## Pseudorandom Generators, Measure Theory, and Natural Proofs (1995)

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Citations: | 29 - 4 self |

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@MISC{Regan95pseudorandomgenerators,,

author = {W. Regan and D. Sivakumar and Jin-yi Cai},

title = { Pseudorandom Generators, Measure Theory, and Natural Proofs},

year = {1995}

}

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### Abstract

We prove that if strong pseudorandom number generators exist, then the class of languages that have polynomialsized circuits (P/poly) is not measurable within exponential time, in terms of the resource-bounded measure theory of Lutz. We prove our result by showing that if P/poly has measure zero in exponential time, then there is a natural proof against P/poly, in the terminology of Razborov and Rudich [25]. We also provide a partial converse of this result.

### Citations

726 | A pseudorandom generator from any one-way function
- H˚astad, Impagliazzo, et al.
- 1999
(Show Context)
Citation Context ...Delta)-hard against uniform adversaries if for everysh(n)-time bounded probabilistic TM, M, and all but finitely many n, M achieves bias at most 1=h(n) on G n . A well-known "robustness" the=-=orem (see [5, 9]) states t-=-hat so longas `(n) = n O(1) , H(G n ) is invariant up to constant factors. As Razborov and Rudich do, we work with PSRGs that stretch n bits to 2n bits. We use "secure against" interchangeab... |

717 | Proof Verification and the Hardness of Approximation Problems
- Arora, Lund, et al.
- 1998
(Show Context)
Citation Context ...he string w as input, give them N = jwj in binary notation on their input tape, and let them query individual bits of w. (Then M is formally the same as the machines used to define the PCP classes in =-=[4, 3].) Me-=-asure time bounds in terms of n = dlog 2 Ne = jNj rather than N. Then the function d(\Delta) belongs to G(P) as defined in [1] if M runs in time n O(1) , and if every node N in the directed "depe... |

628 |
How to construct random functions
- Goldreich, Goldwasser, et al.
- 1986
(Show Context)
Citation Context ...lem is that the natural property we construct in Proposition 6 is nonuniform, and this nonuniformity carries over to the statistical test constructed in the theorem of Razborov and Rudich, drawing on =-=[7]-=-. That is, the property belongs to QP/poly. We have not been able to obtain a QP-natural property under the hypothesiss(P/polyjEXP) = 0---the sticking point is that we have not been able to enforce an... |

258 | Checking computations in polylogarithmic time
- Babai, Fortnow, et al.
- 1991
(Show Context)
Citation Context ...he string w as input, give them N = jwj in binary notation on their input tape, and let them query individual bits of w. (Then M is formally the same as the machines used to define the PCP classes in =-=[4, 3].) Me-=-asure time bounds in terms of n = dlog 2 Ne = jNj rather than N. Then the function d(\Delta) belongs to G(P) as defined in [1] if M runs in time n O(1) , and if every node N in the directed "depe... |

217 | A fast and simple randomized parallel algorithm for the maximal independent set problem - Alon, Babai, et al. - 1986 |

176 | Natural proofs
- Razborov, Rudich
- 1997
(Show Context)
Citation Context ...measures defined by Allender and Strauss [2]. We prove our theorems by showing that the martingales used in defining Lutz's measure theory yield natural properties , as defined by Razborov and Rudich =-=[26], of equiv-=-alent nonuniform complexity that diagonalize against (or "are useful against") the class C on which the martingale succeeds. An improvement of theorems in [26] due to Razborov [25] yields th... |

172 | Almost everywhere high nonuniform complexity
- Lutz
- 1992
(Show Context)
Citation Context ...tural proof against P/poly, in the terminology of Razborov and Rudich [25]. We also provide a partial converse of this result. 1 Introduction The theory of resource-bounded measure, initiated by Lutz =-=[13]-=-, provides a useful framework that links many central problems in complexity theory. Given a measure defined on a large complexity class, such as EXP = DTIME[2 n O(1) ], and a subclass C such as P, NP... |

