## A Degree-Decreasing Lemma for (MOD q - MOD p) Circuits (2001)

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### BibTeX

@MISC{Grolmusz01adegree-decreasing,

author = {Vince Grolmusz},

title = {A Degree-Decreasing Lemma for (MOD q - MOD p) Circuits},

year = {2001}

}

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### Abstract

plus an arbitrary linear function of n input variables. Keywords: Circuit complexity, modular circuits, composite modulus 1 Introduction Boolean circuits are one of the most interesting models of computation. They are widely examined in VLSI design, in general computability theory and in complexity theory context as well as in the theory of parallel computation. Almost all of the strongest and deepest lower bound results for the computational complexity of finite functions were proved using the Boolean circuit model of computation ([13], [22], [9], [14], [15], or see [20] for a survey). Even these famous and sophisticated lower bound results were proven for very restricted circuit classes. Bounded depth and polynomial size is one of the most natural restrictions. Ajtai [1], Furst, Saxe, and Sipser [5] proved that no polynomial sized, constant depth circuit can compute the PARITY function. Yao [22] and Hastad [9] generalized this result

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Citation Context ...n. Almost all of the strongest and deepest lower bound results for the computational complexity of finite functions were proved using the Boolean circuit model of computation ([14], [22], [10], [15], =-=[16]-=-, or see [20] for a survey). Even these famous and sophisticated lower bound results were proven for very restricted circuit classes. Bounded depth and polynomial size is one of the most natural restr... |

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Citation Context ...l computation. Almost all of the strongest and deepest lower bound results for the computational complexity of finite functions were proved using the Boolean circuit model of computation ([14], [22], =-=[10]-=-, [15], [16], or see [20] for a survey). Even these famous and sophisticated lower bound results were proven for very restricted circuit classes. Bounded depth and polynomial size is one of the most n... |

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Citation Context ... + 1. However, the converse is not true. This can be seen by considering the degree-2 polynomial x 1 y 1 + x 2 y 2 + \Delta \Delta \Delta + x n y n over GF(2), which has high communication complexity =-=[5]-=-, while polynomials which are computable by d fan-in 2 MULTIPLICATION gates have low communication complexity for small d's . 3 The Degree-Decreasing Lemma The following lemma is our main tool. It exp... |

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Citation Context ...us and sophisticated lower bound results were proven for very restricted circuit classes. Bounded depth and polynomial size is one of the most natural restrictions. Ajtai [1], Furst, Saxe, and Sipser =-=[6]-=- proved that no polynomial sized, constant depth circuit can compute the PARITY function. Yao [22] and Hastad [10] generalized this result for sub-logarithmic depths. Since the modular gates are very ... |

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Citation Context ...or a survey). Even these famous and sophisticated lower bound results were proven for very restricted circuit classes. Bounded depth and polynomial size is one of the most natural restrictions. Ajtai =-=[1]-=-, Furst, Saxe, and Sipser [6] proved that no polynomial sized, constant depth circuit can compute the PARITY function. Yao [22] and Hastad [10] generalized this result for sub-logarithmic depths. Sinc... |

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Citation Context ...d MOD 3 gates, or MOD 6 gates can compute the n-fan-in OR and AND functions [11], [3]. Consequently, these circuits are more powerful than circuits with MOD p gates only. By the famous theorem of Yao =-=[23]-=- and Beigel and Tarui [4], every polynomial-size, constant-depth circuit with AND, OR, NOT and MODm Grolmusz: A Degree-Decreasing Lemma for (MOD q -- MOD p) Circuits 3 gates can be converted to a dept... |

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Citation Context ...equences: for the Constant Degree Hypothesis of Barrington, Straubing and Th'erien [3], and generalizing the lower bound results of Yan and Parberry [21], Krause and Waack [13] and Krause and Pudl'ak =-=[12]-=-. Perhaps the most important application is an exponential lower bound for the size of (MOD q ; MOD p ) circuits computing the n fan-in AND, where the input of each MOD p gate at the bottom is an arbi... |

