## A Lower Bound for Perceptrons and an Oracle Separation of the PP PH Hierarchy (1997)

Venue: | Journal of Computer and System Sciences |

Citations: | 6 - 0 self |

### BibTeX

@ARTICLE{Berg97alower,

author = {Christer Berg and Staffan Ulfberg},

title = {A Lower Bound for Perceptrons and an Oracle Separation of the PP PH Hierarchy},

journal = {Journal of Computer and System Sciences},

year = {1997},

volume = {56},

pages = {165--172}

}

### OpenURL

### Abstract

We show that there are functions computable by linear size boolean circuits of depth k that require super-polynomial size perceptrons of depth k \Gamma 1, for k ! logn=(6loglogn). This result implies the existence of an oracle A such that S p;A k 6` PP S p;A k\Gamma2 and in particular this oracle separates the levels in the PP PH hierarchy. Using the same ideas, we show a lower bound for another function, which makes it possible to strengthen the oracle separation to D p;A k 6` PP S p;A k\Gamma2 . 1 Introduction There is a strong connection between lower bounds for boolean circuits (consisting of AND, OR, and NOT gates) and relativization results about the polynomial time hierarchy. This fact was first established by Furst, Saxe, and Sipser [5]. Sipser [13] later defined a family of functions that are computable by linear size circuits of depth k, and showed that they require super-polynomial size boolean circuits of depth k \Gamma 1. Yao [14] and Hstad [8, 9] impr...

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Citation Context ...]. Sipser [13] later defined a family of functions that are computable by linear size circuits of depth k, and showed that they require super-polynomial size boolean circuits of depth k \Gamma 1. Yao =-=[14]-=- and Hstad [8, 9] improved Sipser's result by showing that the same functions actually require exponential size circuits of depth k \Gamma 1; this fact implies the existence of an oracle that separate... |

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Citation Context ...could be used to show the separation for all k. In the monotone setting, the separation between depth k and depth k \Gamma 1 perceptrons for all k follows from a stronger result by Hstad and Goldmann =-=[10]-=- that separates boolean circuits of depth k from threshold circuits of depth k \Gamma 1. In this paper we show that there are functions computable by linear size boolean circuits of depth k that requi... |

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Citation Context ... p;A k 6` PP S p;A k\Gamma2 . (We use D p k to denote the complexity class P S p k\Gamma1 .) Beigel, Hemachandra, and Wechsung [2] showed that P NP[log] ` PP, and later Beigel, Reingold, and Spielman =-=[3]-=- proved the even stronger P PP[log] = PP. A relativization of these results show that our result is almost tight. 2 A lower bound for perceptrons We begin this section by defining the function f m k ,... |

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Citation Context ...ng if the first incorrect guess was made in ansstate, and rejecting otherwise. The following relation between alternating oracle Turing machines and perceptrons is similar to for example Lemma 2.1 in =-=[11]-=- and Corollary 2.2 in [5]. Lemma 11. Let M A be a PP S p;A d \Gamma1 oracle machine which runs in time t on input x. Then there is a depth d + 1 perceptron P with unit weights and bottom fan-in t whic... |

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Citation Context ...ts. A depth k perceptron has sub-circuits of depth k \Gamma 1. Linear size perceptrons can compute majority and are thus more powerful than polynomial size constant depth circuits [5]. In 1991, Green =-=[6]-=- used a result by Boppana and Hstad [8] on approximating parity to prove a lower bound for the size of constant depth perceptrons that Department of Numerical Analysis and Computing Science, Royal Ins... |

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Citation Context ...hat there exists an A such that S p;A k 6` PP S p;A k\Gamma2 . The fact that our basis, i.e., the "One-in-a-box" theorem by Minsky and Papert, implies that NP NP 6` PP under an oracle was no=-=ted by Fu [4]-=-. Beigel [1] has strengthened this separation to obtain that P NP 6` PP under an oracle, and in the last section of this paper we use his result as a basis for a lower bound for perceptrons with bound... |

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Citation Context ...h bounded weights. Using this lower bound, we get an oracle A such that D p;A k 6` PP S p;A k\Gamma2 . (We use D p k to denote the complexity class P S p k\Gamma1 .) Beigel, Hemachandra, and Wechsung =-=[2]-=- showed that P NP[log] ` PP, and later Beigel, Reingold, and Spielman [3] proved the even stronger P PP[log] = PP. A relativization of these results show that our result is almost tight. 2 A lower bou... |

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Citation Context ... Royal Institute of Technology, S100 44 Stockholm, Sweden. E-mail: --berg,staffanu@nada.kth.se. 1 compute parity. This bound implies the existence of an oracle that separates \PhiP from PP PH . Green =-=[7]-=- also discussed the question of whether there is an oracle that separates the levels in the PP PH hierarchy. Since this follows from a sufficiently strong lower bound for the size of depth k \Gamma 1 ... |

1 |
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Citation Context ...later defined a family of functions that are computable by linear size circuits of depth k, and showed that they require super-polynomial size boolean circuits of depth k \Gamma 1. Yao [14] and Hstad =-=[8, 9]-=- improved Sipser's result by showing that the same functions actually require exponential size circuits of depth k \Gamma 1; this fact implies the existence of an oracle that separates the levels in t... |