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].
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