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Recursion Theory on the Reals and Continuoustime Computation
 Theoretical Computer Science
, 1995
"... We define a class of recursive functions on the reals analogous to the classical recursive functions on the natural numbers, corresponding to a conceptual analog computer that operates in continuous time. This class turns out to be surprisingly large, and includes many functions which are uncomp ..."
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Cited by 73 (4 self)
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We define a class of recursive functions on the reals analogous to the classical recursive functions on the natural numbers, corresponding to a conceptual analog computer that operates in continuous time. This class turns out to be surprisingly large, and includes many functions which are uncomputable in the traditional sense.
Achilles and the Tortoise climbing up the hyperarithmetical hierarchy
, 1997
"... We pursue the study of the computational power of Piecewise Constant Derivative (PCD) systems started in [5, 6]. PCD systems are dynamical systems defined by a piecewise constant differential equation and can be considered as computational machines working on a continuous space with a continuous tim ..."
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Cited by 26 (6 self)
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We pursue the study of the computational power of Piecewise Constant Derivative (PCD) systems started in [5, 6]. PCD systems are dynamical systems defined by a piecewise constant differential equation and can be considered as computational machines working on a continuous space with a continuous time. We prove that the languages recognized by rational PCD systems in dimension d = 2k + 3 (respectively: d = 2k + 4), k 0, in finite continuous time are precisely the languages of the ! k th (resp. ! k + 1 th ) level of the hyperarithmetical hierarchy. Hence the reachability problem for rational PCD systems of dimension d = 2k + 3 (resp. d = 2k + 4), k 1, is hyperarithmetical and is \Sigma ! kcomplete (resp. \Sigma ! k +1 complete).
Elimination of Constants from Machines over Algebraically Closed Fields
 Journal of Complexity
, 1996
"... Let K be an algebraically closed field of characteristic 0. We show that constants can be removed efficiently from any machine over K solving a problem which is definable without constants. This gives a new proof of the transfer theorem of Blum, Cucker, Shub & Smale for the problem P ? = NP. We h ..."
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Cited by 12 (7 self)
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Let K be an algebraically closed field of characteristic 0. We show that constants can be removed efficiently from any machine over K solving a problem which is definable without constants. This gives a new proof of the transfer theorem of Blum, Cucker, Shub & Smale for the problem P ? = NP. We have similar results in positive characteristic for nonuniform complexity classes. We also construct explicit and correct test sequences (in the sense of Heintz and Schnorr) for the class of polynomials which are easy to compute. An earlier version of this paper appeared as NeuroCOLT Technical Report 9643. The present paper contains in particular a new bound for the size of explicit correct test sequences. 1 A part of this work was done when the author was visiting DIMACS at Rutgers University. 1 Introduction As in discrete complexity theory, the problem P ? = NP is a major open problem in the BlumShubSmale model of computation over the reals [3]. It has been possible to show that P...
Complexity and Real Computation: A Manifesto
 International Journal of Bifurcation and Chaos
, 1995
"... . Finding a natural meeting ground between the highly developed complexity theory of computer science with its historical roots in logic and the discrete mathematics of the integers and the traditional domain of real computation, the more eclectic less foundational field of numerical analysis ..."
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Cited by 11 (0 self)
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. Finding a natural meeting ground between the highly developed complexity theory of computer science with its historical roots in logic and the discrete mathematics of the integers and the traditional domain of real computation, the more eclectic less foundational field of numerical analysis with its rich history and longstanding traditions in the continuous mathematics of analysis presents a compelling challenge. Here we illustrate the issues and pose our perspective toward resolution. This article is essentially the introduction of a book with the same title (to be published by Springer) to appear shortly. Webster: A public declaration of intentions, motives, or views. k Partially supported by NSF grants. y International Computer Science Institute, 1947 Center St., Berkeley, CA 94704, U.S.A., lblum@icsi.berkeley.edu. Partially supported by the LettsVillard Chair at Mills College. z Universitat Pompeu Fabra, Balmes 132, Barcelona 08008, SPAIN, cucker@upf.es. P...
Are Lower Bounds Easier over the Reals?
, 1997
"... We show that proving lower bounds in algebraic models of computation may not be easier than in the standard Turing machine model. For instance, a superpolynomial lower bound on the size of an algebraic circuit solving the real knapsack problem (or on the running time of a real Turing machine) would ..."
