## Hardness hypotheses, derandomization, and circuit complexity (2004)

Venue: | In Proceedings of the 24th Conference on Foundations of Software Technology and Theoretical Computer Science |

Citations: | 18 - 5 self |

### BibTeX

@INPROCEEDINGS{Hitchcock04hardnesshypotheses,,

author = {John M. Hitchcock},

title = {Hardness hypotheses, derandomization, and circuit complexity},

booktitle = {In Proceedings of the 24th Conference on Foundations of Software Technology and Theoretical Computer Science},

year = {2004},

pages = {336--347},

publisher = {Springer-Verlag}

}

### OpenURL

### Abstract

Abstract We consider hypotheses about nondeterministic computation that have been studied in dif-ferent contexts and shown to have interesting consequences: * The measure hypothesis: NP does not have p-measure 0.* The pseudo-NP hypothesis: there is an NP language that can be distinguished from anyDTIME(2 nffl) language by an NP refuter. * The NP-machine hypothesis: there is an NP machine accepting 0 * for which no 2n ffl-time machine can find infinitely many accepting computations. We show that the NP-machine hypothesis is implied by each of the first two. Previously, norelationships were known among these three hypotheses. Moreover, we unify previous work by showing that several derandomizations and circuit-size lower bounds that are known to followfrom the first two hypotheses also follow from the NP-machine hypothesis. In particular, the NPmachine hypothesis becomes the weakest known uniform hardness hypothesis that derandomizesAM. We also consider UP versions of the above hypotheses as well as related immunity and scaled dimension hypotheses. 1 Introduction The following uniform hardness hypotheses are known to imply full derandomization of ArthurMerlin games (NP = AM): * The measure hypothesis: NP does not have p-measure 0 [24].

### Citations

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(Show Context)
Citation Context ...the class of SIZE(n)-circuits. Then we say G is a pseudo-random generator if 8n > 0, |Prx2\Sigma r log n[Cn(Gn(x))] - Prx2\Sigma n[Cn(x)]| <= 1n . 4sThe celebrated result of Impagliazzo and Wigderson =-=[25]-=- states that pseudo-random generators can be constructed from any Boolean function with high circuit complexity. Klivans and van Melkebeek [29] observed that, the construction of Impagliazzo and Wigde... |

173 | Almost everywhere high nonuniform complexity
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Citation Context ...han T (n) time on all but finitely many strings. A set S is t(n)-printable if there exists a t(n)-time-bounded algorithm that on input 0n outputs all elements of Sn. 2.4 Resource-Bounded Measure Lutz =-=[33]-=- developed resource-bounded measure theory, analogous to classical Lebesgue measure, to study the quantitative structure of complexity classes. Here we briefly give the definitions; we refer to the su... |

122 | Gap-definable counting classes
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(Show Context)
Citation Context ... except that in this case an is the unique accepting computation of M (0n). Since an is the unique accepting computation, we can place L in UE " coUE. (2) By the results of Fenner, Fortnow, and Kurtz =-=[13]-=-, UE " coUE ` ESPP. Thus by (1), for every k, there exists a language L in ESPP whose SATk-oracle circuit complexity is bigger than 2ffln, where SATk is the set of satisfiable quantified boolean formu... |

110 | Graph nonisomorphism has subexponential size proofs unless the polynomial hierarchy collapses
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Citation Context ...il. Therefore there are no easy witnesses and we can use the mechanism in the hypothesis to nondeterministically generate witnesses of high circuit complexity that are sufficient for derandomizing AM =-=[37, 29]-=-. Given the similarity in the proofs, it is natural to ask how much more these hypotheses have in common. We show that all three of the above hypotheses imply the following NP-machine hypothesis: Ther... |

94 | The quantitative structure of exponential time
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Citation Context ...esource-bounded measure theory, analogous to classical Lebesgue measure, to study the quantitative structure of complexity classes. Here we briefly give the definitions; we refer to the survey papers =-=[34, 4]-=- for more detail. A martingale is a function d : \Sigma * ! [0, 1) with the property that, for all w 2 \Sigma *, 2d(w) = d(w0) + d(w1). A martingale d succeeds on a language A ` \Sigma * if lim sup n!... |

