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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 ..."
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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.
On the relations between distributive computability and the BSS model
"... This paper presents an equivalence result between computability in the BSS model and in a suitable distributive category. It is proved that the class of functions R m ! R n (with n; m finite and R a commutative, ordered ring) computable in the BSS model, and the functions distributively computab ..."
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Cited by 6 (3 self)
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This paper presents an equivalence result between computability in the BSS model and in a suitable distributive category. It is proved that the class of functions R m ! R n (with n; m finite and R a commutative, ordered ring) computable in the BSS model, and the functions distributively computable over a natural distributive graph based on the operations of R coincide. Using this result, a new structural characterization, based on iteration, of the same functions is given.
COUNTING COMPLEXITY CLASSES FOR NUMERIC COMPUTATIONS I: SEMILINEAR SETS
"... We define a counting class #Padd in the BlumShubSmalesetting of additive computations over the reals. Structural properties of this class are studied, including a characterization in terms of the classical counting class #P introduced by Valiant. We also establish transfer theorems for both dire ..."
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Cited by 5 (4 self)
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We define a counting class #Padd in the BlumShubSmalesetting of additive computations over the reals. Structural properties of this class are studied, including a characterization in terms of the classical counting class #P introduced by Valiant. We also establish transfer theorems for both directions between the real additive and the discrete setting. Then we characterize in terms of completeness results the complexity of computing basic topological invariants of semilinear sets given by additive circuits. It turns out that the computation of the Euler characteristic is FP #Padd addcomplete, while for fixed k, the computation of the kth Betti number is FPARaddcomplete. Thus the latter is more difficult under standard complexity theoretic assumptions. We use all the above to prove some analogous completeness results in the classical setting.
Circuits versus Trees in Algebraic Complexity
 In Proc. STACS 2000
, 2000
"... . This survey is devoted to some aspects of the \P = NP ?" problem over the real numbers and more general algebraic structures. We argue that given a structure M , it is important to nd out whether NPM problems can be solved by polynomial depth computation trees, and if so whether these tre ..."
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Cited by 5 (4 self)
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. This survey is devoted to some aspects of the \P = NP ?" problem over the real numbers and more general algebraic structures. We argue that given a structure M , it is important to nd out whether NPM problems can be solved by polynomial depth computation trees, and if so whether these trees can be eciently simulated by circuits. Point location, a problem of computational geometry, comes into play in the study of these questions for several structures of interest. 1 Introduction In algebraic complexity one measures the complexity of an algorithm by the number of basic operations performed during a computation. The basic operations are usually arithmetic operations and comparisons, but sometimes transcendental functions are also allowed [2123, 26]. Even when the set of basic operations has been xed, the complexity of a problem depends on the particular model of computation considered. The two main categories of interest for this paper are circuits and trees. In section 2 and...
Smallspace analogues of Valiant’s classes
"... Abstract. In the uniform circuit model of computation, the width of a boolean circuit exactly characterises the “space ” complexity of the computed function. Looking for a similar relationship in Valiant’s algebraic model of computation, we propose width of an arithmetic circuit as a possible measur ..."
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Cited by 2 (0 self)
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Abstract. In the uniform circuit model of computation, the width of a boolean circuit exactly characterises the “space ” complexity of the computed function. Looking for a similar relationship in Valiant’s algebraic model of computation, we propose width of an arithmetic circuit as a possible measure of space. We introduce the class VL as an algebraic variant of deterministic logspace L. In the uniform setting, we show that our definition coincides with that of VPSPACE at polynomial width. Further, to define algebraic variants of nondeterministic spacebounded classes, we introduce the notion of “readonce ” certificates for arithmetic circuits. We show that polynomialsize algebraic branching programs can be expressed as a readonce exponential sum over polynomials in VL, i.e. VBP ∈ Σ R · VL. We also show that Σ R · VBP = VBP, i.e. VBPs are stable under readonce exponential sums. Further, we show that readonce exponential sums over a restricted class of constantwidth arithmetic circuits are within VQP, and this is the largest known such subclass of polylogwidth circuits with this property. 1
On NCreal complexity classes for additive circuits and their relations with NC
, 1993
"... Based on the results of Blum, Shub and Smale [1], Meer [5], Cucker and Matamala [3] and Koiran [7], we develop the study of real computation models restricted to additive operations. More specifically we introduce some complexity classes defined by algebraic circuits and we study their relations ..."
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Cited by 1 (0 self)
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Based on the results of Blum, Shub and Smale [1], Meer [5], Cucker and Matamala [3] and Koiran [7], we develop the study of real computation models restricted to additive operations. More specifically we introduce some complexity classes defined by algebraic circuits and we study their relationships with the real computation model. We show that the languages accepted by nonuniform additive circuits of polynomial size and polylogarithmic depth are those accepted by uniform additive circuits of polynomial size and polylogarithmic depth with advice. Moreover, we prove that binary languages accepted by real uniform circuits of polynomial size and polylogarithmic depth, when the test nodes in the circuit are equality test, are the languages belonging to NC; when the test nodes are inequality test, the class obtained is NC/Poly. We also prove that the class defined by family of algebraic circuits with polynomial size and polylogarithmic depth is strictly contained in the class...
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...