149 |
Zufälligkeit und Wahrscheinlichkeit
- Schnorr
- 1971
(Show Context)
Citation Context ...ly discuss conditions for classes to have measure zero. This thesis and [15, 16] show that these measurability conditions can be defined in terms of martingales of the kind studied earlier by Schnorr =-=[29, 30, 31]. A martin-=-gale is a function d from f0; 1g into the nonnegative reals that satisfies the following "exact average law": for all w 2 f0; 1g , d(w) = d(w0) + d(w1) 2 : (1) Let D stand for the nonnegativ... |

91 | The quantitative structure of exponential time
- Lutz
- 1997
(Show Context)
Citation Context ... is not measurable at all. For example, P has measure zero in EXP. In fact, for any fixed c ? 0, DTIME[2 n c ] has measure zero in EXP. The class of P-bi-immune sets in EXP has measure one [19]. Lutz =-=[13, 15]-=- has advanced the hypothesis that NP does not have measure zero, which implies NP 6= P. Indeed, the hypothesis implies that NP has P-bi-immune sets, and that for everysc ? 0, there are languages in NP... |

88 |
A unified approach to the definition of random sequences
- Schnorr
- 1971
(Show Context)
Citation Context ...ly discuss conditions for classes to have measure zero. This thesis and [15, 16] show that these measurability conditions can be defined in terms of martingales of the kind studied earlier by Schnorr =-=[29, 30, 31]. A martin-=-gale is a function d from f0; 1g into the nonnegative reals that satisfies the following "exact average law": for all w 2 f0; 1g , d(w) = d(w0) + d(w1) 2 : (1) Let D stand for the nonnegativ... |

85 |
Process complexity and effective random tests
- Schnorr
- 1973
(Show Context)
Citation Context ...ly discuss conditions for classes to have measure zero. This thesis and [15, 16] show that these measurability conditions can be defined in terms of martingales of the kind studied earlier by Schnorr =-=[29, 30, 31]. A martin-=-gale is a function d from f0; 1g into the nonnegative reals that satisfies the following "exact average law": for all w 2 f0; 1g , d(w) = d(w0) + d(w1) 2 : (1) Let D stand for the nonnegativ... |

80 |
Measure and Category
- Oxtoby
- 1980
(Show Context)
Citation Context ...n ffi against non-uniform (resp. uniform) adversaries. 2.2. Resource-bounded measure The resource-bounded measure theory of Lutz [15, 16] is developed along the lines of classical measure theory (see =-=[23, 6, 24]). Languag-=-es are regarded as points in the topological space 3 whose basic open sets are the "cylinders" Cw , one for each w 2 f0; 1g , and complexity classes are point sets. The general form of Lutz'... |

79 | Efficient cryptographic schemes provably as secure as subset sum
- Impagliazzo, Naor
- 1996
(Show Context)
Citation Context .... If for some fl ? 0 there exists a one-way function of security 2 n fl , then P/poly is not measurable in EXP. Based on assumptions about the hardness of the subset-sum problem, Impagliazzo and Naor =-=[11]-=- show how to construct a pseudorandom generator in NC 1 . Razborov and Rudich note that if there is a pseudorandom function generator of exponential hardness in NC 1 , there is no P/poly-natural proof... |

69 | Efficient Checking of Polynomials and Proofs and the Hardness of Approximation Problems
- Sudan
- 1996
(Show Context)
Citation Context ...he string w as input, give them N = jwj in binary notation on their input tape, and let them query individual bits of w. (Then M is formally the same as the machines used to define the PCP classes in =-=[4, 3, 32]-=-.) Measure time bounds in terms of n = dlog 2 Ne = jN j rather than N . Then the function d(\Delta) belongs to \Gamma(P) as defined in [2] if M runs in time n O(1) , and if every node N in the directe... |

