## The Permanent Requires Large Uniform Threshold Circuits (1999)

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Citations: | 28 - 8 self |

### BibTeX

@MISC{Allender99thepermanent,

author = {Eric Allender},

title = {The Permanent Requires Large Uniform Threshold Circuits},

year = {1999}

}

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### Abstract

We show that the permanent cannot be computed by uniform constant-depth threshold circuits of size T (n) for any function T such that for all k, T (k) (n) = o(2 n ). More generally, we show that any problem that is hard for the complexity class C=P requires circuits of this size (on the uniform constant-depth threshold circuit model). In particular, this lower bound applies to any problem that is hard for the complexity classes PP or #P.

### Citations

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Citation Context ...inistic polynomial time machine M with the property that x 2 A if and only if the number of accepting paths of M on input x is greater than the number of rejecting paths. PP contains both NP and coNP =-=[Gil77]-=-. Another related complexity class is C= P; a set A is in C=P if there is a nondeterministic polynomial time machine M with the property that x 2 A if and only if the number of accepting paths of M on... |

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127 | On Uniformity within NC
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- 1990
(Show Context)
Citation Context ... question of which notion of uniformity is the "right" one to use when studying classes of circuits is not always clear. For the circuit classes considered here, convincing arguments are pre=-=sented in [BIS90]-=-, arguing that a very restrictive notion of uniformity called Dlogtime-uniformity is the correct notion to use. Briefly, a circuit family fCng is Dlogtime-uniform if, given a tuple (n; g; h), a determ... |

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41 |
Finite monoids and the fine structure of NC
- Barrington, Thérien
- 1988
(Show Context)
Citation Context ...braic techniques, and shows that NC 1 corresponds to computation over non-solvable algebras. Barrington also defined the corresponding notion of computation over solvable algebras, and it is shown in =-=[BT88]-=- that this notion corresponds exactly to ACC 0 circuits. To restate these two points: 1. The results of [Bar89] establish intimate connections between circuit complexity and algebraic structure. 2. In... |

36 |
Parallel computation with threshold functions
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(Show Context)
Citation Context ...inal language. 2 Similarly, we will find it very convenient to have a single model of computation that is sufficient for describing both TC 0 and the counting hierarchy. Such a model was described in =-=[PS88]. In their model, wh-=-ich they call a "threshold Turing machine", TC 0 corresponds to O(logn) time and O(1) uses of the "threshold" operation, and the counting hierarchy corresponds to polynomial time a... |

31 | Nondeterministic NC1 computation - Caussinus, McKenzie, et al. - 1998 |

29 |
A primer on the complexity theory of neural networks
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(Show Context)
Citation Context ...he complexity of important natural computational problems such as sorting, counting, and integer multiplication. It is also a good complexity-theoretic model for the "neural net" model of co=-=mputation [Par90]-=-. It is easy to observe that ACC 0 ` TC 0 (for example, see [BIS90]), and thus we have even fewer lower bounds for the threshold circuit model than for ACC 0 circuits. Furthermore, since TC 0 (s(n)) `... |

26 |
Quasi-realtime languages
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- 1970
(Show Context)
Citation Context ...en the uniformity machine will have k tapes, too.) However, by changing the naming convention for the gates in the circuit in a way that makes use of the ideas in the original tape-reduction proof of =-=[BG70]-=- for nondeterministic machines, we can make do with a two-tape deterministic machine checking the uniformity condition. (That is, if M 1 is the k-tape uniformity machine for the original circuit famil... |

24 | A first-order isomorphism theorem
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(Show Context)
Citation Context ...mputable by a circuit with no gates (other than NOT gates). A projection is Dlogtime-uniform if the circuit satisfies the usual Dlogtime-uniformity conditions. For more background and motivation, see =-=[ABI97]-=-. For instance, the standard complete set f(i; x; 0 j ) : M i accepts x in time jg is a good choice for C.) The proof consists of two cases. Case 1: [C requires size greater than t(n) to compute on un... |

23 | A uniform circuit lower bound for the permanent
- Allender, Gore
- 1994
(Show Context)
Citation Context ...t subclass of NC 1 . Although, as mentioned above, it is unknown if small ACC 0 circuits suffice to compute all problems in NTIME(2 n O(1) ), lower bounds for uniform ACC 0 circuits were presented in =-=[AG94]-=-. The techniques of [AG94] (see also [II96]) use diagonalization, which is less useful in the nonuniform setting. Since our results, like those of [CMTV96] and [AG94], concern uniform circuits, it is ... |

21 | bounds on the size of bounded depth networks over a complete basis with logical addition - Lower - 1987 |

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17 | Almost-Everywhere Complexity Hierarchies for Nondeterministic Time
- Allender, Beigel, et al.
- 1993
(Show Context)
Citation Context ...counting hierarchy: Corollary 6.3 Let ffl be greater than 0. Then: ACC 0 is properly contained in ACC 0 (2 n ffl ). TC 0 is properly contained in TC 0 (2 n ffl ). But now we will use the technique of =-=[ABHH93]-=- to get a better separation. Lemma 6.4 Let S be a constructible function, S(n)sn. If ACC 0 = ACC 0 (S(n)), then ACC 0 = ACC 0 (S(S(n))). Proof: Let A be any set in oetime(O(log S(S(n)))). Since a cons... |

16 | Nondeterministic NC computation
- Caussinus, McKenzie, et al.
- 1996
(Show Context)
Citation Context ...old circuit model). In particular, this lower bound applies to any problem that is hard for the complexity classes PP or #P. This extends a recent result by Caussinus, McKenzie, Th'erien, and Vollmer =-=[CMTV96], showing -=-that there are problems in the counting hierarchy that require superpolynomial-size uniform TC 0 circuits. The proof in [CMTV96] uses "leaf languages" as a tool in obtaining their separation... |

15 | The power of the middle bit of a #p function - Green, Köbler, et al. - 1995 |

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5 | A note on uniform circuit lower bounds for the counting hierarchy - Allender - 1996 |

4 | Finite monoids and the ne structure of NC 1 - Barrington, Therien - 1988 |

2 |
Parallel complexity hierarchies based on PRAMs and DLOGTIME-uniform circuits
- Iwama, Iwamoto
- 1996
(Show Context)
Citation Context ... above, it is unknown if small ACC 0 circuits suffice to compute all problems in NTIME(2 n O(1) ), lower bounds for uniform ACC 0 circuits were presented in [AG94]. The techniques of [AG94] (see also =-=[II96]-=-) use diagonalization, which is less useful in the nonuniform setting. Since our results, like those of [CMTV96] and [AG94], concern uniform circuits, it is necessary to briefly discuss uniformity. A ... |

2 | Simple characterization of P(#P) and complete problems - Toda - 1994 |