the possibility that this choice may annoy some readers, but believe that promise problem actually provide the most adequate formulation of natural decisional problems.