## On the power of small-depth threshold circuits (1990)

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Venue: | Proceedings 31st Annual IEEE Symposium on Foundations of Computer Science |

Citations: | 102 - 2 self |

### BibTeX

@INPROCEEDINGS{Hastad90onthe,

author = {Johan Hastad and Mikael Goldmann},

title = {On the power of small-depth threshold circuits},

booktitle = {Proceedings 31st Annual IEEE Symposium on Foundations of Computer Science},

year = {1990},

pages = {610--618}

}

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### Abstract

Abstract. Weinvestigate the power of threshold circuits of small depth. In particular, we give functions that require exponential size unweighted threshold circuits of depth 3 when we restrict the bottom fanin. We also prove that there are monotone functions fk that can be computed in depth k and linear size ^ � _-circuits but require exponential size to compute by a depth k; 1 monotone weighted threshold circuit. Key words. Circuit complexity, monotone circuits, threshold circuits, lower bounds Subject classi cations. 68Q15, 68Q99 1.

### Citations

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Algebraic methods in the theory of lower bounds for boolean circuit complexity
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(Show Context)
Citation Context ...er bounds on circuit-size for any function in NP, several interesting results have beenshown for restricted circuit classes e.g. monotone circuits [4, 16, 3, 13, 14, 15] and circuits of bounded depth =-=[1, 9, 10, 17, 19, 21]-=-. The smallest natural circuit-class that is not known to be strictly contained in NP is TC 0 , the set of functions computable by constant-depth polynomialsize circuits containing threshold gates. Th... |

186 | Unbiased bits from sources of weak randomness and probabilistic communication complexity
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Citation Context ...he circuit is 2 (n=s) . Proof. In this case we will use a two person communication game. Lemma 1 could be used also for this purpose but let us use an older more accurate result by Chor and Goldreich =-=[8]-=-. Lemma 6. (Theorem 21 in [8]) To -evaluate the inner product function (gip1) in a two-person game requires at least at least n ; 3 ; 3log ;1 bits of communication. Now suppose there is a circuit of s... |

183 |
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(Show Context)
Citation Context ...er bounds on circuit-size for any function in NP, several interesting results have beenshown for restricted circuit classes e.g. monotone circuits [4, 16, 3, 13, 14, 15] and circuits of bounded depth =-=[1, 9, 10, 17, 19, 21]-=-. The smallest natural circuit-class that is not known to be strictly contained in NP is TC 0 , the set of functions computable by constant-depth polynomialsize circuits containing threshold gates. Th... |

182 |
Computational limitations of small-depth circuits
- H˚astad
- 1987
(Show Context)
Citation Context ...er bounds on circuit-size for any function in NP, several interesting results have beenshown for restricted circuit classes e.g. monotone circuits [4, 16, 3, 13, 14, 15] and circuits of bounded depth =-=[1, 9, 10, 17, 19, 21]-=-. The smallest natural circuit-class that is not known to be strictly contained in NP is TC 0 , the set of functions computable by constant-depth polynomialsize circuits containing threshold gates. Th... |

149 |
Monotone circuits for connectivity require superlogarithmic depth
- Karchmer, Wigderson
- 1988
(Show Context)
Citation Context ...o successful. While there are still no non-linear lower bounds on circuit-size for any function in NP, several interesting results have beenshown for restricted circuit classes e.g. monotone circuits =-=[4, 16, 3, 13, 14, 15]-=- and circuits of bounded depth [1, 9, 10, 17, 19, 21]. The smallest natural circuit-class that is not known to be strictly contained in NP is TC 0 , the set of functions computable by constant-depth p... |

129 |
Threshold circuits of bounded depth
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- 1993
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Citation Context ...ant depth circuits (with or without modular gates) are not su cient to prove lower bounds for threshold circuits. The best known results about smalldepth threshold circuits are those by Hajnal et al. =-=[12]-=- where, among other results, it is established that depth 3 threshold circuits of polynomial size are more powerful than corresponding circuits of depth 2. To further understand the nature of threshol... |

127 | The monotone circuit complexity of boolean functions
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(Show Context)
Citation Context ...o successful. While there are still no non-linear lower bounds on circuit-size for any function in NP, several interesting results have beenshown for restricted circuit classes e.g. monotone circuits =-=[4, 16, 3, 13, 14, 15]-=- and circuits of bounded depth [1, 9, 10, 17, 19, 21]. The smallest natural circuit-class that is not known to be strictly contained in NP is TC 0 , the set of functions computable by constant-depth p... |

