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Journal ArticleDOI

On the computational power of dynamical systems and hybrid systems

20 Nov 1996-Theoretical Computer Science (Elsevier Science Publishers Ltd.)-Vol. 168, Iss: 2, pp 417-459
TL;DR: It is deduced that many dynamical systems have intrinsically super-Turing capabilities and a new discrete model of computation is introduced: the analog automaton model, characterized as P/poly in polynomial time and as unbounded in exponential time.
About: This article is published in Theoretical Computer Science.The article was published on 1996-11-20 and is currently open access. It has received 64 citations till now. The article focuses on the topics: Dynamical systems theory & Linear dynamical system.
Citations
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Journal ArticleDOI
TL;DR: This paper considers problems related to stability or stabilizability of linear systems with parametric uncertainty, robust control, time-varying linear systems, nonlinear and hybrid systems, and stochastic optimal control.

785 citations


Cites background or methods from "On the computational power of dynam..."

  • ...high-order neural nets,13 Bournez and Cosnard (1996) for simulation by analog automata, Garzon (1995) for simulation by cellular automata, Siegelmann and Sontag (1994) and Siegelmann (1998) for simulation by saturated linear systems that can involve arbitrary real (not necessarily rational) numbers, Branicky (1995b), Branicky (1995a), Ruohonen (1997) and Moore (1991) for simulation by di!erential equations, Moore (1990) for simulation by continuous-time physical devices in dimension three, and Koiran and Moore (1999) for simulation by analytic maps....

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  • ...…systems), Littman, Dean and Kaelbling (1995), Rust (1996), Mundhenk, Goldsmith, Lusena and Allender (1997), (Markov decision theory), Alur et al. (1995) (hybrid systems), Bournez and Cosnard (1996), Moore (1998) (Turing machine as dynamical systems) and Orponen (1994) (recurrent neural networks)....

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  • ...…by 13 In high-order neural nets, activations are combined using multiplication as opposed to just linear combinations. high-order neural nets,13 Bournez and Cosnard (1996) for simulation by analog automata, Garzon (1995) for simulation by cellular automata, Siegelmann and Sontag (1994) and…...

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Journal ArticleDOI
TL;DR: It is proved that basic questions related to their stability and controllability are either undecidable or computationally intractable (NP-hard) in nonlinear systems.

242 citations

Journal ArticleDOI
TL;DR: This Issue is not only general to serve as a survey of recent progress in hybrid systems theory but also specific to introduce interesting and stimulating applications of hybrid systems in biology and medicine.
Abstract: In this introductory article, we survey the contents of this Theme Issue. This Theme Issue deals with a fertile region of hybrid dynamical systems that are characterized by the coexistence of conti...

128 citations


Cites methods from "On the computational power of dynam..."

  • ...…a new generation of hybrid computing has recently been explored using analogue computation with chaotic neuro-dynamics and digital computation through algorithm (Aihara 2003; Horio & Aihara 2008); see also Branicky (1995) and Bournez & Cosnard (1996) for the computational power of hybrid systems....

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Journal ArticleDOI
TL;DR: A sharp delineation between classes of SDSs whose behavior is easy to predict and thosewhose behavior is hard to predict is shown, and new PSPACE-hard lower bounds on the complexity of state reachability problems for these models are shown.

90 citations

Journal ArticleDOI
TL;DR: This paper studies problems such as given a discrete time dynamical system of the form x(t + 1)= f(x(t)) and shows that AM-activity and Stability become decidable in dimension 1 for continuous functions.

82 citations


Cites background from "On the computational power of dynam..."

  • ...In traditional simulations of counter machines or Turing machines by dynamical systems, a machine configuration is encoded by a single point of the dynamical system’s state space [11, 16, 15, 14, 10, 6, 2]....

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References
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Book
01 Jan 1979
TL;DR: This book is a rigorous exposition of formal languages and models of computation, with an introduction to computational complexity, appropriate for upper-level computer science undergraduates who are comfortable with mathematical arguments.
Abstract: This book is a rigorous exposition of formal languages and models of computation, with an introduction to computational complexity. The authors present the theory in a concise and straightforward manner, with an eye out for the practical applications. Exercises at the end of each chapter, including some that have been solved, help readers confirm and enhance their understanding of the material. This book is appropriate for upper-level computer science undergraduates who are comfortable with mathematical arguments.

