Lower bounds for Sum and Sum of Products of Read-once Formulas

Abstract

We study limitations of polynomials computed by depth-2 circuits built over read-once formulas (ROFs). In particular: • We prove a 2 Ω( n ) lower bound for the sum of ROFs computing the 2 n -variate polynomial in VP defined by Raz and Yehudayoff [21]. • We obtain a 2 Ω(√ n ) lower bound on the size of Σ Π [ n 1/15 ] arithmetic circuits built over restricted ROFs of unbounded depth computing the permanent of an n × n matrix (superscripts on gates denote bound on the fan-in). The restriction is that the number of variables with + gates as a parent in a proper sub formula of the ROF has to be bounded by √ n . This proves an exponential lower bound for a subclass of possibly non-multilinear formulas of unbounded depth computing the permanent polynomial. • We also show an exponential lower bound for the above model against a polynomial in VP. • Finally, we observe that the techniques developed yield an exponential lower bound on the size of ΣΠ [ N 1/30 ] arithmetic circuits built over syntactically multi-linear ΣΠΣ [ N 1/4 ] arithmetic circuits computing a product of variable disjoint linear forms on N variables, where the superscripts on gates denote bound on the fan-in. Our proof techniques are built on the measure developed by Kumar et al. [14] and are based on a non-trivial analysis of ROFs under random partitions. Further, our results exhibit strengths and provide more insight into the lower bound techniques introduced by Raz [19].

Limitations of Sums of Bounded-Read Formulas

“…Further, Mahajan and Tawari [22] obtained a tight lower bound on the size of Σ•ROF computing an elementary symmetric polynomial. Ramya and Rao [29] obtained an exponential lower bound on the number of ROFs required to compute a polynomial in VP. In this article, we improve the lower bound in [29] to obtain an exponential separation between read-k formulas and Σ • ROF for a sufficiently large constant k. Formally, we prove: Theorem 1.…”

“…Ramya and Rao [29] obtained an exponential lower bound on the number of ROFs required to compute a polynomial in VP. In this article, we improve the lower bound in [29] to obtain an exponential separation between read-k formulas and Σ • ROF for a sufficiently large constant k. Formally, we prove: Theorem 1. There is constant k > 0 and a family of multilinear polynomials f PRY computable by read-k formulas such that if…”

“…Ramya and Rao [29] obtain an exponential separation between Σ • ROF and multilinear VP. Our result is an extension of this result for the case of a simpler polynomial computable by bounded read formulas.…”

“…In [29], Ramya and Rao show an exponential lower bound for the sum of ROFs computing a polynomial in VP. While this establishes a super polynomial separation between Σ • ROF and syntactic multilinear formulas, it is interesting to see if this separation is exponential.…”

“…Given that there is no promising approach yet to prove super quadratic size lower bounds for smABPs, it is imperative to consider further structural restrictions on smABPs and formulas to develop finer insights into the difficulty of the problem. Following the works in [27,29,28], we study syntactic multilinear formulas and smABPs with restrictions on the number of reads of variables and the order in which variables appear in a smABP.…”