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21
Polynomial differential equations compute all real computable functions on computable compact intervals
, 2007
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The Elementary Computable Functions over the Real Numbers: Applying Two New Techniques. Archives for Mathematical Logic
, 2008
"... The basic motivation behind this work is to tie together various computational complexity classes, whether over different domains such as the naturals or the reals, or whether defined in different manners, via function algebras (Real Recursive Functions) or via Turing Machines (Computable Analysis). ..."
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Cited by 9 (4 self)
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The basic motivation behind this work is to tie together various computational complexity classes, whether over different domains such as the naturals or the reals, or whether defined in different manners, via function algebras (Real Recursive Functions) or via Turing Machines (Computable Analysis). We provide general tools for investigating these issues, using two techniques we call approximation and lifting. We use these methods to obtain two main theorems. First we provide an alternative proof of the result from Campagnolo, Moore and Costa [3], which precisely relates the Kalmar elementary computable functions to a function algebra over the reals. Secondly, we build on that result to extend a result of Bournez and Hainry [1], which provided a function algebra for the C2 real elementary computable functions; our result does not require the restriction to C2 functions. In addition to the extension, we provide an alternative approach to the proof. Their proof involves simulating the operation of a Turing Machine using a function algebra. We avoid this simulation, using a technique we call lifting, which allows us to lift the classic result regarding the elementary computable functions to a result on the reals. The two new techniques bring a different perspective to these problems, and furthermore appear more easily applicable to other problems of this sort. 1
The general purpose analog computer and computable analysis are two equivalent paradigms of analog computation
 Theory and Applications of Models of Computation, Third International Conference, TAMC 2006
, 2006
"... Abstract. In this paper we revisit one of the first models of analog computation, Shannon’s General Purpose Analog Computer (GPAC). The GPAC has often been argued to be weaker than computable analysis. As main contribution, we show that if we change the notion of GPACcomputability in a natural way, ..."
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Abstract. In this paper we revisit one of the first models of analog computation, Shannon’s General Purpose Analog Computer (GPAC). The GPAC has often been argued to be weaker than computable analysis. As main contribution, we show that if we change the notion of GPACcomputability in a natural way, we compute exactly all real computable functions (in the sense of computable analysis). Moreover, since GPACs are equivalent to systems of polynomial differential equations then we show that all real computable functions can be defined by such models. 1
Lipschitz continuous ordinary differential equations are polynomialspace complete
 Comput. Complexity
, 2010
"... ABSTRACT. In answer to Ko’s question raised in 1983, we show that an initial value problem given by a polynomialtime computable, Lipschitz continuous function can have a polynomialspace complete solution. The key insight is simple: the Lipschitz condition means that the feedback in the differentia ..."
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ABSTRACT. In answer to Ko’s question raised in 1983, we show that an initial value problem given by a polynomialtime computable, Lipschitz continuous function can have a polynomialspace complete solution. The key insight is simple: the Lipschitz condition means that the feedback in the differential equation is weak. We define a class of polynomialspace computation tableaux with equally restricted feedback, and show that they are still polynomialspace complete. The same technique also settles Ko’s two later questions on Volterra integral equations.
How much can analog and hybrid systems be proved (super)Turing
 Applied Mathematics and Computation
, 2006
"... Church thesis and its variants say roughly that all reasonable models of computation do not have more power than Turing Machines. In a contrapositive way, they say that any model with superTuring power must have something unreasonable. Our aim is to discuss how much theoretical computer science can ..."
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Cited by 6 (2 self)
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Church thesis and its variants say roughly that all reasonable models of computation do not have more power than Turing Machines. In a contrapositive way, they say that any model with superTuring power must have something unreasonable. Our aim is to discuss how much theoretical computer science can quantify this, by considering several classes of continuous time dynamical systems, and by studying how much they can be proved Turing or superTuring. 1
Computational bounds on polynomial differential equations
, 2008
"... In this paper we study from a computational perspective some properties of the solutions of polynomial ordinary differential equations. We consider elementary (in the sense of Analysis) discretetime dynamical systems satisfying certain criteria of robustness. We show that those systems can be simul ..."
