## Almost Optimal Lower Bounds for Small Depth Circuits (1989)

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Venue: | RANDOMNESS AND COMPUTATION |

Citations: | 241 - 7 self |

### BibTeX

@INPROCEEDINGS{Håstad89almostoptimal,

author = {Johan Håstad},

title = {Almost Optimal Lower Bounds for Small Depth Circuits},

booktitle = {RANDOMNESS AND COMPUTATION},

year = {1989},

pages = {6--20},

publisher = {JAI Press}

}

### Years of Citing Articles

### OpenURL

### Abstract

We give improved lower bounds for the size of small depth circuits computing several functions. In particular we prove almost optimal lower bounds for the size of parity circuits. Fur-ther we show that there are functions computable in polynomial size and depth k but requires ex-ponential size when the depth is restricted to k-1. Our main lemma which is of independent interest states that by using a random restriction we can convert an AND of small ORs to an OR of small ANDs and conversely.

### Citations

184 |
Separating the polynomial-time hierarchy by oracles
- YAO
(Show Context)
Citation Context ...ity and majority. The first superpolynomial lower bounds for the circuits computing parity was obtained by Furst, Saxe and Sipser lESS]. Ajtai [Aj] independently gave slightly stronger bounds and Yao =-=[Y]-=- proved the first exponential lower bounds. (The case of monotone small depth circuits has been studied by Boppana [B] and Klawe, Paul, Pippenger and Yannakakis [KPPY].) We will in this paper give alm... |

184 |
Computational Limitations of Small Depth Circuits
- H˚astad
- 1988
(Show Context)
Citation Context ...ions f n k and in section 6 we briefly mention some more details of the implications for relativized complexity. An earlier version of this paper appeared in [H1]. The paper is also part of my thesis =-=[H2]-=-. 2. Background 2.1 Computational Model We will be working with unbounded fanin circuits of small depth. A typical example looks like this. Figure 1 We can assume that the only negations occur as nega... |

137 |
bounds on the monotone complexity of some Boolean functions, Doklady Akad. Nauk SSSR 282
- Razborov, Lower
- 1985
(Show Context)
Citation Context ...s. The first example is the the case of monotone circuits i.e. circuits just containing AND and OR gates and no negations. Superpolynomial lower bounds were proved for the clique function by Razborov =-=[R]-=- and these were improved to exponential lower bounds by Alon and Boppana [AB]. Andreev [An] independently obtained exponential lower bounds for other NPfunctions. The second model where interesting lo... |

127 | The monotone circuit complexity of boolean functions
- Alon, Boppana
- 1987
(Show Context)
Citation Context ... containing AND and OR gates and no negations. Superpolynomial lower bounds were proved for the clique function by Razborov [R] and these were improved to exponential lower bounds by Alon and Boppana =-=[AB]-=-. Andreev [An] independently obtained exponential lower bounds for other NP-functions. The second model where interesting lower bounds have been proved is the model of small depth circuits. These circ... |

120 |
Parity, circuits, and the polynomial time hierarchy
- Furst, Saxe, et al.
- 1984
(Show Context)
Citation Context ...nd this can be done without increasing the size of the circuit significantly. An easy induction now gives the result. The idea of giving random values to some of the variables was first introduced in =-=[FSS]-=- and weaker versions of our main lemma were used in [FSS] and [Y]. In [FSS] the probability of size not increasing to much was not proved to be exponentially small and Yao only proved that the resulti... |

112 |
Σ1 1-formulae on finite structures
- Ajtai
- 1983
(Show Context)
Citation Context ...work was done while the author visited AT&T Bell Laboratories. 1 majority. The first superpolynomial lower bounds for the circuits computing parity was obtained by Furst, Saxe and Sipser [FSS]. Ajtai =-=[Aj]-=- independently gave slightly stronger bounds and Yao [Y] proved the first exponential lower bounds. (The case of monotone small depth circuits has been studied by Boppana [B], Valiant [V], and Klawe, ... |

73 |
Borel sets and circuit complexity
- Sipser
- 1983
(Show Context)
Citation Context ...of functions f n k of n inputs which have linear size circuits of depth k but require exponential size circuits when restricted to depth k \Gamma 1. These functions f n k were introduced by Sipser in =-=[S]-=-. Sipser proved superpolynomial lower bounds for the size of the circuits when the depth was restricted to be k \Gamma 1. Yao claimed exponential lower bounds for the same situation. 1.3 Small depth c... |

47 | Optimal Bounds for Decision Problems on the CRCW PRAM
- Beame, Hastad
- 1989
(Show Context)
Citation Context ...hat parity requires time\Omega\Gamma log n log log n ) to compute on such a PRAM. Interestingly enough the same bounds can be proved for more powerful PRAMs using extensions of the present techniques =-=[BeH]-=-. 1.5 Outline of paper. In section 3 we prove the main lemma. The necessary background and some motivation are given in section 2. The application to parity circuits is in section 4 and in section 5 w... |

