NDLI: Elusive functions and lower bounds for arithmetic circuits

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Elusive functions and lower bounds for arithmetic circuits

Content Provider	ACM Digital Library
Author	Raz, Ran
Abstract	A basic fact in linear algebra is that the image of the curve $f(x)=(x^{1},x^{2},x^{3},.,x^{m}),$ say over C, is not contained in any m-1 dimensional affine subspace of $C^{m}.$ In other words, the image of f is not contained in the image of any polynomial-mapping $Γ:C^{m-1}$ → $C^{m}$ of degree~1 (that is, an affine mapping). Can one give an explicit example for a polynomial curve f:C → $C^{m},$ such that, the image of f is not contained in the image of any polynomial-mapping $Γ:C^{m-1}$ → $C^{m}$ of degree 2? In this paper, we show that problems of this type are closely related to proving lower bounds for the size of general arithmetic circuits. For example, any explicit f as above (with the right notion of explicitness implies super-polynomial lower bounds for computing the permanent over~C. More generally, we say that a polynomial-mapping $f:F^{n}$ → $F^{m}$ is (s,r)-elusive, if for every polynomial-mapping $Γ:F^{s}$ → $F^{m}$ of degree r, Im(f) ⊄ Im(Γ). We show that for many settings of the parameters n,m,s,r, explicit constructions of elusive polynomial-mappings imply strong (up to exponential) lower bounds for general arithmetic circuits. Finally, for every r < log n, we give an explicit example for a polynomial-mapping $f:F^{n}$ → $F^{n^{2}},$ of degree O(r), that is (s,r)-elusive for s = $n^{1+Ω(1/r)}.$ We use this to construct for any r, an explicit example for an n-variate polynomial of total-degree O(r), with coefficients in {0,1,}such that, any depth r arithmetic circuit for this polynomial (over any field) is of size ≥ $n^{1+Ω(1/r)}.$ In particular, for any constant r, this gives a constant degree polynomial, such that, any depth r arithmetic circuit for this polynomial is of size ≥ $n^{1+Ω(1)}.$ Previously, only lower bounds of the type Ω(n • $λ_{r}$ (n)), where $λ_{r}$ (n) are extremely slowly growing functions (e.g., $λ_{5}(n)$ = log n, and $λ_{7}(n)$ = log* log*n), were known for constant-depth arithmetic circuits for polynomials of constant degree.
Starting Page	711
Ending Page	720
Page Count	10
File Format	PDF
ISBN	9781605580470
DOI	10.1145/1374376.1374479
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2008-05-17
Publisher Place	New York
Access Restriction	Subscribed
Subject Keyword	Lower bounds Bounded depth circuits Arithmetic circuits Circuit complexity
Content Type	Text
Resource Type	Article

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