TY - GEN
T1 - Randomness - A computational complexity perspective
AU - Wigderson, Avi
PY - 2008
Y1 - 2008
N2 - Man has grappled with the meaning and utility of randomness for centuries. Research in the Theory of Computation in the last thirty years has enriched this study considerably. This lecture will describe two main aspects of this research on randomness, demonstrating its power and weakness respectively. Randomness is paramount to computational efficiency: The use of randomness seems to dramatically enhance computation (and do other wonders) for a variety of problems and settings. In particular, examples will be given of probabilistic algorithms (with tiny error) for natural tasks in different areas, which are exponentially faster than their (best known) deterministic counterparts. Computational efficiency is paramount to understanding randomness: We will explain the computationally-motivated definition of "pseudorandom" distributions, namely ones which cannot be distin- guished from the uniform distribution by any efficient procedure from a given class. Using this definition, we show how such pseudorandomness may be generated deterministically, from (appropriate) computationally difficult problems. Consequently, randomness is probably not as powerful as it seems above. We conclude with the power of randomness in other computational settings, such as space complexity and probabilistic proof systems. In particular we'll discuss the remarkable properties of Zero-Knowledge proofs and of Probabilistically Checkable proofs. The bibliography contains several useful books and surveys in which material pertaining to the computational randomness may be found. In particular, we include surveys on topics not covered in the lecture, including extractors (designed to purify weak random sources) and expander graphs (perhaps the most useful "pseudorandom" object).
AB - Man has grappled with the meaning and utility of randomness for centuries. Research in the Theory of Computation in the last thirty years has enriched this study considerably. This lecture will describe two main aspects of this research on randomness, demonstrating its power and weakness respectively. Randomness is paramount to computational efficiency: The use of randomness seems to dramatically enhance computation (and do other wonders) for a variety of problems and settings. In particular, examples will be given of probabilistic algorithms (with tiny error) for natural tasks in different areas, which are exponentially faster than their (best known) deterministic counterparts. Computational efficiency is paramount to understanding randomness: We will explain the computationally-motivated definition of "pseudorandom" distributions, namely ones which cannot be distin- guished from the uniform distribution by any efficient procedure from a given class. Using this definition, we show how such pseudorandomness may be generated deterministically, from (appropriate) computationally difficult problems. Consequently, randomness is probably not as powerful as it seems above. We conclude with the power of randomness in other computational settings, such as space complexity and probabilistic proof systems. In particular we'll discuss the remarkable properties of Zero-Knowledge proofs and of Probabilistically Checkable proofs. The bibliography contains several useful books and surveys in which material pertaining to the computational randomness may be found. In particular, we include surveys on topics not covered in the lecture, including extractors (designed to purify weak random sources) and expander graphs (perhaps the most useful "pseudorandom" object).
KW - Complexity
KW - Derandomnization
KW - Pseudorandom
KW - Randomness
UR - http://www.scopus.com/inward/record.url?scp=44649100862&partnerID=8YFLogxK
U2 - 10.1007/978-3-540-79709-8_1
DO - 10.1007/978-3-540-79709-8_1
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AN - SCOPUS:44649100862
SN - 3540797084
SN - 9783540797081
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 1
EP - 2
BT - Computer Science - Theory and Applications - Third International Computer Science Symposium in Russia, CSR 2008, Proceedings
T2 - 3rd International Computer Science Symposium in Russia, CSR 2008
Y2 - 7 June 2008 through 12 June 2008
ER -