## Non-uniform ACC circuit lower bounds (2010)

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Citations: | 22 - 4 self |

### BibTeX

@TECHREPORT{Williams10non-uniformacc,

author = {Ryan Williams},

title = {Non-uniform ACC circuit lower bounds},

institution = {},

year = {2010}

}

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### Abstract

The class ACC consists of circuit families with constant depth over unbounded fan-in AND, OR, NOT, and MODm gates, where m> 1 is an arbitrary constant. We prove: • NTIME[2 n] does not have non-uniform ACC circuits of polynomial size. The size lower bound can be slightly strengthened to quasi-polynomials and other less natural functions. • ENP, the class of languages recognized in 2O(n) time with an NP oracle, doesn’t have non-uniform ACC circuits of 2no(1) size. The lower bound gives an exponential size-depth tradeoff: for every d there is a δ> 0 such that ENP doesn’t have depth-d ACC circuits of size 2nδ. Previously, it was not known whether EXP NP had depth-3 polynomial size circuits made out of only MOD6 gates. The high-level strategy is to design faster algorithms for the circuit satisfiability problem over ACC circuits, then prove that such algorithms entail the above lower bounds. The algorithm combines known properties of ACC with fast rectangular matrix multiplication and dynamic programming, while the second step requires a subtle strengthening of the author’s prior work [STOC’10]. Supported by the Josef Raviv Memorial Fellowship.

### Citations

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Citation Context ...tial lower bound for computing the majority of n bits with constant-depth circuits made up of AND, OR, NOT, and MOD2 gates. (A MODm gate outputs 1 iff m divides the sum of its inputs.) Then Smolensky =-=[Smo87]-=- proved exponential lower bounds for computing MODq with constant-depth circuits made up of AND, OR, NOT, and MODp gates, for distinct primes p and q. Barrington [Bar89] suggested the next step would ... |

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Citation Context ... NTIME[2 n ] doesn’t have non-uniform polysize C-circuits. 2 Preliminaries We presume the reader has background in circuit complexity and complexity theory in general. The textbook of Arora and Barak =-=[AB09]-=- covers all the necessary material; in particular, Chapter 14 gives an excellent summary of ACC and the frontiers in circuit complexity. On the machine model. An important point about this paper is th... |

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Citation Context ...ynomial size circuits is MAEXP [BFT98]. Later it was shown that the MAEXP lower bound can be improved to half-exponential size functions f which satisfy f(f(n)) ≥ 2n [MVW99]. Kabanets and Impagliazzo =-=[KI04]-=- proved that NEXP RP either doesn’t have polynomial size Boolean circuits (over AND, OR, NOT), or it doesn’t have polynomial size arithmetic circuits (over the integers, with addition and multiplicati... |

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Citation Context ...s, perhaps we could rule out weak circuits for complicated functions. Could one prove that nondeterministic exponential time (NEXP) doesn’t have polynomial size circuits? A series of papers [BFNW93], =-=[KvM99]-=-, [IKW02] showed that even this sort of lower bound would imply derandomization results: in the case of NEXP lower bounds, it would imply that Merlin-Arthur games can be non-trivially simulated with n... |

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Citation Context ...a critical role in this work. Define a SYM + circuit to be a depth-two circuit which computes some symmetric function at the output gate, and computes ANDs of input variables on the second layer. Yao =-=[Yao90]-=- showed that every ACC circuit of s size can be represented by a probabilistic SYM + circuit of s O(logc s) size, where c depends on the depth, and the ANDs have poly(log s) fan-in. Beigel and Tarui [... |

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Citation Context ...ine a SYM + circuit to be a depth-two circuit which computes some symmetric function at the output gate, and computes ANDs of input variables on the second layer. 3 Extending work on AC 0 by Allender =-=[All89]-=-, Yao [Yao90] showed that every ACC circuit of s size can be represented by a probabilistic SYM + circuit of s O(logc s) size, where c depends on the depth, and the ANDs have poly(logs) fan-in. Beigel... |

76 |
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Citation Context ...lass, this is shorthand for the union of all classes where the O(1) is substituted by a fixed constant. For example, the class TIME[2nO(1) ] is shorthand for ⋃ c≥0 TIME[2nc]. Other Prior Work. Kannan =-=[Kan82]-=- showed in 1982 that for any superpolynomial constructible function S : N → N, the class NTIME[S(n)] NP does not have polynomial size circuits. Another somewhat small class known to not have unrestric... |

