## Abstract

in this paper we give reconstruction algorithms for depth-3 arithmetic circuits with k multiplication gates (also known as ∑π∑(k) circuits), where k = O(1). Namely, we give an algorithm that when given a black box holding a ∑π∑(k) circuit C over a field F as input, makes queries to the black box (possibly over a polynomial sized extension field of F) and outputs a circuit C0 computing the same polynomial as C. in particular we obtain the following results. 1) When C is a multilinear ∑π∑(k) circuit (i.e. each of its multiplication gates computes a multilinear polynomial) then our algorithm runs in polynomial time (when k is a constant) and outputs a multilinear ∑π∑(k) circuits computing the same polynomial. 2) in the general case, our algorithm runs in quasipolynomial time and outputs a generalized depth-3 circuit (as defined in [1]) with k multiplication gates. For example, the polynomials computed by generalized depth- 3 circuits can be computed by quasi-polynomial sized depth-3 circuits. in fact, our algorithm works in the slightly more general case where the black box holds a generalized depth-3 circuits. Prior to this work there were reconstruction algorithms for several different models of bounded depth circuits: the well studied class of depth-2 arithmetic circuits (that compute sparse polynomials) and its close by model of depth-3 set-multilinear circuits. For the class of depth-3 circuits only the case of k = 2 (i.e. ∑π∑(2) circuits) was known. Our proof technique combines ideas from [2] and [1] with some new ideas. Our most notable new ideas are: We prove the existence of a unique canonical representation of depth-3 circuits. This enables us to work with a specific representation in mind. Another technical contribution is an isolation lemma for depth-3 circuits that enables us to reconstruct a single multiplication gate of the circuit.

Original language | English |
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Title of host publication | Proceedings of the 2009 24th Annual IEEE Conference on Computational Complexity, CCC 2009 |

Pages | 274-285 |

Number of pages | 12 |

DOIs | |

State | Published - 2009 |

Externally published | Yes |

Event | 2009 24th Annual IEEE Conference on Computational Complexity, CCC 2009 - Paris, France Duration: 15 Jul 2009 → 18 Jul 2009 |

### Publication series

Name | Proceedings of the Annual IEEE Conference on Computational Complexity |
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ISSN (Print) | 1093-0159 |

### Conference

Conference | 2009 24th Annual IEEE Conference on Computational Complexity, CCC 2009 |
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Country/Territory | France |

City | Paris |

Period | 15/07/09 → 18/07/09 |