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Cited by 8 (3 self)
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We show that proving lower bounds in algebraic models of computation may not be easier than in the standard Turing machine model. For instance, a superpolynomial lower bound on the size of an algebraic circuit solving the real knapsack problem (or on the running time of a real Turing machine) would imply a separation of P from PSPACE. A more general result relates parallel complexity classes in boolean and real models of computation. We also propose a few problems in algebraic complexity and topological complexity.
Some bounds on the computational power of Piecewise Constant Derivative systems.
 In Proceeding of ICALP'97
, 1997
"... We study the computational power of Piecewise Constant Derivative (PCD) systems. PCD systems are dynamical systems defined by a piecewise constant differential equation and can be considered as computational machines working on a continuous space with a continuous time. We show that the computation ..."
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Cited by 7 (2 self)
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We study the computational power of Piecewise Constant Derivative (PCD) systems. PCD systems are dynamical systems defined by a piecewise constant differential equation and can be considered as computational machines working on a continuous space with a continuous time. We show that the computation time of these machines can be measured either as a discrete value, called discrete time, or as a continuous value, called continuous time. We relate the two notions of time for general PCD systems. We prove that general PCD systems are equivalent to Turing machines and linear machines in finite discrete time. We prove that the languages recognized by purely rational PCD systems in dimension d in finite continuous time are precisely the languages of the d \Gamma 2 th level of the arithmetical hierarchy. Hence the reachability problem of purely rational PCD systems of dimension d in finite continuous time is \Sigma d\Gamma2 complete. 1 Introduction There has been recently an increasing in...
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 trees ca ..."
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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...
Complexity and Dimension
, 1996
"... Introduction 2 Keywords: algebraic decision trees  sparse sets  computational complexity. 1 Introduction In 1977 Berman and Hartmanis ([1]) conjectured that all NPcomplete sets are polynomially isomorphic. Should this conjecture be proved, we would have as a consequence that no "small" NPcom ..."
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Cited by 2 (1 self)
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Introduction 2 Keywords: algebraic decision trees  sparse sets  computational complexity. 1 Introduction In 1977 Berman and Hartmanis ([1]) conjectured that all NPcomplete sets are polynomially isomorphic. Should this conjecture be proved, we would have as a consequence that no "small" NPcomplete set exists in a precise sense of the word "small". Denote by \Sigma the set f0; 1g and by \Sigma the set of all finite sequences of elements in \Sigma. A set S ` \Sigma is said to be sparse when there is a polynomial p such that for all n 2 IN the subset S n of all elements in S having size n has cardinality at most p(n). If the BermanHartmanis conjecture is
Lower Bounds Are not Easier over the Reals: Inside PH
, 1999
"... We prove that all NP problems over the reals with addition and order can be solved in polynomial time with the help of a boolean NP oracle. As a consequence, the "P = NP?" question over the reals with addition and order is equivalent to the classical question. For the reals with addition and equalit ..."
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Cited by 2 (0 self)
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We prove that all NP problems over the reals with addition and order can be solved in polynomial time with the help of a boolean NP oracle. As a consequence, the "P = NP?" question over the reals with addition and order is equivalent to the classical question. For the reals with addition and equality only, the situation is quite different since P is known to be different from NP. Nevertheless, we prove similar transfer theorems for the polynomial hierarchy.
!()+, ./01 23456
, 1995
"... Computing the maximum bichromatic discrepancy is an interesting theoretical problem with important applications in computational learning theory, computational geometry and computer graphics. In this paper we give algorithms to compute the maximum bichromatic discrepancy for simple geometric ranges, ..."
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Computing the maximum bichromatic discrepancy is an interesting theoretical problem with important applications in computational learning theory, computational geometry and computer graphics. In this paper we give algorithms to compute the maximum bichromatic discrepancy for simple geometric ranges, including rectangles and halfspaces. In addition, we give extensions to other discrepancy problems. 1 Introduction The main theme of this paper is to present efficient algorithms that solve the problem of computing the maximum bichromatic discrepancy for axis oriented rectangles. This problem arises naturally in different areas of computer science, such as computational learning theory, computational geometry and computer graphics ([Ma], [DG]), and has applications in all these areas. In computational learning theory, the problem of agnostic PAClearning with simple geometric hypotheses can be reduced to the problem of computing the maximum bichromatic discrepancy for simple geometric ra...