44 |
With probability one, a random oracle separates PSPACE from the polynomial-time hierarchy
- CAI
- 1989
(Show Context)
Citation Context ...ould imply the existence of an oracle separating PSPACE from the polynomial time hierarchy. Yao [Y] was the first to prove sufficiently good lower bounds to obtain the separation for an oracle A. Cai =-=[C]-=- extended his methods to prove that a random oracle separated the two complexity classes with probability 1. In [S] Sipser proved the corresponding theorem that the same lower bounds for the functions... |

42 |
Simulation of parallel random access machines by circuits
- Stockmeyer, Vishkin
- 1984
(Show Context)
Citation Context ...this separation is still open. 2 1.4 Relations to PRAMs The model of small depth circuits has relations to computation by parallel random access machines (PRAM). In particular, Stockmeyer and Vishkin =-=[SV]-=- proved that any function that can be computed on a slightly limited PRAM with a polynomial number of processors in time T can also be computed by polynomial size unbounded fanin circuits of depth O(T... |

16 |
Threshold functions and bounded depth monotone circuits
- Boppana
- 1984
(Show Context)
Citation Context ... and Sipser lESS]. Ajtai [Aj] independently gave slightly stronger bounds and Yao [Y] proved the first exponential lower bounds. (The case of monotone small depth circuits has been studied by Boppana =-=[B]-=- and Klawe, Paul, Pippenger and Yannakakis [KPPY].) We will in this paper give almost optimal lower bounds for the size of circuits computing parity. However it is quite likely that the longer lasting... |

10 |
On monotone formulae with restricted depth
- Klawe, Paul, et al.
- 1984
(Show Context)
Citation Context ...stronger bounds and Yao [Y] proved the rst exponential lower bounds. (The case of monotone small depth circuits has been studied by Boppana [B], Valiant [V], and Klawe, Paul, Pippenger and Yannakakis =-=[KPPY]-=-.) We will in this paper give almost optimal lower bounds for the size of circuits computing parity. However it is quite likely that the longer lasting contribution will be our main lemma. The main le... |

9 |
Exponential lower bounds for restricted monotone circuits
- VALIANT
(Show Context)
Citation Context ... [FSS]. Ajtai [Aj] independently gave slightly stronger bounds and Yao [Y] proved the rst exponential lower bounds. (The case of monotone small depth circuits has been studied by Boppana [B], Valiant =-=[V]-=-, and Klawe, Paul, Pippenger and Yannakakis [KPPY].) We will in this paper give almost optimal lower bounds for the size of circuits computing parity. However it is quite likely that the longer lastin... |

3 |
the Polynomial Time Hierarchy
- Furst, Saxe, et al.
- 1981
(Show Context)
Citation Context ...** Some of the work was done while the author visited AT&T Bell Laboratories. 1majority. The rst superpolynomial lower bounds for the circuits computing parity was obtained by Furst, Saxe and Sipser =-=[FSS]-=-. Ajtai [Aj] independently gave slightly stronger bounds and Yao [Y] proved the rst exponential lower bounds. (The case of monotone small depth circuits has been studied by Boppana [B], Valiant [V], a... |

1 |
Z~-Formulae on Finite Structures", Annals of Pure and Applied Logic 24(1983) 1-48 tAB] Alon N. and Boppana R. "The Monotone Circuit Complexity of Boolean Functions
- Ajtai
(Show Context)
Citation Context ... Functions considered have been simple functions like parity and majority. The first superpolynomial lower bounds for the circuits computing parity was obtained by Furst, Saxe and Sipser lESS]. Ajtai =-=[Aj]-=- independently gave slightly stronger bounds and Yao [Y] proved the first exponential lower bounds. (The case of monotone small depth circuits has been studied by Boppana [B] and Klawe, Paul, Pippenge... |

1 |
On one method of obtaining lower bounds of individual monotone function complexity
- Andreev
- 1985
(Show Context)
Citation Context ...D and OR gates and no negations. Superpolynomial lower bounds were proved for the clique function by Razborov [R] and these were improved to exponential lower bounds by Alon and Boppana [AB]. Andreev =-=[An]-=- independently obtained exponential lower bounds for other NPfunctions. The second model where interesting lower bounds have been proved is the model of small depth circuits. These circuits have the f... |

1 | With l)robal)ility One, a Random Oracle Separates I'SI)ACE J'rom tile l)olynomial - Time llierarchy" These proceedings - Cai |

1 |
Approximation Properties of Constant Depth Circuits", manuscript in preparation
- Boppana, Hastad
(Show Context)
Citation Context ...is result one needs to establish that a small circuit makes an error when trying to compute parity on a random input with a probability close to 1 2 . This problem and related problems are studied in =-=[BoH]-=-. To prove that a random oracle separates the different levels within the polynomial hierarchy one would have to strengthen Theorem 4 to say that no depth k \Gamma 1 circuit computes a function which ... |