Abstract
One powerful theme in complexity theory and pseudorandomness in the past few decades has been the use of lower bounds to give pseudorandom generators (PRGs). However, the general results using this hardness vs. randomness paradigm suffer from a quantitative loss in parameters, and hence do not give nontrivial implications for models where we don’t know super-polynomial lower bounds but do know lower bounds of a fixed polynomial. We show that when such lower bounds are proved using random restrictions, we can construct PRGs which are essentially best possible without in turn improving the lower bounds.
More specifically, say that a circuit family has shrinkage exponent Γ if a random restriction leaving a p fraction of variables unset shrinks the size of any circuit in the family by a factor of pΓ + o(1). Our PRG uses a seed of length s1/(Γ + 1) + o(1) to fool circuits in the family of size s. By using this generic construction, we get PRGs with polynomially small error for the following classes of circuits of size s and with the following seed lengths:
(1) For de Morgan formulas, seed length s1/3+o(1);
(2) For formulas over an arbitrary basis, seed length s1/2+o(1);
(3) For read-once de Morgan formulas, seed length s.234...;
(4) For branching programs of size s, seed length s1/2+o(1).
The previous best PRGs known for these classes used seeds of length bigger than n/2 to output n bits, and worked only for size s=O(n) [8].
- Miklós Ajtai and Avi Wigderson. 1985. Deterministic simulation of probabilistic constant depth circuits (preliminary version). In Proceedings of the 26th Annual IEEE Symposium on Foundations of Computer Science. 11--19. Google ScholarDigital Library
- Noga Alon, László Babai, and Alon Itai. 1986. A fast and simple randomized parallel algorithm for the maximal independent set problem. J. Algorithms 7, 4 (1986), 567--583. Google ScholarDigital Library
- N. Alon and J. H. Spencer. 2011. The Probabilistic Method. Wiley. Google ScholarDigital Library
- A. E. Andreev. 1987. On a method for obtaining more than quadratic effective lower bounds for the complexity of π-schemes. Moscow Univ. Math. Bull. 42, 1 (1987), 63--66.Google Scholar
- Sergei Artemenko and Ronen Shaltiel. 2011. Lower bounds on the query complexity of non-uniform and adaptive reductions showing hardness amplification. In Proceedings of APPROX-RANDOM. 377--388. Google ScholarDigital Library
- László Babai, Lance Fortnow, Noam Nisan, and Avi Wigderson. 1993. BPP has subexponential time simulations unless EXPTIME has publishable proofs. Computational Complexity 3 (1993), 307--318. Google ScholarDigital Library
- Manuel Blum and Silvio Micali. 1984. How to generate cryptographically strong sequences of pseudo-random bits. SIAM J. Comput. 13, 4 (1984), 850--864. Google ScholarDigital Library
- Andrej Bogdanov, Periklis A. Papakonstantinou, and Andrew Wan. 2011. Pseudorandomness for read-once formulas. In Proceedings of the 52nd Annual IEEE Symposium on Foundations of Computer Science. 240--246. Google ScholarDigital Library
- Eshan Chattopadhyay, Pooya Hatami, Omer Reingold, and Avishay Tal. 2017. Improved pseudorandomness for unordered branching programs through local monotonicity. Electronic Colloquium on Computational Complexity (ECCC) 24 (2017), 171.Google Scholar
- V. Guruswami, C. Umans, and S. Vadhan. 2009. Unbalanced expanders and randomness extractors from Parvaresh-Vardy codes. J. ACM 56 (2009), 1--34. Google ScholarDigital Library
- Dan Gutfreund and Salil P. Vadhan. 2008. Limitations of hardness vs. randomness under uniform reductions. In Proceedings of APPROX-RANDOM. 469--482. Google ScholarDigital Library
- Johan Håstad. 1998. The shrinkage exponent of de Morgan Formulas is 2. SIAM J. Comput. 27, 1 (1998), 48--64. Google ScholarDigital Library
- Johan Håstad, Alexander A. Razborov, and Andrew Chi-Chih Yao. 1995. On the shrinkage exponent for read-once formulae. Theor. Comput. Sci. 141, 1&2 (1995), 269--282. Google ScholarDigital Library
- Russell Impagliazzo, Valentine Kabanets, and Avi Wigderson. 2002. In search of an easy witness: Exponential time vs. probabilistic polynomial time. J. Comput. Syst. Sci. 65, 4 (2002), 672--694. Google ScholarDigital Library
- Russell Impagliazzo and Noam Nisan. 1993. The effect of random restrictions on formula size. Random Struct. Algorithms 4, 2 (1993), 121--134.Google ScholarCross Ref
- Russell Impagliazzo, Noam Nisan, and Avi Wigderson. 1994. Pseudorandomness for network algorithms. In Proceedings of the 26th Annual ACM Symposium on Theory of Computing. 356--364. Google ScholarDigital Library
- Valentine Kabanets and Russell Impagliazzo. 2004. Derandomizing polynomial identity tests means proving circuit lower bounds. Computational Complexity 13, 1--2 (2004), 1--46. Google ScholarDigital Library
