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

### Cached

### Download Links

- [dmtcs.loria.fr]
- [dmtcs.loria.fr]
- [www.maths.tcd.ie]
- [www.cs.elte.hu]
- DBLP

### Other Repositories/Bibliography

Citations: | 1 - 1 self |

### BibTeX

@MISC{Grolmusz01adegree-decreasing,

author = {Vince Grolmusz},

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

year = {2001}

}

### OpenURL

### 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

### Citations

290 | Algebraic methods in the theory of lower bounds for Boolean circuit complexity
- Smolensky
- 1987
(Show Context)
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... |

239 | Almost optimal lower bounds for small depth circuits
- Hastad
- 1989
(Show Context)
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... |

192 | PP is as hard as the polynomial-time hierarchy - Toda - 1991 |

188 | Unbiased Bits from Sources of Weak Randomness and Probabilistic
- Chor, Goldreich
- 1988
(Show Context)
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... |

184 |
Separating the polynomial-time hierarchy by oracles
- YAO
- 1985
(Show Context)
Citation Context ...arallel 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 ... |

136 |
Lower bounds on the monotone complexity of some Boolean functions. Dokl. Akad. Nauk SSSR (in Russian), 281(4):798–801
- Razborov
- 1985
(Show Context)
Citation Context ...y 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 (=-=[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 o... |

120 |
Parity, circuits, and the polynomial time hier- archy," 22nd FOCS
- Furst, Saxe, et al.
- 1981
(Show Context)
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 ... |

109 |
Σ 1 1-formulae on finite structures
- Ajtai
- 1983
(Show Context)
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... |

93 |
On ACC and threshold circuits
- Yao
- 1990
(Show Context)
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... |

85 | 1 -formulae on finite structures - Ajtai - 1983 |

56 | On the computational power of depth 2 circuits with threshold and modulo gates
- Krause, Pudlák
- 1994
(Show Context)
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... |

54 | Representing boolean functions as polynomials modulo composite numbers
- Barrington, Beigel, et al.
- 1994
(Show Context)
Citation Context ...ower 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 c... |

46 |
Nonuniform automata over groups
- Barrington, Straubing, et al.
- 1990
(Show Context)
Citation Context ...ar polynomials on the input-level with the price of increasing the 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 siz... |

42 |
Variation ranks of communication matrices and lower bounds for depth two circuits having symmetric gates with unbounded fan--in
- Waack
- 1991
(Show Context)
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... |

37 |
Handbook of Theoretical Computer Science”. Vol. A,B, Algorithms and Complexity
- Leeuwen
- 1990
(Show Context)
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... |

28 | Leeuwen (editor), Handbook of theoretical computer - van - 1990 |

21 |
bounds on the size of bounded depth networks over a complete basis with logical addition
- Lower
- 1987
(Show Context)
Citation Context ...utation. 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... |

20 | Separating the communication complexities of MOD m and MOD p circuits
- Grolmusz
- 1992
(Show Context)
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... |

20 | A lower bound on the MOD 6 degree of the OR function
- Tardos, Barrington
- 1998
(Show Context)
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... |

12 |
On interpolation by analytic functions with special properties and some weak lower bounds on the size of circuits with symmetric gates
- Smolensky
- 1990
(Show Context)
Citation Context ... [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, since the... |

12 |
Functions with Bounded Symmetric Communication Complexity and Circuits with MOD m Gates
- Szegedy
(Show Context)
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... |

10 | A weight-size trade-off for circuits with mod m gates
- Grolmusz
- 1994
(Show Context)
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 |
Meshulam: On mod p Transversals
- Kahn, R
(Show Context)
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 |
Exponential size lower bounds for some depth three circuits
- Yan, Parberry
- 1994
(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 m representation of boolean functions
- Grolmusz
- 1995
(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... |