63 | Rectangular matrix multiplication revisited - COPPERSMITH |

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(Show Context)
Citation Context ...e lower bounds of this work. There has also been other substantial work on representing ACC [BT88], [AAD00], [Han06], [KH09] as well as many lower bounds in restricted cases [BST90], [Thé94], [YP94], =-=[KP94]-=-, [BS95], [Cau96], [Gro98], [GT00], [CGPT06], [CW09]. Significant work has gone into the constant degree hypothesis [BST90] that a certain type of low-depth ACC circuit requires exponential size to co... |

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Citation Context ...n be accepted by a nondeterministic algorithm in O(2nn10 /nk) time. (Here, 10 is a substitute for a small universal constant.) When k > 10 this contradicts the nondeterministic time hierarchy theorem =-=[SFM78]-=-, [Zak83], so one of the assumptions must be false. Two known facts are applied in the proof. First, there is a polynomial-time reduction from any L ∈ NTIME[2n ] to the NEXP-complete problem SUCCINCT ... |

50 |
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Citation Context ...uraged by the progress on AC 0 , attention turned to lower bounds for what seemed to be minor generalizations. The most natural generalization was to grant AC 0 the parity function for free. Razborov =-=[Raz87]-=- proved an exponential lower bound for computing the majority of n bits with constant-depth circuits made up of AND, OR, NOT, and MOD2 gates. (A MODm gate outputs 1 iff m divides the sum of its inputs... |

47 |
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Citation Context ... KH09] as well as many lower bounds in restricted cases [BST90, Thé94, YP94, KP94, BS95, Cau96, Gro98, GT00, CGPT06, CW09]. Significant work has gone into understanding the constant degree hypothesis =-=[BST90]-=- that a certain type of low-depth ACC circuit requires exponential size to compute the AND function. The hypothesis is still open. All prior works on non-uniform ACC lower bounds attack the problem in... |

45 | Nonrelativizing separations
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Citation Context ...nomial constructible function S : N → N, the class NTIME[S(n)] NP does not have polynomial size circuits. Another somewhat small class known to not have unrestricted polynomial size circuits is MAEXP =-=[BFT98]-=-. Later it was shown that the MAEXP lower bound can be improved to half-exponential size functions f which satisfy f(f(n)) ≥ 2n [MVW99]. Kabanets and Impagliazzo [KI04] proved that NEXP RP either does... |

45 |
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Citation Context ...e underlying decompressed formula is satisfiable. For natural reasons, call FC the decompression of C, and call C the compression of FC. The SUCCINCT 3SAT problem is a canonical NEXP-complete problem =-=[PY86]-=-. Fact 3.1 There is a constant c > 0 such that for every L ∈ NTIME[2 n ], there is a reduction from L to SUCCINCT 3SAT which on input x of length n runs in poly(n) time and produces a circuit Cx with ... |

40 | Finite monoids and the fine structure of NC1 - Barrington, Thérien - 1988 |

37 | Satis is Quasilinear Complete in NQL
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Citation Context ...epending on L), outputs the ith clause of the resulting 3SAT formula in O((log n) c ) time. The proofs in the above references build on even earlier work of Schnorr, Cook, Gurevich-Shelah, and Robson =-=[Sch78]-=-, [Coo88], [GS89], [Rob91]. In a nutshell, all of these proofs exploit the locality of computation: every nondeterministic computation running in linear time can be represented with a nondeterministic... |

34 |
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Citation Context ...wer bounds for solving clique with monotone circuits (i.e., general circuits without NOT gates), and the bound was improved to exponential size by Alon and Boppana [AB87]. However, it was later shown =-=[Raz89]-=- that the monotone techniques probably would not extend to general circuits. Encouraged by the progress on AC 0 , attention turned to lower bounds for what seemed to be minor generalizations. The most... |

33 |
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Citation Context ...then by gradually lifting the restrictions over time, superpolynomial size unrestricted lower bounds for NP could be attained, proving P ̸= NP. Furst, Saxe, and Sipser [FSS81] and independently Ajtai =-=[Ajt83]-=- showed that functions such as the parity of n bits cannot be computed by polynomial size AC 0 circuits, i.e., polynomial size circuit families of constant depth over the usual basis of AND, OR, and N... |

32 | Algebrization: A new barrier in complexity theory
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(Show Context)
Citation Context ...han statements like “There are no strong pseudorandom functions implementable with C circuits”.More conclusively, the approach of this work definitely avoids relativization [BGS75] and algebrization =-=[AW09]-=- because there are oracles A relative to which NEXP A ⊂ ACC A , and even NEXP Ã ⊂ ACC A (Scott Aaronson, personal communication). The ACC SAT algorithm used in the lower bounds relies on non-relativiz... |