- V. M. Khrapchenko. 1971. Complexity of the realization of a linear function in the class of π-circuits. Math. Notes Acad. Sciences USSR 9 (1971), 21--23.Google Scholar
- Ilan Komargodski and Ran Raz. 2013. Average-case lower bounds for formula size. In Proceedings of the 45th Annual ACM Symposium on Theory of Computing. 171--180. Google ScholarDigital Library
- I. Komargodski, R. Raz, and A. Tal. 2013. Improved average-case lower bounds for DeMorgan formula size. In Proceedings of the 54th Annual IEEE Symposium on Foundations of Computer Science. 588--597. Google ScholarDigital Library
- Noam Nisan. 1991. Pseudorandom bits for constant depth circuits. Combinatorica 11, 1 (1991), 63--70.Google ScholarCross Ref
- Noam Nisan. 1992. Pseudorandom generators for space-bounded computation. Combinatorica 12, 4 (1992), 449--461.Google ScholarCross Ref
- Noam Nisan and Avi Wigderson. 1994. Hardness vs Randomness. J. Comput. Syst. Sci. 49, 2 (1994), 149--167. Google ScholarDigital Library
- Noam Nisan and David Zuckerman. 1996. Randomness is linear in space. J. Comput. Syst. Sci. 52, 1 (1996), 43--52. Google ScholarDigital Library
- Mike Paterson and Uri Zwick. 1993. Shrinkage of de Morgan formulae under restriction. Random Struct. Algorithms 4, 2 (1993), 135--150.Google ScholarCross Ref
- O. Reingold, T. Steinke, and S. P. Vadhan. 2013. Pseudorandomness for regular branching programs via fourier analysis. In Proceedings of APPROX-RANDOM. 655--670.Google Scholar
- Jeanette P. Schmidt, Alan Siegel, and Aravind Srinivasan. 1995. Chernoff-Hoeffding bounds for applications with limited independence. SIAM J. Discrete Math. 8, 2 (1995), 223--250. Google ScholarDigital Library
- Ronen Shaltiel and Emanuele Viola. 2008. Hardness amplification proofs require majority. In Proceedings of the 40th Annual ACM Symposium on Theory of Computing. 589--598. Google ScholarDigital Library
- Thomas Steinke, Salil P. Vadhan, and Andrew Wan. 2014. Pseudorandomness and Fourier growth bounds for width-3 branching programs. In Proceedings of APPROX-RANDOM. 885--899.Google Scholar
- B. A. Subbotovskaya. 1961. Realizations of linear functions by formulas using +, *, −. Sov. Math. Dokl. 2 (1961), 110--112.Google Scholar
- Avishay Tal. 2014. Shrinkage of De Morgan formulae by spectral techniques. In Proceedings of the 55th Annual IEEE Symposium on Foundations of Computer Science. 551--560. Google ScholarDigital Library
- Luca Trevisan and Tongke Xue. 2013. A derandomized switching lemma and an improved derandomization of AC0. In Proceedings of the 28th Conference on Computational Complexity. 242--247.Google ScholarCross Ref
- Leslie G. Valiant. 1984. Short monotone formulae for the majority function. J. Algorithms 5, 3 (1984), 363--366.Google ScholarCross Ref
- Thomas Watson. 2011. Query complexity in errorless hardness amplification. In Proceedings of APPROX-RANDOM. 688--699. Google ScholarDigital Library
- Ryan Williams. 2010. Improving exhaustive search implies superpolynomial lower bounds. In Proceedings of the 42nd Annual ACM Symposium on Theory of Computing. 231--240. Google ScholarDigital Library
- Andrew Chi-Chih Yao. 1982. Theory and applications of trapdoor functions (extended abstract). In Proceedings of the 23rd Annual IEEE Symposium on Foundations of Computer Science. 80--91. Google Scholar
- David Zuckerman. 1997. Randomness-optimal oblivious sampling. Random Struct. Algorithms 11, 4 (1997), 345--367. Google ScholarDigital Library
Index Terms
- Pseudorandomness from Shrinkage
Recommendations
Pseudorandom generators for width-3 branching programs
STOC 2019: Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of ComputingWe construct pseudorandom generators of seed length Õ(log(n)· log(1/є)) that є-fool ordered read-once branching programs (ROBPs) of width 3 and length n. For unordered ROBPs, we construct pseudorandom generators with seed length Õ(log(n) · poly(1/є)). ...
Pseudorandomness from Shrinkage
FOCS '12: Proceedings of the 2012 IEEE 53rd Annual Symposium on Foundations of Computer ScienceOne powerful theme in complexity theory and pseudorandom ness in the past few decades has been the use lower bounds to give pseudorandom generators (PRGs). However, the general results using this hardness vs.\ randomness paradigm suffer a quantitative ...
Improved Average-Case Lower Bounds for DeMorgan Formula Size
FOCS '13: Proceedings of the 2013 IEEE 54th Annual Symposium on Foundations of Computer ScienceWe give an explicit function h:0, 1nto0, 1 such that every deMorgan formula of size n3-o(1)/r2 agrees with h on at most a fraction of 12+2-Ω(r) of the inputs. This improves the previous average-case lower bound of Komargodski and Raz (STOC, 2013). Our ...
Comments