Reconstruction of Full Rank Algebraic Branching Programs

Abstract

An algebraic branching program (ABP) A can be modelled as a product expression X 1 • X 2. .. X d , where X 1 and X d are 1×w and w ×1 matrices respectively, and every other X k is a w ×w matrix; the entries of these matrices are linear forms in m variables over a field F (which we assume to be either Q or a field of characteristic poly(m)). The polynomial computed by A is the entry of the 1 × 1 matrix obtained from the product d k=1 X k. We say A is a full rank ABP if the w 2 (d − 2) + 2w linear forms occurring in the matrices X 1 , X 2 ,. .. , X d are F-linearly independent. Our main result is a randomized reconstruction algorithm for full rank ABPs: Given blackbox access to an m-variate polynomial f of degree at most m, the algorithm outputs a full rank ABP computing f if such an ABP exists, or outputs 'no full rank ABP exists' (with high probability). The running time of the algorithm is polynomial in m and β, where β is the bit length of the coefficients of f. The algorithm works even if X k is a w k−1 × w k matrix (with w 0 = w d = 1), and w = (w 1 ,. .. , w d−1) is unknown.

Randomized polynomial-time equivalence between determinant and trace-IMM equivalence tests

“…But, do they have similar equivalence testing complexity? Our work here, in conjunction with [GGKS19] and [KNST19], gives an answer to this question. 7 The problem is well-posed even if f is given verbosely as a list of coefficients and it is not required to output an invertible transformation A in the 'yes' case.…”

“…1. No knowledge of w: The algorithm requires no knowledge of w, if the input polynomial f is equivalent to Tr-IMM w,d for some w ∈ N then the algorithm finds such a w. The algorithm in Theorem 1 first reduces TRACE to TRACE-TI (finding w in this step), and then solves TRACE-TI using DET oracle over F. The reduction from TRACE to TRACE-TI (which resembles a similar reduction used in the equivalence test for IMM [KNST19]) does not require oracle access to DET. A randomized polynomial-time algorithm for TRACE-TI over C was given in [Gro12], but the algorithm there does not reduce TRACE-TI to DET.…”

Derandomization and absolute reconstruction for sums of powers of linear forms

“…Polynomial factorization is an important subroutine in many if not most reconstruction algorithms for arithmetic circuits, see e.g. [18,19,35,36,38,39,48]. It may even seem unavoidable for some problems: reconstruction of ΠΣ circuits is nothing but the problem of factorization into products of linear forms, and reconstruction of ΠΣΠ circuits is factorization into products of sparse polynomials.…”

“…It has found applications to the elimination of redundant variables [36], the computation of the Lie algebra of a polynomial [37], the reconstruction of random arithmetic formulas [25], full rank algebraic programs [38] and nondegenerate depth 3 circuits [39].…”

“…Equivalence to P d for arbitrary d was studied by Kayal [36]. This paper also begins a study of equivalence to other specific polynomials such as the elementary symmetric polynomials; this study is continued in [20,23,24,37,38], in particular for the permanent and determinant polynomials. The contributions of the present paper are twofold:…”