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Lower Bounds on Arithmetic Circuits via Partial Derivatives
 COMPUTATIONAL COMPLEXITY
, 1995
"... In this paper we describe a new technique for obtaining lower bounds on restriced classes of nonmonotone arithmetic circuits. The heart of this technique is a complexity measure for multivariate polynomials, based on the linear span of their partial derivatives. We use the technique to obtain new lo ..."
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Cited by 40 (6 self)
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In this paper we describe a new technique for obtaining lower bounds on restriced classes of nonmonotone arithmetic circuits. The heart of this technique is a complexity measure for multivariate polynomials, based on the linear span of their partial derivatives. We use the technique to obtain new lower bounds for computing symmetric polynomials and iterated matrix products.
Descriptive Complexity Theory over the Real Numbers
 LECTURES IN APPLIED MATHEMATICS
, 1996
"... We present a logical approach to complexity over the real numbers with respect to the model of Blum, Shub and Smale. The logics under consideration are interpreted over a special class of twosorted structures, called Rstructures: They consist of a finite structure together with the ordered field ..."
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Cited by 24 (8 self)
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We present a logical approach to complexity over the real numbers with respect to the model of Blum, Shub and Smale. The logics under consideration are interpreted over a special class of twosorted structures, called Rstructures: They consist of a finite structure together with the ordered field of reals and a finite set of functions from the finite structure into R. They are a special case of the metafinite structures introduced recently by Grädel and Gurevich. We argue that Rstructures provide the right class of structures to develop a descriptive complexity theory over R. We substantiate this claim by a number of results that relate logical definability on Rstructures with complexity of computations of BSSmachines.
A New Method to Obtain Lower Bounds for Polynomial Evaluation
, 1999
"... We present a new method to obtain lower bounds for the time complexity of polynomial evaluation procedures. Time, denoted by L, is measured in terms of nonscalar arithmetic operations. In contrast with known methods for proving lower complexity bounds, our method is purely combinatorial and does not ..."
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Cited by 2 (2 self)
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We present a new method to obtain lower bounds for the time complexity of polynomial evaluation procedures. Time, denoted by L, is measured in terms of nonscalar arithmetic operations. In contrast with known methods for proving lower complexity bounds, our method is purely combinatorial and does not require powerful tools from algebraic or diophantine geometry.
MULTIPLICATIVE COMPLEXITY OF DIRECT SUM OF QUADRATIC SYSTEMS
"... We consider the quadratic complexity of certain sets of quadratic forms. We study classes of direct sums of quadratic forms. For these classes of problems we show that the complexity of one direct sum is the sum of the complexities of the summands and that every minimal quadratic algorithm for compu ..."
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We consider the quadratic complexity of certain sets of quadratic forms. We study classes of direct sums of quadratic forms. For these classes of problems we show that the complexity of one direct sum is the sum of the complexities of the summands and that every minimal quadratic algorithm for computing the direct sums is a directsum algorithm.