Abstract

An arithmetic read-once formula (ROF for short) is a formula (a circuit whose underlying graph is a tree) in which the operations aref+;g and each 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 xi with a univariate polynomial Ti(xi). We obtain a deterministic non-adaptive reconstruction algorithm for PROFs, that is, an algorithm that, given black-box access to a PROF, constructs a PROF computing the same polynomial. The running time of the algorithm is (nd) O(log n) for PROFs of individual degrees at most d. To the best of our knowledge our results give the first subexponential-time deterministic reconstruction algorithms for ROFs. Another question that we study is the following generalization of the polynomial identity testing (PIT) problem. Given an arithmetic circuit computing a polynomial P( ¯ x), decide whether there is a PROF computing P( ¯ x), and find one if one exists. We call this question the read-once testing problem (ROT for short). Previous (randomized) algorithms for reconstruction of ROFs imply that there exists a randomized algorithm for the ROT problem.

Highlights

  • Let F be a field and C a class of arithmetic circuits

  • Another question that we study is the following generalization of the polynomial identity testing (PIT) problem

  • We focus in this work on ROFs and, as a result, give the first deterministic sub-exponential time reconstruction algorithm for PROFs

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Summary

Introduction

Let F be a field and C a class of arithmetic circuits. The reconstruction problem for the class C is defined as follows. In the black-box model, the algorithm is not given access to the circuit and instead it can only query the value of the circuit on different inputs. A deterministic black-box PIT algorithm for a circuit class C has to compute a hitting-set for that class, that is, a set of points H such that if a circuit from C evaluates to zero over H it must compute the zero polynomial. One of the main properties of a hitting set H is that the values of any circuit from C on H describe the polynomial computed by that circuit uniquely, in the sense that two circuits that agree on H must compute the same polynomial since their difference evaluates to zero over H Such a set does not provide us with an efficient algorithm for reconstructing circuits from C. We focus in this work on ROFs and, as a result, give the first deterministic sub-exponential time reconstruction algorithm for PROFs

Our results
Reconstruction of preprocessed read-once formulas
Read-once testing
Related work
Techniques
Subsequent work
Organization
Preliminaries
Partial derivatives
Commutator
Polynomials and circuit classes
ROFs and ROPs
The gate-graph of ROFs and ROPs
Factors of ROPs
Commutators of ROPs
From PIT to justifying assignments
ROF graph-related algorithms
Factoring a ROF
Counting the number of monomials in a PROP
Reconstruction of a PROF
Constructing the gate-graph
Conclusion
Reconstruction of a general PROF
Randomized PROF reconstruction
Deterministic PROF reconstruction
Non-adaptive PROF reconstruction
Generic scheme
Read-once verification
Sparse polynomials
Depth-3 circuits
Multilinear depth-4 circuits
Multilinear read-k formulas
Full Text
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