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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 28 (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 Measure of Space for Computing over the Reals
, 2006
"... Abstract. We propose a new complexity measure of space for the BSS model of computation. We define LOGSPACEW and PSPACEW complexity classes over the reals. We prove that LOGSPACEW is included in NC 2 R ∩ PW, i.e. is small enough for being relevant. We prove that the Real Circuit Decision Problem is ..."
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Abstract. We propose a new complexity measure of space for the BSS model of computation. We define LOGSPACEW and PSPACEW complexity classes over the reals. We prove that LOGSPACEW is included in NC 2 R ∩ PW, i.e. is small enough for being relevant. We prove that the Real Circuit Decision Problem is PRcomplete under LOGSPACEW reductions, i.e. that LOGSPACEW is large enough for containing natural algorithms. We also prove that PSPACEW is included in PARR.
Query Languages For SemiAlgebraic Databases Based On Descriptive Complexity Over R
, 1998
"... . We propose the study of query languages for databases involving real numbers as data (called semialgebraic databases in the sequel). As main new aspect our approach is based on real number complexity theory as introduced in [8] and descriptive complexity for the latter developed in [17]. Usin ..."
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. We propose the study of query languages for databases involving real numbers as data (called semialgebraic databases in the sequel). As main new aspect our approach is based on real number complexity theory as introduced in [8] and descriptive complexity for the latter developed in [17]. Using this formal framework a uniform treatment of semialgebraic query languages is obtained. Precise results about both the data and the expressioncomplexity of several such query languages are proved. More explicitly, relying on descriptive complexity theory over R gives the possibility to derive a hierarchy of complete languages for most of the important real number complexity classes. A clear correspondence between different logics and such complexity classes is established. In particular, it is possible to formalize queries involving in a uniform manner real spaces of different dimensions. This can be done in such a way that the logical description exactly reflects the computation...