83 | Oracles and queries that are sufficient for exact learning
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(Show Context)
Citation Context ...machine hypothesis implies AM = NP. Thus NEXP = NP which is a contradiction of the nondeterministic time hierarchy theorem. Therefore NEXP 6` P/poly. (5) By the results of Kannan [28], Bshouty et al. =-=[10]-=-, and Kobler and Watanabe [30], ZPPNP 6` io-SIZE(nk) for any k > 0. Since the NP-machine hypothesis implies BPPNP = PNP, PNP 6` io-SIZE(nk) for every k > 0. 1For the definition of easy on average we r... |

67 | Derandomizing Arthur-Merlin Games using Hitting Sets
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(Show Context)
Citation Context ...il. Therefore there are no easy witnesses and we can use the mechanism in the hypothesis to nondeterministically generate witnesses of high circuit complexity that are sufficient for derandomizing AM =-=[37, 29]-=-. Given the similarity in the proofs, it is natural to ask how much more these hypotheses have in common. We show that all three of the above hypotheses imply the following NP-machine hypothesis: Ther... |

58 | In search of an easy witness: Exponential time vs. probabilistic polynomial time
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(Show Context)
Citation Context ... 0 [24]. * The pseudo-NP hypothesis: NP has a language that can be distinguished from any DTIME(2n ffl ) language by an NP refuter [32]. * NE " coNE cannot infinitely-often be decided in 22 ffln time =-=[23]-=-. While the hypotheses are quite different, each of these results rely on the ingenious "easy witness method" of Kabanets [26]: try to show that the hypothesis is false by searching for an easy witnes... |

58 | New collapse consequences of np having small circuits
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(Show Context)
Citation Context ...= NP. Thus NEXP = NP which is a contradiction of the nondeterministic time hierarchy theorem. Therefore NEXP 6` P/poly. (5) By the results of Kannan [28], Bshouty et al. [10], and Kobler and Watanabe =-=[30]-=-, ZPPNP 6` io-SIZE(nk) for any k > 0. Since the NP-machine hypothesis implies BPPNP = PNP, PNP 6` io-SIZE(nk) for every k > 0. 1For the definition of easy on average we refer to [11]. 10s(6) Let ffl0 ... |

54 | Almost every set in exponential time is P-bi-immune
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Citation Context ...re hypothesis implies that NP contains a DTIME(2n ffl )-bi-immune language that does not have superpolynomial gaps. Proof. The measure hypothesis actually yields a much stronger conclusion. Mayordomo =-=[36]-=- showed that the class X of all languages that are not DTIME(2cn)-bi-immune has p-measure 0. It is well known that the strong law of large numbers holds for p-measure; in particular, the set Y = aeA f... |

50 |
Bi-immune sets for complexity classes
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- 1985
(Show Context)
Citation Context ...gs to C. We say that L is C-bi-immune if both L and L are C-immune. We write T (n)-immune and T (n)-bi-immune for DTIME(T (n))-immune and DTIME(T (n))-bi-immune, respectively. Balc'azar and Sch"oning =-=[8]-=- observed that a language L is T (n)-bi-immune if and only if every machine that decides L takes more than T (n) time on all but finitely many strings. A set S is t(n)-printable if there exists a t(n)... |

48 |
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- 1973
(Show Context)
Citation Context ...sy witness method, it can be viewed as having the essential character needed to apply the method. This is because NP-machine hypothesis has an equivalent formulation in terms of Levin's Kt complexity =-=[31]-=- and Allender and Ronneberger [2] showed that Kt complexity has an equivalence with E-oracle circuit complexity. The derandomization of AM is just one of many consequences of the measure hypothesis (s... |

46 | On Pseudorandomness and Resource-Bounded Measure
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(Show Context)
Citation Context ...e x of a BP * PP language, guess a computation path p of the NP-machine. If p is accepting, it has high circuit complexity, so build a generator with it and derandomize the BP * PP computation (as in =-=[5]-=-) for x. Let d be the length of a computation path that implements this derandomization. If p is rejecting, make a computation tree of depth d that is half accepting paths and half rejecting paths. (8... |

43 | Easiness assumptions and hardness tests: Trading time for zero error
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(Show Context)
Citation Context ...uter [32]. * NE " coNE cannot infinitely-often be decided in 22 ffln time [23]. While the hypotheses are quite different, each of these results rely on the ingenious "easy witness method" of Kabanets =-=[26]-=-: try to show that the hypothesis is false by searching for an easy witness, a witness that has low circuit complexity when viewed as the truth-table of a Boolean function. If *This research was suppo... |

