Hardness-Randomness Tradeoffs for Bounded Depth Arithmetic Circuits

Dvir, Zeev; Shpilka, Amir; Yehudayoff, Amir

doi:10.1137/080735850

Cited by 59 publications

(124 citation statements)

References 35 publications

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“…In [KI04] Kabanets and Impagliazzo proved that PIT can be derandomized if and only if NEXP does not have small arithmetic circuits. Later, [DSY09] observed a similar result for bounded depth circuits. By combining their result with our Observation 1 we obtain the following corollary.…”

Section: Extensions Of Theorem 11supporting

confidence: 61%

“…Another line of works that is related to our results is that of Kabanets and Impagliazzo [KI04] and of [DSY09]. There it was shown that the question of derandomizing the PIT problem is closely related to the problem of proving lower bounds for arithmetic circuits (Corollary 4.7, given in Section 4, is an analogous result).…”

Section: Related Workmentioning

confidence: 79%

“…In [KI04], Kabanets and Impagliazzo proved that the following three conditions cannot be true simultaneously for arithmetic circuits (the bounded depth version was later observed in [DSY09]). …”

Section: Extensions Of Theorem 11mentioning

confidence: 99%

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On the Relation between Polynomial Identity Testing and Finding Variable Disjoint Factors

Shpilka

Volkovich

2010

Automata, Languages and Programming

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We say that a polynomial f (x 1 , . . . , x n ) is indecomposable if it cannot be written as a product of two polynomials that are defined over disjoint sets of variables. The polynomial decomposition problem is defined to be the task of finding the indecomposable factors of a given polynomial. Note that for multilinear polynomials, factorization is the same as decomposition, as any two different factors are variable disjoint.In this paper we show that the problem of derandomizing polynomial identity testing is essentially equivalent to the problem of derandomizing algorithms for polynomial decomposition. More accurately, we show that for any reasonable circuit class there is a deterministic polynomial time (black-box) algorithm for polynomial identity testing of that class if and only if there is a deterministic polynomial time (black-box) algorithm for factoring a polynomial, computed in the class, to its indecomposable components.An immediate corollary is that polynomial identity testing and polynomial factorization are equivalent (up to a polynomial overhead) for multilinear polynomials. In addition, we observe that derandomizing the polynomial decomposition problem is equivalent, in the sense of Kabanets and Impagliazzo [KI04], to proving arithmetic circuit lower bounds to NEXP.Our approach uses ideas from [SV08], that showed that the polynomial identity testing problem for a circuit class C is essentially equivalent to the problem of deciding whether a circuit from C computes a polynomial that has a read-once arithmetic formula.

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Section: Extensions Of Theorem 11supporting

confidence: 61%

Section: Related Workmentioning

confidence: 79%

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On the Relation between Polynomial Identity Testing and Finding Variable Disjoint Factors

Shpilka

Volkovich

2010

Automata, Languages and Programming

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“…PIT is a central problem in algebraic complexity. Deterministically solving PIT is known to imply lower bounds for arithmetic circuits [HS80,Agr05,KI04,DSY09]. PIT also has some algorithmic implications.…”

Section: Every Other Gate Of C Is Labeled By Either '×' or '+' And mentioning

confidence: 99%

Testing Equivalence of Polynomials under Shifts

Dvir

Oliveira

Shpilka

2014

Automata, Languages, and Programming

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Two polynomials f, g ∈ F[x 1 , . . . , x n ] are called shift-equivalent if there exists a vector (a 1 , . . . , a n ) ∈ F n such that the polynomial identity f (x 1 + a 1 , . . . , x n + a n ) ≡ g(x 1 , . . . , x n ) holds. Our main result is a new randomized algorithm that tests whether two given polynomials are shift equivalent. Our algorithm runs in time polynomial in the circuit size of the polynomials, to which it is given black box access. This complements a previous work of Grigoriev [Gri97] who gave a deterministic algorithm running in time n O(d) for degree d polynomials.Our algorithm uses randomness only to solve instances of the Polynomial Identity Testing (PIT) problem. Hence, if one could de-randomize PIT (a long-standing open problem in complexity) a de-randomization of our algorithm would follow. This establishes an equivalence between de-randomizing shift-equivalence testing and de-randomizing PIT (both in the black-box and the white-box setting). For certain restricted models, such as Read Once Branching Programs, we already obtain a deterministic algorithm using existing PIT results.

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“…Kabanets and Impagliazzo [26] showed that, if the permanent requires large arithmetic circuits, then the probabilistic algorithm to test if two arithmetic formulae (or, more generally, two arithmetic circuits of polynomial degree) are equivalent can be simulated by a quick deterministic algorithm. Subsequently, Dvir et al [27] built on the techniques of Kabanets and Impagliazzo, to show that, if one could present a multilinear polynomial (such as the permanent) that requires depthd arithmetic formulae of size 2 n e , then the probabilistic algorithm to test if two arithmetic circuits of depth d − 5 are equivalent (where, in addition, the variables in these circuits have degree at most log O(1) n) can be derandomized to obtain a 2 log O(1) n deterministic algorithm for the problem.…”

Section: Introductionmentioning

confidence: 99%

Uniform derandomization from pathetic lower bounds

Allender

Arvind

Santhanam

et al. 2012

Phil. Trans. R. Soc. A.

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The notion of probabilistic computation dates back at least to Turing, who also wrestled with the practical problems of how to implement probabilistic algorithms on machines with, at best, very limited access to randomness. A more recent line of research, known as derandomization, studies the extent to which randomness is superfluous. A recurring theme in the literature on derandomization is that probabilistic algorithms can be simulated quickly by deterministic algorithms, if one can obtain impressive (i.e. superpolynomial, or even nearly exponential) circuit size lower bounds for certain problems. In contrast to what is needed for derandomization, existing lower bounds seem rather pathetic. Here, we present two instances where 'pathetic' lower bounds of the form n 1+e would suffice to derandomize interesting classes of probabilistic algorithms. We show the following:-If the word problem over S 5 requires constant-depth threshold circuits of size n 1+e for some e > 0, then any language accepted by uniform polynomial size probabilistic threshold circuits can be solved in subexponential time (and, more strongly, can be accepted by a uniform family of deterministic constantdepth threshold circuits of subexponential size). -If there are no constant-depth arithmetic circuits of size n 1+e for the problem of multiplying a sequence of n 3 × 3 matrices, then, for every constant d, black-box identity testing for depth-d arithmetic circuits with bounded individual degree can be performed in subexponential time (and even by a uniform family of deterministic constant-depth AC 0 circuits of subexponential size).

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Hardness-Randomness Tradeoffs for Bounded Depth Arithmetic Circuits

Cited by 59 publications

References 35 publications

On the Relation between Polynomial Identity Testing and Finding Variable Disjoint Factors

On the Relation between Polynomial Identity Testing and Finding Variable Disjoint Factors

Testing Equivalence of Polynomials under Shifts

Uniform derandomization from pathetic lower bounds

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