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Cited by 5 (3 self)
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In this paper we study from a computational perspective some properties of the solutions of polynomial ordinary differential equations. We consider elementary (in the sense of Analysis) discretetime dynamical systems satisfying certain criteria of robustness. We show that those systems can be simulated with elementary and robust continuoustime dynamical systems which can be expanded into fully polynomial ordinary differential equations in Q[π]. This sets a computational lower bound on polynomial ODEs since the former class is large enough to include the dynamics of arbitrary Turing machines. We also apply the previous methods to show that the problem of determining whether the maximal interval of definition of an initialvalue problem defined with polynomial ODEs is bounded or not is in general undecidable, even if the parameters of the system are computable and comparable and if the degree of the corresponding polynomial is at most 56. Combined with earlier results on the computability of solutions of polynomial ODEs, one can conclude that there is from a computational point of view a close connection between these systems and Turing machines.
Characterizing Computable Analysis with Differential Equations
, 2008
"... The functions of Computable Analysis are defined by enhancing the capacities of normal Turing Machines to deal with real number inputs. We consider characterizations of these functions using function algebras, known as Real Recursive Functions. Bournez and Hainry 2006 [5] used a function algebra to ..."
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The functions of Computable Analysis are defined by enhancing the capacities of normal Turing Machines to deal with real number inputs. We consider characterizations of these functions using function algebras, known as Real Recursive Functions. Bournez and Hainry 2006 [5] used a function algebra to characterize the twice continuously differentiable functions of Computable Analysis, restricted to certain compact domains. In a similar model, Shannon’s General Purpose Analog Computer, Bournez et. al. 2007 [3] also characterize the functions of Computable Analysis. We combine the results of [5] and Graça et. al. [13], to show that a different function algebra also yields Computable Analysis. We believe that our function algebra is an improvement due to its simple definition and because the operations in our algebra are less obviously designed to mimic the operations in the usual definition of the recursive functions using the primitive recursion and minimization operators. 1
Differential Recursion
 ACM TRANSACTIONS ON COMPUTATIONAL LOGIC
, 2009
"... We present a redevelopment of the theory of realvalued recursive functions that was introduced by C. Moore in 1996 by analogy with the standard formulation of the integervalued recursive functions. While his work opened a new line of research on analog computation, the original paper contained som ..."
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We present a redevelopment of the theory of realvalued recursive functions that was introduced by C. Moore in 1996 by analogy with the standard formulation of the integervalued recursive functions. While his work opened a new line of research on analog computation, the original paper contained some technical inaccuracies. We discuss possible attempts to remove the ambiguity in the behaviour of the operators on partial functions, with a focus on his “primitive recursive” functions generated by the differential recursion operator that solves initial value problems. Under a reasonable reformulation, the functions in this class are shown to be analytic and computable in a strong sense in Computable Analysis. Despite this wellbehavedness, the class turns out to be too big to have the originally purported relation to differentially algebraic functions, and hence to C. E. Shannon’s model of analog computation.
Differential recursion and differentially algebraic functions ∗
, 704
"... Moore introduced a class of realvalued “recursive ” functions by analogy with Kleene’s formulation of the standard recursive functions. While his concise definition inspired a new line of research on analog computation, it contains some technical inaccuracies. Focusing on his “primitive recursive ” ..."
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Moore introduced a class of realvalued “recursive ” functions by analogy with Kleene’s formulation of the standard recursive functions. While his concise definition inspired a new line of research on analog computation, it contains some technical inaccuracies. Focusing on his “primitive recursive ” functions, we pin down what is problematic and discuss possible attempts to remove the ambiguity regarding the behavior of the differential recursion operator on partial functions. It turns out that in any case the purported relation to differentially algebraic functions, and hence to Shannon’s model of analog computation, fails. 1.
The Extended Analog Computer and Functions Computable in a Digital Sense
"... In this paper we compare the computational power of the Extended Analog Computer (EAC) with partial recursive functions. We first give a survey of some part of computational theory in discrete and in real space. In the last section we show that the EAC can generate any partial recursive function def ..."
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In this paper we compare the computational power of the Extended Analog Computer (EAC) with partial recursive functions. We first give a survey of some part of computational theory in discrete and in real space. In the last section we show that the EAC can generate any partial recursive function defined over N. Moreover we conclude that the classical halting problem for partial recursive functions is an equivalent of testing by EAC if sets are empty or not.