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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 ..."
Abstract

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.
Variations by complexity theorists on three themes of
 Computational Complexity
, 2005
"... This paper surveys some connections between geometry and complexity. A main role is played by some quantities —degree, Euler characteristic, Betti numbers — associated to algebraic or semialgebraic sets. This role is twofold. On the one hand, lower bounds on the deterministic time (sequential and pa ..."
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Cited by 12 (4 self)
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This paper surveys some connections between geometry and complexity. A main role is played by some quantities —degree, Euler characteristic, Betti numbers — associated to algebraic or semialgebraic sets. This role is twofold. On the one hand, lower bounds on the deterministic time (sequential and parallel) necessary to decide a set S are established as functions of these quantities associated to S. The optimality of some algorithms is obtained as a consequence. On the other hand, the computation of these quantities gives rise to problems which turn out to be hard (or complete) in different complexity classes. These two kind of results thus turn the quantities above into measures of complexity in two quite different ways. 1
The complexity of factors of multivariate polynomials
 In Proc. 42th IEEE Symp. on Foundations of Comp. Science
, 2001
"... The existence of string functions, which are not polynomial time computable, but whose graph is checkable in polynomial time, is a basic assumption in cryptography. We prove that in the framework of algebraic complexity, there are no such families of polynomial functions ofbounded degree over field ..."
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Cited by 5 (2 self)
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The existence of string functions, which are not polynomial time computable, but whose graph is checkable in polynomial time, is a basic assumption in cryptography. We prove that in the framework of algebraic complexity, there are no such families of polynomial functions ofbounded degree over fields of characteristic zero. The proof relies on a polynomial upper bound on the approximative complexity of a factor of a polynomial in terms of the (approximative) complexity of and the degree of the factor. This extends a result by Kaltofen (STOC 1986). The concept of approximative complexity allows to cope with the case that a factor has an exponential multiplicity, by using a perturbation argument. Our result extends to randomized (twosided error) decision complexity. 1
On Randomized Algebraic Test Complexity
, 1992
"... We investigate the impact of randomization on the complexity of deciding membership in a (semi)algebraic subset X ae R m . Examples are exhibited where allowing for a certain error probability ffl in the answer of the algorithms the complexity of decision problems decreases. A randomized(\Omega ..."
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Cited by 1 (1 self)
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We investigate the impact of randomization on the complexity of deciding membership in a (semi)algebraic subset X ae R m . Examples are exhibited where allowing for a certain error probability ffl in the answer of the algorithms the complexity of decision problems decreases. A randomized(\Omega k ; f=; g) decision tree (k ` R a subfield) over m will be defined as a pair (T ; ¯) where ¯ a probability measure on some R n and T is a(\Omega k ; f=; g)decision tree over m+n. We prove a general lower bound on the average decision complexity for testing membership in an irreducible algebraic subset X ae R m and apply it to kgeneric complete intersection of polynomials of the same degree, extending results in [4, 6]. We also give applications to nongeneric cases, such as graphs of elementary symmetric functions, SL(m; R), and determinant varieties, extending results in [Li 90]. 1 Dept. of Computer Science, University of Bonn, 5300 Bonn 1. Sponsored by the Schweizerischer Nat...