Non-commutative arithmetic circuits: depth reduction and size lower bounds

Allender, Eric; Jia, Jiao; Mahajan, Meena; Vinay, V.

doi:10.1016/s0304-3975(97)00227-2

Cited by 78 publications

(70 citation statements)

References 32 publications

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“…We can assume, without loss of generality, that all the intermediate polynomials computed inside the circuit C have degree bounded by d [22]. In [20,2] it is shown that C can be transformed to a circuit D of degree d, size M O(1) and depth O(log d) with multiplication gates of fan-in two. We do a careful analysis of the transformation in [2] to obtain a circuit D with more structural properties.…”

Section: Lemma 22 For Any N and K(mentioning

confidence: 99%

“…In [20,2] it is shown that C can be transformed to a circuit D of degree d, size M O(1) and depth O(log d) with multiplication gates of fan-in two. We do a careful analysis of the transformation in [2] to obtain a circuit D with more structural properties. In particular, we will be interested in getting good bounds on the degree of the gates.…”

Section: Lemma 22 For Any N and K(mentioning

confidence: 99%

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Arithmetic Circuits: A Chasm at Depth Four

Agrawal

Vinay

2008

2008 49th Annual IEEE Symposium on Foundations of Computer Science

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We show that proving exponential lower bounds on depth four arithmetic circuits imply exponential lower bounds for unrestricted depth arithmetic circuits. In other words, for exponential sized circuits additional depth beyond four does not help.We then show that a complete black-box derandomization of Identity Testing problem for depth four circuits with multiplication gates of small fanin implies a nearly complete derandomization of general Identity Testing.

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Section: Lemma 22 For Any N and K(mentioning

confidence: 99%

Section: Lemma 22 For Any N and K(mentioning

confidence: 99%

Arithmetic Circuits: A Chasm at Depth Four

Agrawal

Vinay

2008

2008 49th Annual IEEE Symposium on Foundations of Computer Science

Self Cite

167

219

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“…But #NC 1 is a class of functions from {0, 1} * to N, while VPA may have arbitrary input alphabets. Using standard terminology (see for instance [2,10]), we assume that the leaves of the #NC 1 circuits can be labeled by predicates of the form…”

Section: Definition 2 (#Vpa)mentioning

confidence: 99%

Counting Paths in VPA Is Complete for #NC 1

Krebs

Limaye

Mahajan

2010

Lecture Notes in Computer Science

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Abstract. We give a #NC 1 upper bound for the problem of counting accepting paths in any fixed visibly pushdown automaton. Our algorithm involves a non-trivial adaptation of the arithmetic formula evaluation algorithm of Buss, Cook, Gupta, Ramachandran ([9]). We also show that the problem is #NC 1 hard. Our results show that the difference between #BWBP and #NC 1 is captured exactly by the addition of a visible stack to a nondeterministic finite-state automaton.

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“…Some years later, Allender, Jiao, Mahajan and Vinay [2] studied this parallelization method and showed it could be done uniformly. Using the proof of these results, Agrawal and Vinay proved [1] that if an n-variate polynomial f of degree d = O(n) has a circuit of size 2 o(d+d log( …”

Section: Introductionmentioning

confidence: 99%

Improved bounds for reduction to depth 4 and depth 3

Tavenas

2015

Information and Computation

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Abstract. Koiran [8] showed that if an n-variate polynomial fn of degree d (with d = n O(1) ) is computed by a circuit of size s, then it is also computed by a homogeneous circuit of depth four and of size 2. Using this result, Gupta, Kamath, Kayal and Saptharishi [7] found an upper bound for the size of a depth three circuit computing fn. We improve here Koiran's bound. Indeed, we show that it is possible to transform an arithmetic circuit into a depth four circuit of size. Then, mimicking the proof in [7], it also impliesupper bound for depth three circuits. This new bound is not far from optimal in the sense that Gupta, Kamath, Kayal and Saptharishi [6] also showed a 2 Ω( √ d) lower bound for the size of homogeneous depth four circuits such that gates at the bottom have fan-in at most √ d. Finally, we show that this last lower bound also holds if the fan-in is at least √ d.

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Non-commutative arithmetic circuits: depth reduction and size lower bounds

Cited by 78 publications

References 32 publications

Arithmetic Circuits: A Chasm at Depth Four

Arithmetic Circuits: A Chasm at Depth Four

Counting Paths in VPA Is Complete for #NC 1

Improved bounds for reduction to depth 4 and depth 3

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