## Lower Bounds and Separations for Constant Depth Multilinear Circuits

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Citations: | 17 - 7 self |

### BibTeX

@MISC{Raz_lowerbounds,

author = {Ran Raz and Amir Yehudayoff},

title = {Lower Bounds and Separations for Constant Depth Multilinear Circuits},

year = {}

}

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### Abstract

We prove an exponential lower bound for the size of constant depth multilinear arithmetic circuits computing either the determinant or the permanent (a circuit is called multilinear, if the polynomial computed by each of its gates is multilinear). We also prove a super-polynomial separation between the size of product-depth 1 d and product-depth d + 1 multilinear circuits (where d is constant). That is, there exists a polynomial f such that • There exists a multilinear circuit of product-depth d + 1 and of polynomial size computing f. • Every multilinear circuit of product-depth d computing f has super-polynomial size. 1

### Citations

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184 | Separating the polynomial-time hierarchy by oracles - YAO - 1985 |

182 | Computational limitations of small-depth circuits - H˚astad - 1987 |

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

77 | Lower bounds for the size of circuits of bounded depth with basis - Razborov - 1987 |

50 | Multi-linear formulas for Permanent and Determinant are of superpolynomial size
- Raz
- 2004
(Show Context)
Citation Context ...e study of multilinear circuits was initiated by Nisan and Wigderson in [NW96]. A super-polynomial lower bound for the size of multilinear formulas for the determinant and the permanent was proved in =-=[R04a]-=-. Then, [R04b] proved a super-polynomial separation between the size of multilinear formulas and the size of multilinear circuits ([RY] simplified the proof of this separation). Later, [RSY] proved a ... |

45 | Lower bounds for non-commutative computation - Nisan - 1991 |

42 | Lower bounds on arithmetic circuits via partial derivatives
- Nisan, Wigderson
- 1996
(Show Context)
Citation Context ... for every product gate in Φ and for every two gates v1, v2 ∈ child(v), the two sets Xv1 and Xv2 are disjoint. 1.2 Background The study of multilinear circuits was initiated by Nisan and Wigderson in =-=[NW96]-=-. A super-polynomial lower bound for the size of multilinear formulas for the determinant and the permanent was proved in [R04a]. Then, [R04b] proved a super-polynomial separation between the size of ... |

39 | Exponential lower bounds for depth-3 arithmetic circuits in algebras of functions over finite fields - Grigoriev, Razborov |

32 | Die berechnungskomplexitat von elementarsymmetrischen funktionen und von interpolationskoeffizienten - Strassen - 1973 |

16 | Separation of Multilinear Circuit and Formula Size. Theory Of Computing 2(6) (2006) (preliminary version
- Raz
(Show Context)
Citation Context ...tilinear circuits was initiated by Nisan and Wigderson in [NW96]. A super-polynomial lower bound for the size of multilinear formulas for the determinant and the permanent was proved in [R04a]. Then, =-=[R04b]-=- proved a super-polynomial separation between the size of multilinear formulas and the size of multilinear circuits ([RY] simplified the proof of this separation). Later, [RSY] proved a roughly n 4/3 ... |

16 | Depth-3 arithmetic formulae over fields of characteristic zero
- Shpilka, Wigderson
- 2001
(Show Context)
Citation Context ...s. Constant depth arithmetic circuits have been studied extensively. Over finite fields [GK, GR] proved an exponential lower bound for the size of depth 3 circuits. Over fields of characteristic zero =-=[SW]-=- proved a roughly n 2 lower bound for the size of depth 3 circuits. For arithmetic circuits of arbitrary constant depth [SS, R07] proved a roughly n 1+1/d lower bound for the size of depth d arithmeti... |

14 | A lower bound for the size of syntactically multilinear arithmetic circuits
- Raz, Shpilka, et al.
- 2007
(Show Context)
Citation Context ...roved in [R04a]. Then, [R04b] proved a super-polynomial separation between the size of multilinear formulas and the size of multilinear circuits ([RY] simplified the proof of this separation). Later, =-=[RSY]-=- proved a roughly n 4/3 lower bound for the size of syntactically multilinear arithmetic circuits. Constant depth arithmetic circuits have been studied extensively. Over finite fields [GK, GR] proved ... |

14 | Lower bounds for polynomial evaluation and interpolation problems, Comput. Complexity 6(4 - Shoup, Smolensky - 1997 |

13 | Elusive functions and lower bounds for arithmetic circuits - Raz - 2010 |

7 | Balancing syntactically multilinear arithmetic circuits
- Raz, Yehudayoff
(Show Context)
Citation Context ...r formulas for the determinant and the permanent was proved in [R04a]. Then, [R04b] proved a super-polynomial separation between the size of multilinear formulas and the size of multilinear circuits (=-=[RY]-=- simplified the proof of this separation). Later, [RSY] proved a roughly n 4/3 lower bound for the size of syntactically multilinear arithmetic circuits. Constant depth arithmetic circuits have been s... |