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26
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 27 (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).
On the Computational Power of Dynamical Systems and Hybrid Systems
 Theoretical Computer Science
, 1996
"... We explore the simulation and computational capabilities of discrete and continuous dynamical systems. We introduce and compare several notions of simulation between discrete and continuous systems. We give a general framework that allows discrete and continuous dynamical systems to be considered as ..."
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Cited by 22 (5 self)
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We explore the simulation and computational capabilities of discrete and continuous dynamical systems. We introduce and compare several notions of simulation between discrete and continuous systems. We give a general framework that allows discrete and continuous dynamical systems to be considered as computational machines. We introduce a new discrete model of computation: the analog automaton model. We characterize the computational power of this model as P=poly in polynomial time and as unbounded in exponential time. We prove that many very simple dynamical systems from literature are able to simulate analog automata. From this results we deduce that many dynamical systems have intrinsically superTuring capabilities. 1 Introduction The computational power of abstract machines which compute over the reals in unbounded precision in constant time is still an open problem. We refer the reader to [18] for an upto date survey. Indeed, a basic model for their computations has been propose...
Hypercomputation and the Physical ChurchTuring Thesis
, 2003
"... A version of the ChurchTuring Thesis states that every e#ectively realizable physical system can be defined by Turing Machines (`Thesis P'); in this formulation the Thesis appears an empirical, more than a logicomathematical, proposition. We review the main approaches to computation beyond Tu ..."
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Cited by 21 (0 self)
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A version of the ChurchTuring Thesis states that every e#ectively realizable physical system can be defined by Turing Machines (`Thesis P'); in this formulation the Thesis appears an empirical, more than a logicomathematical, proposition. We review the main approaches to computation beyond Turing definability (`hypercomputation'): supertask, nonwellfounded, analog, quantum, and retrocausal computation. These models depend on infinite computation, explicitly or implicitly, and appear physically implausible; moreover, even if infinite computation were realizable, the Halting Problem would not be a#ected. Therefore, Thesis P is not essentially di#erent from the standard ChurchTuring Thesis.
On the expressiveness and decidability of ominimal hybrid systems
 Journal of Complexity
"... Abstract. This paper is driven by a general motto: bisimulate a hybrid system by a finite symbolic dynamical system. In the case of ominimal hybrid systems, the continuous and discrete components can be decoupled, and hence, the problem reduces in building a finite symbolic dynamical system for the ..."
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Cited by 12 (6 self)
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Abstract. This paper is driven by a general motto: bisimulate a hybrid system by a finite symbolic dynamical system. In the case of ominimal hybrid systems, the continuous and discrete components can be decoupled, and hence, the problem reduces in building a finite symbolic dynamical system for the continuous dynamics of each location. We show that this can be done for a quite general class of hybrid systems defined on ominimal structures. In particular, we recover the main result of a paper by Lafferriere G., Pappas G.J. and Sastry S. on ominimal hybrid systems. We also study related decidability questions. Mathematics Subject Classification: 68Q60, 03C64, 03D15. 1
On the computational power and superTuring capabilities of dynamical systems
, 1995
"... We explore the simulation and computational capabilities of dynamical systems. We first introduce and compare several notions of simulation between discrete systems. We give a general framework that allows dynamical systems to be considered as computational machines. We introduce a new discrete ..."
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Cited by 10 (0 self)
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We explore the simulation and computational capabilities of dynamical systems. We first introduce and compare several notions of simulation between discrete systems. We give a general framework that allows dynamical systems to be considered as computational machines. We introduce a new discrete model of computation: the analog automaton model. We determine the computational power of this model and prove that it does have superTuring capabilities. We then prove that many very simple dynamical systems from literature are actually able to simulate analog automata. From this result we deduce that many dynamical systems have intrinsically superTuring capabilities.
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 8 (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...
Logics Which Capture Complexity Classes Over the Reals
 Journal of Symbolic Logic
"... this paper we continue the work initiated in [7]. Our starting point is the absence of a meaningful class of polynomial space over the reals. Michaux proved in [10] that such a class would contain all decidable problems. There are two natural candidates to ocuppy this vacancy. The class PAR IR of se ..."
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Cited by 8 (4 self)
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this paper we continue the work initiated in [7]. Our starting point is the absence of a meaningful class of polynomial space over the reals. Michaux proved in [10] that such a class would contain all decidable problems. There are two natural candidates to ocuppy this vacancy. The class PAR IR of sets decidable in parallel polynomial time and the class (EXP IR ; PSPACE IR ) of sets decidable by machines which work simultaneously in exponential time and polynomial space. Both classes yield PSPACE over IF 2 but it turns out that the first one is weaker over IR. We give logics for IRstructures which capture these two classes. Previously, we recall the basic concepts about IRstructures and their logics. Scattered along these sections some other complexity classes are characterized in terms of descriptive complexity. Among them, we find NC
An explicit solution to Post’s Problem over the reals
, 2008
"... In the BSS model of real number computations we prove a concrete and explicit semidecidable language to be undecidable yet not reducible from (and thus strictly easier than) the real Halting Language. This solution to Post’s Problem over the reals significantly differs from its classical, discrete ..."
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Cited by 6 (3 self)
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In the BSS model of real number computations we prove a concrete and explicit semidecidable language to be undecidable yet not reducible from (and thus strictly easier than) the real Halting Language. This solution to Post’s Problem over the reals significantly differs from its classical, discrete variant where advanced diagonalization techniques are only known to yield the existence of such intermediate Turing degrees. Then we strengthen the above result and show as well the existence of an uncountable number of incomparable semidecidable Turing degrees below the real Halting Problem in the BSS model. Again, our proof will give concrete such problems representing these different degrees. Finally we show the corresponding result for the linear BSS model, that is over (R, +, −,<)rather than (R, +, −, ×, ÷,<).
On the structure of NPC
 C . SIAM Journal on Computing
, 1999
"... This paper deals with complexity classes PC and NP C , as they were introduced over the complex numbers by Blum, Shub and Smale [3] . Under the assumption PC 6= NP C the existence of noncomplete problems in NP C , not belonging to PC , is established. ..."
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Cited by 3 (1 self)
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This paper deals with complexity classes PC and NP C , as they were introduced over the complex numbers by Blum, Shub and Smale [3] . Under the assumption PC 6= NP C the existence of noncomplete problems in NP C , not belonging to PC , is established.
Isomorphism Theorem for BSS Recursively Enumerable Sets over Real Closed Fields
, 1998
"... The main result of this paper lies in the framework of BSS computability : it shows roughly that any recursively enumerable set S in R N 6 1, where R is a real closed field, is isomorphic to R dimS by a bijection ' which is decidable over S. Moreover the map S 7! ' is computable. ..."
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Cited by 1 (0 self)
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The main result of this paper lies in the framework of BSS computability : it shows roughly that any recursively enumerable set S in R N 6 1, where R is a real closed field, is isomorphic to R dimS by a bijection ' which is decidable over S. Moreover the map S 7! ' is computable.