30 | The permanent requires large uniform threshold circuits
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(Show Context)
Citation Context ...rm ACC circuit family can be simulated by subexponential uniform SYM + circuits. This was applied to show that the Permanent does not have uniform ACC circuits of subexponential size. Later, Allender =-=[All99]-=- improved the Permanent lower bound to polynomial size uniform TC 0 circuits. However, these proofs require uniformity, and the difference between uniformity and non-uniformity may well be vast (e.g.,... |

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Citation Context ...all of its possible inputs. It remains to show how to compute g efficiently. Given f, the function g can be computed in O(2n · poly(n)) time by a dynamic programming algorithm of Yates from 1937 (cf. =-=[BHK09]-=-, Section 2.2). For i = 0, . . . , n, define gi : 2 [n] → N by g0(T ) = f(T ), and { gi−1(T ) + gi−1(T \ {i}) if i ∈ T, gi(T ) = gi−1(T ) otherwise. It follows that each gi+1 can be obtained from gi i... |

28 | How to multiply matrices faster - Pan - 1984 |

25 |
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Citation Context ...ak circuits, perhaps we could rule out weak circuits for complicated functions. Could one prove that nondeterministic exponential time (NEXP) doesn’t have polynomial size circuits? A series of papers =-=[BFNW93]-=-, [KvM99], [IKW02] showed that even this sort of lower bound would imply derandomization results: in the case of NEXP lower bounds, it would imply that Merlin-Arthur games can be non-trivially simulat... |

25 | Time-Space Lower Bounds for Satisfiability
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Citation Context ...f 2 n · poly(n) size is satisfiable. Fact 3.1 follows from several prior works concerned with the complexity of the Cook-Levin theorem [Tou01, FLvMV05]: Theorem 3.3 (Tourlakis [Tou01], Fortnow et al. =-=[FLvMV05]-=-) There is a c > 0 such that for all L ∈ NTIME[n], L reduces to 3SAT in O(n(log n) c ) time. Moreover there is an algorithm (with random access to its input) that, given an instance of L with length n... |

25 |
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Citation Context ...eview the prior state-of-the-art for Circuit SAT algorithms. It is known that CNF SAT can be solved in 2n−Ω(n/ ln(m/n)) poly(m) time, where m is the number of clauses and n is the number of variables =-=[Sch05]-=-, [CIP06], [DH08]. Calabro, Impagliazzo, and Paturi have recently shown that AC 0 SAT can be solved by a randomized algorithm in 2n−n1−o(1) time, on circuits with n1+o(1) gates [CIP09]. Recently, Sant... |

24 |
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Citation Context ...on L), outputs the ith clause of the resulting 3SAT formula in O((log n) c ) time. The proofs in the above references build on even earlier work of Schnorr, Cook, Gurevich-Shelah, and Robson [Sch78], =-=[Coo88]-=-, [GS89], [Rob91]. In a nutshell, all of these proofs exploit the locality of computation: every nondeterministic computation running in linear time can be represented with a nondeterministic circuit ... |

24 | Fast algorithms with preprocessing for matrix-vector multipication problems - Gohberg, Olshevsky - 1994 |

23 | A uniform circuit lower bound for the permanent
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Citation Context ...stic SYM + circuit of s O(logc s) size, where c depends on the depth, and the ANDs have poly(log s) fan-in. Beigel and Tarui [BT94] showed how to remove the probabilistic condition. Allender and Gore =-=[AG94]-=- showed that every subexponential uniform ACC circuit family can be simulated by subexponential uniform SYM + circuits. This was applied to show that the Permanent does not have uniform ACC circuits o... |

21 | H.: Superlinear lower bounds for bounded-width branching programs
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(Show Context)
Citation Context ...bounds of this work. There has also been other substantial work on representing ACC [BT88], [AAD00], [Han06], [KH09] as well as many lower bounds in restricted cases [BST90], [Thé94], [YP94], [KP94], =-=[BS95]-=-, [Cau96], [Gro98], [GT00], [CGPT06], [CW09]. Significant work has gone into the constant degree hypothesis [BST90] that a certain type of low-depth ACC circuit requires exponential size to compute AN... |

20 |
On TC0, AC0, and arithmetic circuits
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(Show Context)
Citation Context ...on which returns the middle bit of the sum of its inputs. This representation may also be used in the lower bounds of this work. There has also been other substantial work on representing ACC [BT88], =-=[AAD00]-=-, [Han06], [KH09] as well as many lower bounds in restricted cases [BST90], [Thé94], [YP94], [KP94], [BS95], [Cau96], [Gro98], [GT00], [CGPT06], [CW09]. Significant work has gone into the constant deg... |

