## Circuit lower bounds for Merlin-Arthur classes (2007)

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Venue: | In Proc. ACM STOC |

Citations: | 12 - 1 self |

### BibTeX

@INPROCEEDINGS{Santhanam07circuitlower,

author = {Rahul Santhanam},

title = {Circuit lower bounds for Merlin-Arthur classes},

booktitle = {In Proc. ACM STOC},

year = {2007},

pages = {275--283},

publisher = {Manuscript}

}

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### Abstract

We show that for each k> 0, MA/1 (MA with 1 bit of advice) doesn’t have circuits of size nk. This implies the first superlinear circuit lower bounds for the promise versions of the classes MA, AM and ZPP NP We extend our main result in several ways. For each k, we give an explicit language in (MA ∩ coMA)/1 which doesn’t have circuits of size nk. We also adapt our lower bound to the average-case setting, i.e., we show that MA/1 cannot be solved on more than 1/2 + 1/nk fraction of inputs of length n by circuits of size nk. Furthermore, we prove that MA does not have arithmetic circuits of size nk for any k. As a corollary to our main result, we obtain that derandomization of MA with O(1) advice implies the existence of pseudo-random generators computable using O(1) bits of advice. 1

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Citation Context ...)/1 �⊆ heur 1−1/n a − SIZE(n k ). We give the proof in the Appendix. Next we amplify the hardness of the average-case hard language in Theorem 18. The tool we use for this purpose is the Yao XOR Lemma=-=[Yao82]-=-. Lemma 19. Let f ⊕t denote the Boolean function which on input of the form x1x2 . . . xt, |x1| = |x2| . . . = |xt| = n, outputs the parity of f(x1), f(x2) . . . f(xt), and outputs 0 on all other inpu... |

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Citation Context ...y, the permanent has interesting properties in the Boolean setting, such as being complete for the class of functions counting the number of accepting paths of a non-deterministic machine. Theorem 7. =-=[Val79b]-=- 0-1-PER over Z is complete for ♯P . Kabanets and Impagliazzo [KI04] showed that checking if an arithmetic circuit computes PER over Z is in coRP. Lemma 8. [KI04] The language {〈C, 1 n 〉|C computes P ... |

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Citation Context ...lynomial-size Boolean circuits 1 . Using Theorem 8, we get that ♯P ⊆ SIZE(poly), and hence PPP = P♯P ⊆ SIZE(poly). By a Karp-Lipton style argument using function-restricted interactive proofs for PPP =-=[LFKN92]-=-, we get that PPP ⊆ MA. Toda [Tod89] showed that PH ⊆ PPP , hence we get that PH ⊆ MA. Kannan [Kan82] exhibited a language in PH without circuits of size nk , thus we derive that MA �⊆ SIZE(nk ). In t... |

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Citation Context ...hers have attempted to understand the difficulty of proving lower bounds by formalizing “obstacles” to traditional techniques, such as the relativization obstacle [BGS75] and the naturalness obstacle =-=[RR97]-=-. Given these obstacles, we are forced to temper our ambitions. There are two distinct lines of research which have made incremental progress over the years toward the ultimate goal of a non-trivial c... |

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Citation Context ...ndom generators. Pseudo-random generators suffice for derandomization; a natural question is whether they are also necessary. Impagliazzo, Kabanets and Wigderson [IKW02], and Kabanets and Impagliazzo =-=[KI04]-=- show results of the form that derandomization implies circuit lower bounds, but the circuit lower bounds obtained are not strong enough to yield pseudo-random generators. In this section, we use Theo... |

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Citation Context ...is at most poly(n). We will work only with polynomials of feasible degree. One of the most well-studied polynomials is the permanent, which is conjectured not have polynomial-size arithmetic circuits =-=[Val79a]-=-. The permanent PER is defined on n2 variables {xij}, i = 1 . . . n, j = 1 . . . n as follows: P ER(�x) = � n� xiπ(i), where the sum is taken over all permutations π i=1 π on {1 . . . n}. We also defi... |

