NDLI: A super-polynomial lower bound for regular arithmetic formulas

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A super-polynomial lower bound for regular arithmetic formulas

Content Provider	ACM Digital Library
Author	Saptharishi, Ramprasad Saha, Chandan Kayal, Neeraj
Abstract	We consider arithmetic formulas consisting of alternating layers of addition (+) and multiplication (×) gates such that the fanin of all the gates in any fixed layer is the same. Such a formula Φ which additionally has the property that its formal/syntactic degree is at most twice the (total) degree of its output polynomial, we refer to as a regular formula. As usual, we allow arbitrary constants from the underlying field F on the incoming edges to a + gate so that a + gate can in fact compute an arbitrary F-linear combination of its inputs. We show that there is an $(n^{2}$ + 1)-variate polynomial of degree 2n in VNP such that any regular formula computing it must be of size at least nΩ(log n). Along the way, we examine depth four (ΣΠΣΠ) regular formulas wherein all multiplication gates in the layer adjacent to the inputs have fanin a and all multiplication gates in the layer adjacent to the output node have fanin b. We refer to such formulas as $ΣΠ^{[b]}ΣΠ^{[a]}-formulas.$ We show that there exists an $n^{2}-variate$ polynomial of degree n in VNP such that any $ΣΠ^{[O(√n)]}ΣΠ^{[√n]}-formula$ computing it must have top fan-in at least 2Ω(√n·log n). In comparison, Tavenas [Tav13] has recently shown that every $n^{O(1)}-variate$ polynomial of degree n in VP admits a $ΣΠ^{[O(√n)]}ΣΠ^{[√n]}-formula$ of top fan-in 2O(√n·log n). This means that any further asymptotic improvement in our lower bound for such formulas (to say 2ω(√n log n)) will imply that VP is different from VNP.
Starting Page	146
Ending Page	153
Page Count	8
File Format	PDF MP4
ISBN	9781450327107
DOI	10.1145/2591796.2591847
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2014-05-31
Publisher Place	New York
Access Restriction	Subscribed
Subject Keyword	Lower bounds Depth-4 circuits Arithmetic circuits Vnp Vp
Content Type	Audio Text
Resource Type	Article

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