41 | Resource-bounded measure and randomness
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(Show Context)
Citation Context ...esource-bounded measure theory, analogous to classical Lebesgue measure, to study the quantitative structure of complexity classes. Here we briefly give the definitions; we refer to the survey papers =-=[34, 4]-=- for more detail. A martingale is a function d : \Sigma * ! [0, 1) with the property that, for all w 2 \Sigma *, 2d(w) = d(w0) + d(w1). A martingale d succeeds on a language A ` \Sigma * if lim sup n!... |

37 | On inverting onto functions
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(Show Context)
Citation Context ... speaking, this hypothesis says that there is an NP search problem that cannot be solved in subexponential time. This hypothesis and several variations have been used a few times in complexity theory =-=[16, 12, 14, 40]-=-. The NP-machine hypothesis in this form is due to Pavan and Selman [40], who showed that it implies a separation of NP-completeness notions. The easy witness method readily applies to show that the N... |

32 | The Complexity of Decision Versus Search
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(Show Context)
Citation Context ...s in io-DTIME(2n ffl0 )/nffi0 . As in the proof of item (6) in Theorem 4.3, we can cycle through all advice strings of length nffi0 and 2For the definition of "search reduces to decision" we refer to =-=[9]-=-. 14scompute the accepting computation of M (0n) for infinitely many n. This process takes at most 2n ffl time. Thus UP " coUP 6` io-DTIME(2n ffl0 )/nffi0 . (4) Kannan [28] showed that for every k > 0... |

31 |
Twelve problems in resource-bounded measure
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(Show Context)
Citation Context ...ender and Ronneberger [2] showed that Kt complexity has an equivalence with E-oracle circuit complexity. The derandomization of AM is just one of many consequences of the measure hypothesis (see e.g. =-=[35]-=-). We show that many of the same or slightly weaker consequences also follow from the NP-machine hypothesis. For example: * PNP = BPPNP. * PP = BP * PP, which implies PH ` PP. * ENP does not have sube... |

30 |
BPP has subexponential simulations unless EXPTIME has publishable proofs
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(Show Context)
Citation Context ...By Theorem 6.2 we have AM ` NP/n. (6.1) Corollary 8 of [23] tells us that if NEXP ` P/poly, then EXP 6` io-[NTIME(2n)/n]. (6.2) Also EXP ` NEXP, so we have EXP ` P/poly which yields EXP ` MA (6.3) by =-=[6]-=-. Putting (6.1) and (6.3) together, we have EXP ` NP/n, a contradiction of (6.2). 7 Conclusion The following figure summarizes relations among several hypotheses and their consequences. It is interest... |

29 | MAX3SAT is exponentially hard to approximate if NP has positive dimension
- Hitchcock
(Show Context)
Citation Context ... pseudo-RP hypothesis. (4) ZPP = EXP. 6 Scaled Dimension In addition to the measure hypothesis on NP, hypotheses on the resource-bounded dimension and scaled dimension of NP have also been considered =-=[17, 22]-=-. While these hypotheses are easily seen to be weaker than the measure hypothesis, they seem incomparable with the other hypotheses considered in this paper. In this section we consider a hypothesis o... |

26 | Scaled dimension and nonuniform complexity
- Hitchcock, Lutz, et al.
- 2003
(Show Context)
Citation Context ...er a hypothesis on the scaled dimension of NP 15sand its consequences for derandomization of NP and circuit-complexity lower bounds for NEXP. For technical background on scaled dimension, we refer to =-=[20, 19, 21]-=-. In the following, we consider the hypothesis that dim(-3)p (NP), the -3rd-order scaled polynomialtime dimension of NP, is positive. If up(NP) 6= 0, then dim(-3)p (NP) > 0. We first show that this sc... |

25 |
Some consequences of the existence of pseudorandom generators
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(Show Context)
Citation Context ...achine hypothesis makes these consequences possible. We conclude this section with some observations about the NP-machine hypothesis and sets of strings that have high Kolmogorov complexity. Allender =-=[3, 1]-=- defined for any language L the function KtL(n) = min{Kt(x) | x 2 L=n}. If L is empty at length n, then KtL(n) is undefined. In [2], the following conditions are shown equivalent: * There is a languag... |