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Citation Context ... is to “approach NP from above” by proving fixed polynomial size circuit lower bounds in the general model for smaller and smaller classes containing NP. This line of research was initiated by Kannan =-=[Kan82]-=- who showed that for each k, there is a language in Σ2 ∩ Π2 which doesn’t have circuits of size 1sn k . This was improved by Köbler and Watanabe [KW98], who used the learning algorithm of Bshouty et a... |

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Citation Context ...lizing language is in C, implying a lower bound. Thus the strength of the result we obtain is directly related to the strength of the collapse consequence of NP having small circuits. Karp and Lipton =-=[KL82]-=- showed that NP ⊆ SIZE(poly) implies PH ⊆ Σ2 ∩ Π2 - this yields Kannan’s result [Kan82]. Strengthenings of the Karp-Lipton theorem [BCG + 96, Cai01] yield the lower bounds for ZPP NP and S2P respectiv... |

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Citation Context .... The first obstacle applies to techniques that relativize, i.e., the technique works even when the circuits and the machines defining the complexity class are given access to the same oracle. Wilson =-=[Wil85]-=- constructed an oracle relative to which NP has linear size circuits, hence no successful techique showing circuit lower bounds for NP can relativize. The second obstacle applies to techniques which a... |

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Citation Context ...e computed in NTIME(2 O(s(n)) )∩coNTIME(2 O(s(n)) ). G is a strong non-deterministic PRG using advice a(n) if fG can be computed in (NTIME(2 O(s(n)) ) ∩ coNTIME(2 O(s(n)) ))/a(n). Nisan and Wigderson =-=[NW94]-=- showed how to construct PRGs from hard functions. Theorem 24. [NW94] If (NE∩coNE)/a(n) �⊆ i.o.SIZE(poly) (resp. (NE∩coNE)/a(n) �⊆ SIZE(poly)), then for each ɛ > 0 there is a strong nondeterministic P... |

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Citation Context ...L in Lemma 14. If PER over Z has polynomial-sized arithmetic circuits, then it also has polynomial-size Boolean circuits. In this case, we can show that P PP = MA, and it follows from results of Toda =-=[Tod89]-=- and Kannan [Kan82] that MA doesn’t have Boolean circuits of size n k for any k. If PER over Z doesn’t have polynomial-size arithmetic circuits, then we use a translation argument as in the proof of T... |

58 | New collapse consequences of np having small circuits
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(Show Context)
Citation Context .... This line of research was initiated by Kannan [Kan82] who showed that for each k, there is a language in Σ2 ∩ Π2 which doesn’t have circuits of size 1sn k . This was improved by Köbler and Watanabe =-=[KW98]-=-, who used the learning algorithm of Bshouty et al. [BCG + 96] to prove n k size circuit lower bounds for ZPP NP , which is contained in Σ2 ∩ Π2. This was further improved to a lower bound for the cla... |

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Citation Context ...lapse of PH to MA. However, this is a notoriously difficult problem that has long been open; also, unlike known Karp-Lipton style consequences of NP ⊆ SIZE(poly), such a collapse would not relativize =-=[BFT98]-=-. We circumvent this problem entirely and adopt a different, somewhat counter-intuitive approach. Instead of reasoning based on whether NP ⊆ SIZE(poly), we reason based on whether C ⊆ SIZE(poly) for a... |

43 | Probabilistic Algorithms for Deciding Equivalence of Straight-Line Programs
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Citation Context ...is Arithmetic Circuit Identity Testing (ACIT) - testing if the circuit computes the zero polynomial. The Schwartz-Zippel lemma [Sch80, R.E79] can be used to show that ACIT over Z is in coRP. Lemma 7. =-=[IM83]-=- ACIT over Z is in coRP. One of the most well-studied polynomials is the permanent, which is conjectured not have polynomial-size arithmetic circuits [Val79a]. The permanent PER is defined on n2 varia... |

42 |
Salil Vadhan, Pseudorandom generators without the XOR Lemma
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Citation Context ...e-case hardness result - in order to get a stronger one, we amplify hardness further using standard techniques. We need the following lemma which follows by applying a standard hardness amplification =-=[STV01]-=- to a hard language in PSPACE obtained using direct diagonalization. Lemma 17. For each constant k, PSPACE �⊆ heur 1/2+1/n k − SIZE(n k ). Theorem 18. There is a constant a > 0 such that for each cons... |

