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].