Characterizing Arithmetic Read-Once Formulae

Volkovich, Ilya

doi:10.1145/2858783

Cited by 12 publications

(9 citation statements)

References 24 publications

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“…In particular, low-defect polynomials are in fact read-once polynomials, as considered in [16] for instance. See Sections 2 and 3 for more on these polynomials.…”

Section: Introductionmentioning

confidence: 99%

Integer complexity: Representing numbers of bounded defect

Altman¹

2016

Theoretical Computer Science

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Abstract. Define n to be the complexity of n, the smallest number of ones needed to write n using an arbitrary combination of addition and multiplication. John Selfridge showed that n ≥ 3 log 3 n for all n. Based on this, this author and Zelinsky defined [4] the "defect" of n, δ(n) := n − 3 log 3 n, and this author showed that the set of all defects is a well-ordered subset of the real numbers [1]. This was accomplished by showing that for a fixed real number r, there is a finite set S of polynomials called "low-defect polynomials" such that for any n with δ(n) < r, n has the form f (3 k 1 , . . . , 3 kr )3 k r+1 for some f ∈ S. However, using the polynomials produced by this method, many extraneous n with δ(n) ≥ r would also be represented. In this paper we show how to remedy this and modify S so as to represent precisely the n with δ(n) < r and remove anything extraneous. Since the same polynomial can represent both n with δ(n) < r and n with δ(n) ≥ r, this is not a matter of simply excising the appropriate polynomials, but requires "truncating" the polynomials to form new ones.

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“…In particular, low-defect polynomials are in fact read-once polynomials, as considered in [16] for instance. See Sections 2 and 3 for more on these polynomials.…”

Section: Introductionmentioning

confidence: 99%

Integer complexity: Representing numbers of bounded defect

Altman¹

2016

Theoretical Computer Science

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“…Also note that augmented low-defect polynomials are never themselves low-defect polynomials; as we will see in a moment (Proposition 2.10), low-defect polynomials always have nonzero constant term, whereas augmented low-defect polynomials always have zero constant term. We can also observe that low-defect polynomials are in fact read-once polynomials as discussed in for instance [21].…”

Section: 2mentioning

confidence: 63%

Integer complexity: the integer defect

Altman

2019

Moscow J. Comb. Number Th.

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Define n to be the complexity of n, the smallest number of ones needed to write n using an arbitrary combination of addition and multiplication. John Selfridge showed that n ≥ 3 log 3 n for all n, leading this author and Zelinsky to define the defect of n, δ(n), to be the difference n − 3 log 3 n. Meanwhile, in the study of addition chains, it is common to consider s(n), the number of small steps of n, defined as ℓ(n) − ⌊log 2 n⌋, an integer quantity. So here we analogously define D(n), the integer defect of n, an integer version of δ(n) analogous to s(n). Note that D(n) is not the same as ⌈δ(n)⌉.We show that D(n) has additional meaning in terms of the defect wellordering considered in [3], in that D(n) indicates which powers of ω the quantity δ(n) lies between when one restricts to n with n lying in a specified congruence class modulo 3. We also determine all numbers n with D(n) ≤ 1, and use this to generalize a result of Rawsthorne [18].

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“…ROFs are syntactic multilinear by definition and have received wide attention in the literature. Volkovich [40] gave a complete characterization of polynomials computed by ROFs. Further, Minahan and Volkovich [24] obtained a complete derandomization of the polynomial identity testing problem on ROFs.…”

Section: Models and Results: (1) Sum Of Rofsmentioning

confidence: 99%

“…Further, Minahan and Volkovich [24] obtained a complete derandomization of the polynomial identity testing problem on ROFs. While most of the multilinear polynomials are not computable by ROFs [40], sum of ROFs, denoted by Σ•ROF is a natural model of computation for multilinear polynomials. Shpilka and Volkovich showed that a restricted form of Σ • ROF requires linear summands to compute the monomial x 1 x 2 • • • x n .…”

Section: Models and Results: (1) Sum Of Rofsmentioning

confidence: 99%

Limitations of Sums of Bounded-Read Formulas

Ghosal¹,

Rao²

2020

Preprint

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Proving super polynomial size lower bounds for various classes of arithmetic circuits computing explicit polynomials is a very important and challenging task in algebraic complexity theory. We study representation of polynomials as sums of weaker models such as read once formulas (ROFs) and read once oblivious algebraic branching programs (ROABPs). We prove:(1) An exponential separation between sum of ROFs and read-k formulas for some constant k.(2) A sub-exponential separation between sum of ROABPs and syntactic multilinear ABPs.Our results are based on analysis of the partial derivative matrix under different distributions. These results highlight richness of bounded read restrictions in arithmetic formulas and ABPs.Finally, we consider a generalization of multilinear ROABPs known as strict-interval ABPs defined in [Ramya-Rao, MFCS2019]. We show that strict-interval ABPs are equivalent to ROABPs upto a polynomial size blow up. In contrast, we show that interval formulas are different from ROFs and also admit depth reduction which is not known in the case of strict-interval ABPs.

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Characterizing Arithmetic Read-Once Formulae

Cited by 12 publications

References 24 publications

Integer complexity: Representing numbers of bounded defect

Integer complexity: Representing numbers of bounded defect

Integer complexity: the integer defect

Limitations of Sums of Bounded-Read Formulas

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