67 | Pseudorandom bits for constant depth circuits - Nisan - 1991 |

57 | Cook versus Karp-Levin: Separating completeness notions if NP is not small
- Lutz, Mayordomo
- 1996
(Show Context)
Citation Context ... implies NP 6= P. Indeed, the hypothesis implies that NP has P-bi-immune sets, and that for everysc ? 0, there are languages in NP that require deterministic time more than 2 n c . Lutz and Mayordomo =-=[17]-=- showed another plausible implication: there would be NP-complete sets under Turing (Cook) reductions that are not complete under many-one (Karp) reductions. In view of this, it is important to seek t... |

56 | Unprovability of lower bounds on circuit size in certain fragments of bounded arithmetic
- Razborov
- 1995
(Show Context)
Citation Context ...nd Rudich [26], of equivalent nonuniform complexity that diagonalize against (or "are useful against") the class C on which the martingale succeeds. An improvement of theorems in [26] due to=-= Razborov [25] yields th-=-e conclusion. The main technical problem solved in our proof is that the martingales are defined on "characteristic prefixes" that define membership in a language L of all strings up to a gi... |

53 |
Pseudo-random generators under uniform assumptions
- H˚astad
- 1990
(Show Context)
Citation Context ...r every e ? 0, for sufficiently large values of k, H(G k )s2 k e . This follows from the above three lemmas and Theorem 2. From the known equivalence of strong PSRGs and strong one-way functions (see =-=[10, 8, 9, 25]-=-), we also have: Corollary 8 If for some g ? 0 there exists a one-way function of security 2 n g , then P/poly does not have measure zero in EXP. 3.1 Non-measurability of P/poly We strengthen the conc... |

53 | Almost every set in exponential time is P-bi-immune
- Mayordomo
- 1994
(Show Context)
Citation Context ... or perhaps is not measurable at all. For example, P has measure zero in EXP. In fact, for any fixed c ? 0, DTIME[2 n c ] has measure zero in EXP. The class of P-bi-immune sets in EXP has measure one =-=[19]-=-. Lutz [13, 15] has advanced the hypothesis that NP does not have measure zero, which implies NP 6= P. Indeed, the hypothesis implies that NP has P-bi-immune sets, and that for everysc ? 0, there are ... |

49 | Measure on small complexity classes with applications for BPP
- Allender, Strauss
- 1994
(Show Context)
Citation Context ...s that "appear random" to all 2 n c time-bounded machines has measure one in EXP. Lutz [15] showed a measure-one class in which every member is a pseudorandom source for BPP, and Allender an=-=d Strauss [1]-=- extended this for measures on DTIME[2 n e ], for every e ? 0. Related work is [14, 18, 11, 12, 20, 31]. Our main theorem relates a measure question directly to PSRGs and the class P/poly of languages... |

46 | The complexity and distribution of hard problems
- Juedes, Lutz
- 1995
(Show Context)
Citation Context ...Lutz [15] showed a measure-one class in which every member is a pseudorandom source for BPP, and Allender and Strauss [1] extended this for measures on DTIME[2 n e ], for every e ? 0. Related work is =-=[14, 18, 11, 12, 20, 31]-=-. Our main theorem relates a measure question directly to PSRGs and the class P/poly of languages having polynomial-sized circuits: Theorem 1 If strong PSRGs exist, then P/poly is not measurable in EX... |

44 | Measure, stochasticity, and the density of hard languages - Lutz, Mayordomo - 1994 |

39 |
Contributions to the study of resource-bounded measure
- Mayordomo
- 1994
(Show Context)
Citation Context ... show much more far-reaching consequences from the existence of PSRGs than are currently known. 1. Introduction The theory of resource-bounded measure, initiated by Lutz [14, 15, 16] and furthered in =-=[20, 12, 17, 18, 19, 13, 1, 21]-=-, has provided a useful framework that links many central problems in complexity theory. Classes that have measure zero are small in a quantitative sense described by Lutz in [16]. Lutz et al. [16, 12... |