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36 | Threshold computation and cryptographic security
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Citation Context ...Theorem 13 shows that Promise − AM requires circuits of size n k , for each k > 0. It is also known that Promise − MA ⊆ Promise − ZPP NP � [NW94, AK97, GZ97] and that Promise − MA ⊆ Promise − BPPpath =-=[HHT97]-=-. Thus Theorem 13 also yields circuit lower bounds for Promise − ZPP NP � and Promise − BPPpath. In fact, for each of these classes, the natural analogue of Theorem 1 holds, i.e., the language for whi... |

28 | A probabilistic-time hierarchy theorem for “Slightly Non-uniform” algorithms - Barak - 2002 |

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25 |
Avi Wigderson. In search of an easy witness: exponential time vs. probabilistic polynomial time
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(Show Context)
Citation Context ...usses on the construction of pseudo-random generators. Pseudo-random generators suffice for derandomization; a natural question is whether they are also necessary. Impagliazzo, Kabanets and Wigderson =-=[IKW02]-=-, and Kabanets and Impagliazzo [KI04] show results of the form that derandomization implies circuit lower bounds, but the circuit lower bounds obtained are not strong enough to yield pseudo-random gen... |

22 | and Lance Fortnow. Random-self-reducibility of complete sets - Feigenbaum - 1993 |

15 |
On the power of PP
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Citation Context ...ng circuit lower bounds for NP. Moreover, such a result would simulataneously strengthen all known lower bounds in this line of research, since it is known that MA ⊆ S2P [RS98, GZ97] and that MA ⊆ PP =-=[Ver92]-=- In the present work, we do not quite achieve this, but we achieve something very close. We show n k size lower bounds for languages accepted by Merlin-Arthur machines running in polynomial time and u... |

14 | Fortnow and Rahul Santhanam. Hierarchy theorems for probabilistic polynomial time - Lance - 2004 |

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10 |
s p 2 ⊆ zppNP
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Citation Context ... learning algorithm of Bshouty et al. [BCG + 96] to prove n k size circuit lower bounds for ZPP NP , which is contained in Σ2 ∩ Π2. This was further improved to a lower bound for the class S2P by Cai =-=[Cai01]-=-, based on an observation by Sengupta. Both S2P and ZPP NP are believed to equal P NP , under a strong enough derandomization assumption. An incomparable result was recently obtained by Vinodchandran ... |

10 | From logarithmic advice to single-bit advice - Goldreich, Sudan, et al. - 2004 |

10 | Another proof that BPP subseteq PH (and more - Goldreich, Zuckerman - 1997 |

9 | On Proving Circuit Lower Bounds Against PH and Some Related Lower Bounds for Constant Depth Circuits
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(Show Context)
Citation Context ...ke progress on other problems in the area of fixed polynomial circuit lower bounds. One interesting question is to find explicit languages with circuit lower bounds - this is known for the case of Σ2 =-=[CW04]-=- but not for any lower class. We improve the situation considerably for promise problems, by giving explicit promise problems in MA ∩ coMA with circuit lower bounds. Theorem 2. For each k, there is an... |

6 | Oracles are subtle but not malicious
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(Show Context)
Citation Context ...result was recently obtained by Vinodchandran [Vin05], who showed n k size lower bounds for the class PP of languages accepted by probabilistic polynomial time machines with unbounded error. Aaronson =-=[Aar05]-=- gave a different proof of the same result. The logical next step would be to prove circuit lower bounds for the class MA, the probabilistic version of the class NP. Such lower bounds would be of inte... |

2 |
A note on the circuit complexity of pp
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(Show Context)
Citation Context ..., based on an observation by Sengupta. Both S2P and ZPP NP are believed to equal P NP , under a strong enough derandomization assumption. An incomparable result was recently obtained by Vinodchandran =-=[Vin05]-=-, who showed n k size lower bounds for the class PP of languages accepted by probabilistic polynomial time machines with unbounded error. Aaronson [Aar05] gave a different proof of the same result. Th... |