20 | A duality between clause width and clause density for SAT
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(Show Context)
Citation Context ... prior state-of-the-art for Circuit SAT algorithms. It is known that CNF SAT can be solved in 2n−Ω(n/ ln(m/n)) poly(m) time, where m is the number of clauses and n is the number of variables [Sch05], =-=[CIP06]-=-, [DH08]. Calabro, Impagliazzo, and Paturi have recently shown that AC 0 SAT can be solved by a randomized algorithm in 2n−n1−o(1) time, on circuits with n1+o(1) gates [CIP09]. Recently, Santhanam [Sa... |

19 | Improving exhaustive search implies superpolynomial lower bounds
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- 2010
(Show Context)
Citation Context ...isfiability problem, the second question would be settled. A. An Overview of the Proofs Let us sketch how these new lower bounds are proved, giving a roadmap for the rest of the paper. In recent work =-=[Wil10]-=-, the author suggested a research program for proving non-uniform circuit lower bounds for NEXP. It was 2 Note that slightly larger classes such as MAEXP and NEXP NP are known to not have polynomial s... |

17 | A Circuit-based Proof of Toda's Theorem - Kannan, Venkateswaran, et al. - 1993 |

16 |
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(Show Context)
Citation Context ...returns the middle bit of the sum of its inputs. This representation may also be used in the lower bounds of this work. There has also been other substantial work on representing ACC [BT88], [AAD00], =-=[Han06]-=-, [KH09] as well as many lower bounds in restricted cases [BST90], [Thé94], [YP94], [KP94], [BS95], [Cau96], [Gro98], [GT00], [CGPT06], [CW09]. Significant work has gone into the constant degree hypot... |

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Citation Context ...ions in the middle, and additions at the bottom level, where each input wire to an addition gate may also multiply the input by a scalar. From the duality of bilinear matrix multiplication algorithms =-=[HM73]-=-, a bilinear algorithm for multiplying N × N and N × M directly implies a bilinear algorithm for multiplying N × M and M × N. Furthermore, Coppersmith’s algorithm is explicit, in that it can be execut... |

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(Show Context)
Citation Context ...C circuits of depth d and size 2nδ. Recall that the lowest complexity class for which we know exponential (unrestricted) circuit lower bounds is ∆ EXP 3 , the third level of the exponential hierarchy =-=[MVW99]-=-. Extending the approach of this paper to settle the second frontier question may be difficult, but this prospect does not look as implausible as it did before. If polynomial unrestricted circuits cou... |

15 | The power of the middle bit of a #P function
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(Show Context)
Citation Context ...s. However, these proofs require uniformity, and the difference between uniformity and non-uniformity may well be vast (e.g., it is clear that P ̸= NEXP, but open whether NEXP ⊆ P/poly). Green et al. =-=[GKRST95]-=- showed that the symmetric function can be assumed to be the specific function which returns the middle bit of the sum of its inputs. This representation may also be used in the lower bounds of this p... |

15 |
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(Show Context)
Citation Context ...e ith clause of the resulting 3SAT formula in O((log n) c ) time. The proofs in the above references build on even earlier work of Schnorr, Cook, Gurevich-Shelah, and Robson [Sch78], [Coo88], [GS89], =-=[Rob91]-=-. In a nutshell, all of these proofs exploit the locality of computation: every nondeterministic computation running in linear time can be represented with a nondeterministic circuit of size O(n·poly(... |

14 |
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Citation Context ...ntradiction. (Note that such a result is not known for the deterministic setting.) A random access Turing machine can also simulate a log-cost random access machine with only constant factor overhead =-=[PR81]-=-. Hence in our proof by contradiction, we may assume that the source algorithm we’re simulating is only a multitape TM, while the target algorithm has all the power we need to perform typical computat... |

12 |
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Citation Context ...m works in the case where the “middle” dimension of the matrices is polynomially smaller than the other two. In this case, matrix multiplication can be done nearly optimally. 4 Lemma 4.3 (Coppersmith =-=[Cop82]-=-) For all sufficiently large N, multiplication of an N ×N .1 matrix with an N .1 × N matrix can be done in O(N 2 log 2 N) arithmetic operations. More precisely, Coppersmith shows that there is a const... |

11 | The complexity of satisfiability of small depth circuits
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(Show Context)
Citation Context ...er of variables [Sch05], [CIP06], [DH08]. Calabro, Impagliazzo, and Paturi have recently shown that AC 0 SAT can be solved by a randomized algorithm in 2n−n1−o(1) time, on circuits with n1+o(1) gates =-=[CIP09]-=-. Recently, Santhanam [San10] applied ideas inspired by formula size lower bounds to show that for a fixed k, Boolean formula SAT can be determined in O(2n−n/ck) time on formulas of size cn. Unfortuna... |