25 | Separation of NP-completeness notions
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(Show Context)
Citation Context ... speaking, this hypothesis says that there is an NP search problem that cannot be solved in subexponential time. This hypothesis and several variations have been used a few times in complexity theory =-=[16, 12, 14, 40]-=-. The NP-machine hypothesis in this form is due to Pavan and Selman [40], who showed that it implies a separation of NP-completeness notions. The easy witness method readily applies to show that the N... |

23 |
The fractal geometry of complexity classes
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Citation Context ...er a hypothesis on the scaled dimension of NP 15sand its consequences for derandomization of NP and circuit-complexity lower bounds for NEXP. For technical background on scaled dimension, we refer to =-=[20, 19, 21]-=-. In the following, we consider the hypothesis that dim(-3)p (NP), the -3rd-order scaled polynomialtime dimension of NP, is positive. If up(NP) 6= 0, then dim(-3)p (NP) > 0. We first show that this sc... |

19 | When worlds collide: Derandomization, lower bounds, and kolmogorov complexity
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(Show Context)
Citation Context ...d as having the essential character needed to apply the method. This is because NP-machine hypothesis has an equivalent formulation in terms of Levin's Kt complexity [31] and Allender and Ronneberger =-=[2]-=- showed that Kt complexity has an equivalence with E-oracle circuit complexity. The derandomization of AM is just one of many consequences of the measure hypothesis (see e.g. [35]). We show that many ... |

18 | Time Bounded Kolmogorov Complexity in Complexity Theory in the Book Kolmogorov Complexity and Computational Complexity, chapter 2
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Citation Context ...achine hypothesis makes these consequences possible. We conclude this section with some observations about the NP-machine hypothesis and sets of strings that have high Kolmogorov complexity. Allender =-=[3, 1]-=- defined for any language L the function KtL(n) = min{Kt(x) | x 2 L=n}. If L is empty at length n, then KtL(n) is undefined. In [2], the following conditions are shown equivalent: * There is a languag... |

16 | Easy sets and hard certificate schemes
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(Show Context)
Citation Context ... speaking, this hypothesis says that there is an NP search problem that cannot be solved in subexponential time. This hypothesis and several variations have been used a few times in complexity theory =-=[16, 12, 14, 40]-=-. The NP-machine hypothesis in this form is due to Pavan and Selman [40], who showed that it implies a separation of NP-completeness notions. The easy witness method readily applies to show that the N... |

15 |
A zero-one law for RP
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Citation Context ...on hypotheses. 1 Introduction The following uniform hardness hypotheses are known to imply full derandomization of ArthurMerlin games (NP = AM): * The measure hypothesis: NP does not have p-measure 0 =-=[24]-=-. * The pseudo-NP hypothesis: NP has a language that can be distinguished from any DTIME(2n ffl ) language by an NP refuter [32]. * NE " coNE cannot infinitely-often be decided in 22 ffln time [23]. W... |

15 | Derandomizing Arthur-Merlin games under uniform assumptions
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(Show Context)
Citation Context ...games (NP = AM): * The measure hypothesis: NP does not have p-measure 0 [24]. * The pseudo-NP hypothesis: NP has a language that can be distinguished from any DTIME(2n ffl ) language by an NP refuter =-=[32]-=-. * NE " coNE cannot infinitely-often be decided in 22 ffln time [23]. While the hypotheses are quite different, each of these results rely on the ingenious "easy witness method" of Kabanets [26]: try... |

15 |
Derandomizing complexity classes
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(Show Context)
Citation Context ...re we only insist that no UP machine distinguishes L from L0. 2.6 Derandomization Next we briefly review definitions of pseudorandom generators. We refer the reader to the recent surveys of Miltersen =-=[38]-=- and Kabanets [27] for more details. Let G = {Gn : {0, 1}r log n ! {0, 1}n}n be a family of functions, and let C = {Cn}n the class of SIZE(n)-circuits. Then we say G is a pseudo-random generator if 8n... |

14 | Partial Bi-immunity, Scaled Dimension, and NPCompleteness. Theory of Computing Systems
- Hitchcock, Pavan, et al.
(Show Context)
Citation Context ... pseudo-RP hypothesis. (4) ZPP = EXP. 6 Scaled Dimension In addition to the measure hypothesis on NP, hypotheses on the resource-bounded dimension and scaled dimension of NP have also been considered =-=[17, 22]-=-. While these hypotheses are easily seen to be weaker than the measure hypothesis, they seem incomparable with the other hypotheses considered in this paper. In this section we consider a hypothesis o... |