38 |
Measure Theory
- Doob
- 1994
(Show Context)
Citation Context ...e "secure against" interchangeably with "hard against." 2.1 Resource-bounded measure The resource-bounded measure theory of Lutz [13, 15] is developed along the lines of classical =-=measure theory (see [22, 6, 23]-=-). Languages are regarded as points in the topological space whose basic open sets are the cylindersC w , one for each w 2 f0;1g , and complexity classes are point sets. The general form of Lutz's the... |

23 |
Pseudo-random generators and complexity classes
- Boppana, Hirschfeld
- 1989
(Show Context)
Citation Context ...Delta)-hard against uniform adversaries if for everysh(n)-time bounded probabilistic TM, M, and all but finitely many n, M achieves bias at most 1=h(n) on G n . A well-known "robustness" the=-=orem (see [5, 9]) states t-=-hat so longas `(n) = n O(1) , H(G n ) is invariant up to constant factors. As Razborov and Rudich do, we work with PSRGs that stretch n bits to 2n bits. We use "secure against" interchangeab... |

19 | Relative to a random oracle, NP is not small
- Kautz, Miltersen
- 1996
(Show Context)
Citation Context ...Lutz [15] showed a measure-one class in which every member is a pseudorandom source for BPP, and Allender and Strauss [1] extended this for measures on DTIME[2 n e ], for every e ? 0. Related work is =-=[14, 18, 11, 12, 20, 31]-=-. Our main theorem relates a measure question directly to PSRGs and the class P/poly of languages having polynomial-sized circuits: Theorem 1 If strong PSRGs exist, then P/poly is not measurable in EX... |

17 |
Introduction to probability and measure
- Parthasarathy
- 1978
(Show Context)
Citation Context ...n ffi against non-uniform (resp. uniform) adversaries. 2.2. Resource-bounded measure The resource-bounded measure theory of Lutz [15, 16] is developed along the lines of classical measure theory (see =-=[23, 6, 24]). Languag-=-es are regarded as points in the topological space 3 whose basic open sets are the "cylinders" Cw , one for each w 2 f0; 1g , and complexity classes are point sets. The general form of Lutz'... |

15 | Circuit size relative to pseudorandom oracles
- Lutz, Schmidt
- 1993
(Show Context)
Citation Context ...Lutz [15] showed a measure-one class in which every member is a pseudorandom source for BPP, and Allender and Strauss [1] extended this for measures on DTIME[2 n e ], for every e ? 0. Related work is =-=[14, 18, 11, 12, 20, 31]-=-. Our main theorem relates a measure question directly to PSRGs and the class P/poly of languages having polynomial-sized circuits: Theorem 1 If strong PSRGs exist, then P/poly is not measurable in EX... |

13 | A pseudorandom oracle characterization of BPP
- Lutz
- 1993
(Show Context)
Citation Context ...th NP in place of P/poly would show much more far-reaching consequences from the existence of PSRGs than are currently known. 1. Introduction The theory of resource-bounded measure, initiated by Lutz =-=[14, 15, 16]-=- and furthered in [20, 12, 17, 18, 19, 13, 1, 21], has provided a useful framework that links many central problems in complexity theory. Classes that have measure zero are small in a quantitative sen... |

5 | Improved resource-bounded BorelCantelli and stochasticity theorems
- Regan, Sivakumar
- 1995
(Show Context)
Citation Context ...r the martingale to succeed on C. If \Pi is merely i.o. diagonalizing, then the above factor seems insufficient. By a modification of the Borel-Cantelli lemma as applied to martingales [15] (see also =-=[28]-=-), it can be shown that if P n (1 \Gamma ae(\Pi n )) converges, then a successful martingale of equivalent nonuniform complexity can be constructed. For example, an i.o.-natural property \Pi of densit... |

2 | One-way functions and balanced NP - Lutz - 1992 |