119 |
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Citation Context |

116 |
Log depth circuits for division and related problems
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Citation Context ... set of functions computable by constant-depth polynomialsize circuits containing threshold gates. Threshold gates are quite powerful and many fairly complicated functions (like division, implicit in =-=[6]-=-) are in TC 0 . It also seems like the techniques used for proving lower bounds for usual constant depth circuits (with or without modular gates) are not su cient to prove lower bounds for threshold c... |

109 | Σ 1 1-formulae on finite structures - Ajtai - 1983 |

108 |
Constant depth reducibility
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- 1984
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Citation Context ...to emphasize that we speak about the more general model we call them weighted threshold circuits. It is interesting to note that TC0 remains the same even if we restrict the circuits to be unweighted =-=[7]-=-. 3. Lower Bounds for Depth 3 Circuits The function we will consider is the \generalized inner product function" considered in [5]. We use doubly indexed variables xi�j where i ranges from 1 to n and ... |

93 | On ACC and threshold circuits - Yao - 1990 |

85 |
A note on the power of threshold circuits
- Allender
- 1989
(Show Context)
Citation Context ...e corresponding distributions pk and qk. It is interesting to note that while the functions fk cannot be computed 1 n 2k by depth k ; 1 monotone threshold circuits of size 2 , by a result of Allender =-=[2]-=- (which is based on work by Toda [20]), they can be computed by depth 3 general threshold circuits of size 2O((log n)k ). This might be taken as another piece of evidence that monotonicity is a severe... |

76 | Monotone Circuits for Matching Require Linear Depth
- Raz, Wigderson
- 1992
(Show Context)
Citation Context ...o successful. While there are still no non-linear lower bounds on circuit-size for any function in NP, several interesting results have beenshown for restricted circuit classes e.g. monotone circuits =-=[4, 16, 3, 13, 14, 15]-=- and circuits of bounded depth [1, 9, 10, 17, 19, 21]. The smallest natural circuit-class that is not known to be strictly contained in NP is TC 0 , the set of functions computable by constant-depth p... |

72 |
Borel sets and circuit complexity
- Sipser
- 1983
(Show Context)
Citation Context ...m the bottom consists of ^-gates if i is even and otherwise it consists of _-gates. The fanin at the top and bottom levels is N and at all other levels it is N 2 . This function was used by Sipser in =-=[18]-=- who showed that it requires superpolynomial size depth k ; 1 circuits over the basis f^� _� :g. It will be convenient to also consider the functions fk, the negations of fk. Clearly fk is computed by... |

60 |
On the computational power of PP and ⊕P , in
- Toda
- 1989
(Show Context)
Citation Context ... qk. It is interesting to note that while the functions fk cannot be computed 1 n 2k by depth k ; 1 monotone threshold circuits of size 2 , by a result of Allender [2] (which is based on work by Toda =-=[20]-=-), they can be computed by depth 3 general threshold circuits of size 2O((log n)k ). This might be taken as another piece of evidence that monotonicity is a severe restriction, and that new techniques... |

53 |
Multiparty protocols and logspace-hard pseudorandom sequences, 21st STOC
- Babai, Nisan, et al.
- 1989
(Show Context)
Citation Context ...explicit function that cannot be computed by small depth 3 unweighted threshold circuits of small bottom fanin. The proof of this is based on a communication game analyzed by Babai, Nisan and Szegedy =-=[5]-=-. In this communication game s +1players (which we call Pj�j = 1�:::�s+1) participate and share some variables which are partitioned into s +1 groups (the j th group being Gj). The player Pj knows all... |

44 |
On a method for obtaining lower bounds for the complexity of individual monotone functions
- Andreev
- 1985
(Show Context)
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31 |
Lower bounds on the monotone network complexity of the logical permanent
- Razborov
- 1985
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30 | On the computational power of PP and \PhiP - Toda - 1989 |

21 | Wigderson: Probabilistic Communication Complexity of Boolean Relations
- Raz, A
- 1989
(Show Context)
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20 |
Circuits and local computation
- Yao
- 1989
(Show Context)
Citation Context ...ther results, it is established that depth 3 threshold circuits of polynomial size are more powerful than corresponding circuits of depth 2. To further understand the nature of threshold circuits Yao =-=[22]-=- studied monotone threshold circuits. In particular, Yao was interested in the question whether it is true for monotone circuits that depth k and polynomial size is more powerful than depth k ; 1 and ... |

8 |
bounds for the size of circuits of bounded depth with basis and, xor
- Lower
- 1987
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1 |
P1 1-formulae on Logic, 24:1{48
- Ajtai
- 1983
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