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Citation Context ...his result has numerous consequences: for the Constant Degree Hypothesis of Barrington, Straubing and Th'erien [3], and generalizing the lower bound results of Yan and Parberry [21], Krause and Waack =-=[13]-=- and Krause and Pudl'ak [12]. Perhaps the most important application is an exponential lower bound for the size of (MOD q ; MOD p ) circuits computing the n fan-in AND, where the input of each MOD p g... |

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Citation Context ... of the strongest and deepest lower bound results for the computational complexity of finite functions were proved using the Boolean circuit model of computation ([14], [22], [10], [15], [16], or see =-=[20]-=- for a survey). Even these famous and sophisticated lower bound results were proven for very restricted circuit classes. Bounded depth and polynomial size is one of the most natural restrictions. Ajta... |

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Citation Context ... The existing lower bound results use diverse techniques from Fourier-analysis, communication complexity theory, grouptheory and several forms of random restrictions (see [3], [12], [18], [19], [17], =-=[9]-=-, [7], [8], [2], [11]). It is not difficult to see that constant-depth circuits with MOD p gates only, (p prime), cannot compute even simple functions: the n-fan-in OR or AND functions, since they can... |

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Citation Context ...s (see [3]). The existing lower bound results use diverse techniques from Fourier-analysis, communication complexity theory, grouptheory and several forms of random restrictions (see [3], [12], [18], =-=[19]-=-, [17], [9], [7], [8], [2], [11]). It is not difficult to see that constant-depth circuits with MOD p gates only, (p prime), cannot compute even simple functions: the n-fan-in OR or AND functions, sin... |

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Citation Context ... graphs (see [3]). The existing lower bound results use diverse techniques from Fourier-analysis, communication complexity theory, grouptheory and several forms of random restrictions (see [3], [12], =-=[18]-=-, [19], [17], [9], [7], [8], [2], [11]). It is not difficult to see that constant-depth circuits with MOD p gates only, (p prime), cannot compute even simple functions: the n-fan-in OR or AND function... |

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Citation Context ...existing lower bound results use diverse techniques from Fourier-analysis, communication complexity theory, grouptheory and several forms of random restrictions (see [3], [12], [18], [19], [17], [9], =-=[7]-=-, [8], [2], [11]). It is not difficult to see that constant-depth circuits with MOD p gates only, (p prime), cannot compute even simple functions: the n-fan-in OR or AND functions, since they can only... |

9 |
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Citation Context ...bound results use diverse techniques from Fourier-analysis, communication complexity theory, grouptheory and several forms of random restrictions (see [3], [12], [18], [19], [17], [9], [7], [8], [2], =-=[11]-=-). It is not difficult to see that constant-depth circuits with MOD p gates only, (p prime), cannot compute even simple functions: the n-fan-in OR or AND functions, since they can only compute constan... |

8 |
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(Show Context)
Citation Context ... size of the circuit. This result has numerous consequences: for the Constant Degree Hypothesis of Barrington, Straubing and Th'erien [3], and generalizing the lower bound results of Yan and Parberry =-=[21]-=-, Krause and Waack [13] and Krause and Pudl'ak [12]. Perhaps the most important application is an exponential lower bound for the size of (MOD q ; MOD p ) circuits computing the n fan-in AND, where th... |

7 | On the weak mod � representation of Boolean functions
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(Show Context)
Citation Context ...ing lower bound results use diverse techniques from Fourier-analysis, communication complexity theory, grouptheory and several forms of random restrictions (see [3], [12], [18], [19], [17], [9], [7], =-=[8]-=-, [2], [11]). It is not difficult to see that constant-depth circuits with MOD p gates only, (p prime), cannot compute even simple functions: the n-fan-in OR or AND functions, since they can only comp... |