14 |
Comparison of reductions and completeness notions
- Pavan
(Show Context)
Citation Context ...is is true, then NP has a language L with no superpolynomial gaps that has the following property P: Every predictor M that correctly decides L takes more than 2n time on all but finitely many inputs =-=[7, 39]-=-. Recall that by Theorem 3.3, if NP has a bi-immune language with no super-polynomial gaps, then the NP-machine hypothesis is true. Thus the difference between the measure hypothesis and the NP-machin... |

13 |
Derandomization: A brief overview. Bulletin of the European Association for Theoretical
- Kabanets
(Show Context)
Citation Context ...that no UP machine distinguishes L from L0. 2.6 Derandomization Next we briefly review definitions of pseudorandom generators. We refer the reader to the recent surveys of Miltersen [38] and Kabanets =-=[27]-=- for more details. Let G = {Gn : {0, 1}r log n ! {0, 1}n}n be a family of functions, and let C = {Cn}n the class of SIZE(n)-circuits. Then we say G is a pseudo-random generator if 8n > 0, |Prx2\Sigma ... |

10 | Properties of NP-complete sets
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- 2004
(Show Context)
Citation Context |

6 |
On polynomial time Turing and many-one completeness
- Watanabe, Tang
- 1992
(Show Context)
Citation Context ..., K2 nffl (w) > log n + O(1). Pavan and Selman [40] showed that NP-machine hypothesis implies that <=pT-completeness is different from <=pm-completeness for NP. An earlier result of Watanabe and Tang =-=[41]-=- achieved the same separation for PSPACE, under a Kolmogorov complexity hypothesis that is strikingly similar to the statements in Theorem 4.10 that are equivalent to the NP-machine hypothesis. Theore... |

5 |
A note on genericty and bi-immunity
- Balcazar, Mayordomo
- 1995
(Show Context)
Citation Context ...is is true, then NP has a language L with no superpolynomial gaps that has the following property P: Every predictor M that correctly decides L takes more than 2n time on all but finitely many inputs =-=[7, 39]-=-. Recall that by Theorem 3.3, if NP has a bi-immune language with no super-polynomial gaps, then the NP-machine hypothesis is true. Thus the difference between the measure hypothesis and the NP-machin... |

3 | Upward separations and weaker hypotheses in resource-bounded measure
- Harkins, Hitchcock
- 2007
(Show Context)
Citation Context ...conjecture that the NP-machine hypothesis is much weaker than the measure hypothesis. There is a relativized world in which the NP-machine hypothesis is true, but the measure hypothesis does not hold =-=[15]-=-. The hypothesis of the following theorem was considered by Impagliazzo, Kabanets, and Wigderson [23]. They used the easy witness method to show that it implies NP = AM. Theorem 3.5. If NE " coNE 6` i... |

2 |
Some reuslts on derandomization. Theory of Computing Systems
- Buhrman, Fortnow, et al.
- 2005
(Show Context)
Citation Context ...bler and Watanabe [30], ZPPNP 6` io-SIZE(nk) for any k > 0. Since the NP-machine hypothesis implies BPPNP = PNP, PNP 6` io-SIZE(nk) for every k > 0. 1For the definition of easy on average we refer to =-=[11]-=-. 10s(6) Let ffl0 = ffl/3 and ffi0 = ffl/6k. Suppose NP ` io-DTIME(2n ffi0 )/nffl0. Consider the following language in NP. L0 = aeh0n, yi fifififi |y| <= n k, there exists w such that yw is an accepti... |

2 |
The size of SPP. Theoretical Computer Science
- Hitchcock
- 2004
(Show Context)
Citation Context ... there exists a language L in ESPP whose SATk-oracle circuit complexity is bigger than 2ffln, where SATk is the set of satisfiable quantified boolean formula with k quantifiers. This implies PH ` SPP =-=[18]-=-. (3) The proof is similar to the proof of item 6 in Theorem 4.3. Consider the following language. L = {h0n, ii | the ith bit of the accepting computation of M (0n) is 1}. Observe that L is in UP " co... |