## Abstract

An arithmetic read-once formula (ROF for short) is a formula (a circuit whose underlying graph is a tree) in which the operations are (Formula presented.) and such that every input variable labels at most one leaf. A preprocessed ROF (PROF for short) is a ROF in which we are allowed to replace each variable x_{i} with a univariate polynomial T_{i}(x_{i}). In this paper, we study the problems of designing deterministic identity testing algorithms for models related to preprocessed ROFs. Our main result gives PIT algorithms for the sum of k preprocessed ROFs, of individual degrees at most d (i.e., each T_{i}(x_{i}) is of degree at most d), that run in time (Formula presented.) in the white-box setting and in time (Formula presented.) in the black-box setting. We also obtain better algorithms when the formulas have a small depth that lead to an improvement in the best PIT algorithm for multilinear depth-3 (Formula presented.) circuits. Our main technique is to prove a hardness of representation result, namely a theorem showing a relatively mild lower bound on the sum of k PROFs. We then use this lower bound in order to design our PIT algorithm.

Original language | English |
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Pages (from-to) | 477-532 |

Number of pages | 56 |

Journal | Computational Complexity |

Volume | 24 |

Issue number | 3 |

DOIs | |

State | Published - 21 Sep 2015 |

## Keywords

- Derandomization
- arithmetic circuits
- bounded-depth circuits
- identity testing
- read-once formulas