A lower bound for the formula size of rational functions

Kalorkoti, K.

doi:10.1007/bfb0012780

Cited by 14 publications

(16 citation statements)

References 2 publications

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“…Kalorkoti proved an Ω(n 3 ) lower bound on the formula-size of determinant [Kal85]. His argument can be thought of as the algebraic analog of Nechiporuk's argument (for the Boolean setting), that gives a lower bound on the Boolean formula size of a Boolean function f by counting the number of different sub-functions of f .…”

Section: Sketchmentioning

confidence: 99%

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Arithmetic Circuits: A survey of recent results and open questions

Shpilka

Yehudayoff

2009

FNT in Theoretical Computer Science

293

275

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A large class of problems in symbolic computation can be expressed as the task of computing some polynomials; and arithmetic circuits form the most standard model for studying the complexity of such computations. This algebraic model of computation attracted a large amount of research in the last five decades, partially due to its simplicity and elegance. Being a more structured model than Boolean circuits, one could hope that the fundamental problems of theoretical computer science, such as separating P from NP, will be easier to solve for arithmetic circuits. However, in spite of the appearing simplicity and the vast amount of mathematical tools available, no major breakthrough has been seen. In fact, all the fundamental questions are still open for this model as well. Nevertheless, there has been a lot of progress in the area and beautiful results have been found, some in the last few years. As examples we mention the connection between polynomial identity testing and lower bounds of Kabanets and Impagliazzo, the lower bounds of Raz for multilinear formulas, and two new approaches for proving lower bounds: Geometric Complexity Theory and Elusive Functions.The goal of this monograph is to survey the field of arithmetic circuit complexity, focusing mainly on what we find to be the most interesting and accessible research directions. We aim to cover the main results and techniques, with an emphasis on works from the last two decades. In particular, we discuss the recent lower bounds for multilinear circuits and formulas, the advances in the question of deterministically checking polynomial identities and the results regarding reconstruction of arithmetic circuits. We do, however, also cover part of the classical works on arithmetic circuits. In order to keep this monograph at a reasonable length, we do not give full proofs of most theorems, but rather try to convey the main ideas behind each proof and demonstrate it, where possible, by proving some special cases.

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Section: Sketchmentioning

confidence: 99%

“…In particular, we explain Strassen's degree bound that gives a super-linear lower bound for general circuits [Str73b] and Kalorkoti's quadratic lower bound on the size of general formulas [Kal85]. We discuss the lower bounds for the size of bounded depth circuits [SS91,Raz08].…”

Section: Lower Bounds For Arithmetic Circuitsmentioning

confidence: 99%

Arithmetic Circuits: A survey of recent results and open questions

Shpilka

Yehudayoff

2009

FNT in Theoretical Computer Science

293

275

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“…While the conjectured lower bound for formulas computing Det n is superpoly(n), the best known lower bound for the same is (n 3 ) (Kalorkoti, 1985). (A slightly better (N 2 ) bound is known for formulas computing an N-variate VNP-polynomial) 1 .…”

Section: Some Known Formula Lower Boundsmentioning

confidence: 99%

Improved Lower Bound for Multi-R-IC Depth Four Circuits as a Function of the Number of Input Variables

Hegde

2017

Proceedings of the Indian National Science Academy

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An arithmetic circuit (respectively, formula) is a rooted graph (respectively, tree) whose nodes are addition or multiplication gates and input variables/nodes. It computes a polynomial in a natural way. The formal degree of an addition (respectively, multiplication) gate with respect to a variable x is defined as the maximum (respectively, sum) of the formal degrees of its children, with respect to x. The formal degree of an input node with respect to x is 1 if the node is labelled with x, and 0 otherwise. In a multi-r-ic formula, the formal degree of every gate with respect to every variable is at most r. Multi-r-ic formulas make an intermediate model between multilinear formulas (the r = 1 case), for which lower bounds are relatively well-understood, and general formulas (the unbounded-r case), which are conjectured to have superpolynomial size lower bound.On depth four multi-r-ic formulas/circuits computing IMM n,d -the product of d symbolic metrices of size n x n each,

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“…Up to now, there is no known tool for obtaining a non-trivial lower bound for the circuit size of the permanent. The only known non-trivial lower bound for the formula size of the permanent is due to Kalorkoti [7], which says that over any field, the formula size of P er n (F) is at least Ω(n 3 ). (Kalorkoti proved the same lower bound for the formula size of the determinant, and Pavel Hrubeš told me that Kalorkoti's proof works also for the formula size of the permanent.)…”

Section: Introductionmentioning

confidence: 99%

Lower Bounds for the Circuit Size of Partially Homogeneous Polynomials

Lê

2017

J Math Sci

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Abstract. In this paper we associate to each multivariate polynomial f that is homogeneous relative to a subset of its variables a series of polynomial families P λ (f ) of m-tuples of homogeneous polynomials of equal degree such that the circuit size of any member in P λ (f ) is bounded from above by the circuit size of f . This provides a method for obtaining lower bounds for the circuit size of f by proving (s, r)-(weak) elusiveness of the polynomial mapping associated with P λ (f ). We discuss some algebraic methods for proving the (s, r)-(weak) elusiveness. We also improve estimates in the normal homogeneous-form of an arithmetic circuit obtained by Raz in [12] which results in better lower bounds for circuit size (Lemma 6.7, Remark 6.8). Our methods yield non-trivial lower bound for the circuit size of several classes of multivariate homogeneous polynomials (Corollary 6.9, Example 6.10).

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A lower bound for the formula size of rational functions

Cited by 14 publications

References 2 publications

Arithmetic Circuits: A survey of recent results and open questions

Arithmetic Circuits: A survey of recent results and open questions

Improved Lower Bound for Multi-R-IC Depth Four Circuits as a Function of the Number of Input Variables

Lower Bounds for the Circuit Size of Partially Homogeneous Polynomials

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