13,779 citations


"On the computational power of dynam..." refers background in this paper

  • ...It is well known [11] that there exist some languages L ⊂ {0, 1}+ which cannot be recognized by Turing machines....

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  • ...Because a discrete two stack automaton is an analog two stack automaton, and since discrete two stack automata can simulate Turing machines [11], analog automata are able to simulate Turing machines....

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Journal ArticleDOI
Charles H. Bennett1
TL;DR: This result makes plausible the existence of thermodynamically reversible computers which could perform useful computations at useful speed while dissipating considerably less than kT of energy per logical step.
Abstract: The usual general-purpose computing automaton (e.g.. a Turing machine) is logically irreversible- its transition function lacks a single-valued inverse. Here it is shown that such machines may he made logically reversible at every step, while retainillg their simplicity and their ability to do general computations. This result is of great physical interest because it makes plausible the existence of thermodynamically reversible computers which could perform useful computations at useful speed while dissipating considerably less than kT of energy per logical step. In the first stage of its computation the logically reversible automaton parallels the corresponding irreversible automaton, except that it saves all intermediate results, there by avoiding the irreversible operation of erasure. The second stage consists of printing out the desired output. The third stage then reversibly disposes of all the undesired intermediate results by retracing the steps of the first stage in backward order (a process which is only possible because the first stage has been carried out reversibly), there by restoring the machine (except for the now-written output tape) to its original condition. The final machine configuration thus contains the desired output and a reconstructed copy of the input, but no other undesired data. The foregoing results are demonstrated explicitly using a type of three-tape Turing machine. The biosynthesis of messenger RNA is discussed as a physical example of reversible computation.

3,497 citations


"On the computational power of dynam..." refers background in this paper

  • ...Since we know that a Turing machine can always be simulated, modulo a polynomial time overhead, by a reversible one (see for example: [7]), we claim that M , from second step, can be built reversible....

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Journal ArticleDOI
TL;DR: A general framework for the formal specification and algorithmic analysis of hybrid systems is presented, which considers symbolic model-checking and minimization procedures that are based on the reachability analysis of an infinite state space.

2,091 citations


"On the computational power of dynam..." refers background in this paper

  • ...formal definitions have been proposed in literature: see for example [1, 9, 21]....

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  • ...We will need also the following definition [1]: Definition 4....

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  • ...Alur and al. propose in [1] the following definition: Definition 4.3 (Hybrid System [1]) A hybrid system is made of 6-components: H = (Loc, V ar, Lab, Edg,Act, Inv) where: • Loc is a finite set of vertices called locations....

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  • ...4 • A hybrid system H is time-deterministic [1] if for every l ∈ Loc and every v ∈ V , there is at most one function f ∈ Act(l) with f(0) = v....

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  • ...3 (Hybrid System [1]) A hybrid system is made of 6-components:...

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Book ChapterDOI
01 Jan 1993
TL;DR: This work presents two semidecision procedures for verifying safety properties of piecewiselinear hybrid automata, in which all variables change at constant rates, and demonstrates that for many of the typical workshop examples, the procedures do terminate and thus provide an automatic way for verifying their properties.
Abstract: We introduce the framework of hybrid automata as a model and specification language for hybrid systems. Hybrid automata can be viewed as a generalization of timed automata, in which the behavior of variables is governed in each state by a set of differential equations. We show that many of the examples considered in the workshop can be defined by hybrid automata. While the reachability problem is undecidable even for very restricted classes of hybrid automata, we present two semidecision procedures for verifying safety properties of piecewiselinear hybrid automata, in which all variables change at constant rates. The two procedures are based, respectively, on minimizing and computing fixpoints on generally infinite state spaces. We show that if the procedures terminate, then they give correct answers. We then demonstrate that for many of the typical workshop examples, the procedures do terminate and thus provide an automatic way for verifying their properties.

1,260 citations


"On the computational power of dynam..." refers background in this paper

  • ...See [1, 2] for the definitions of the following special cases of linear hybrid systems: discrete systems, finite-state systems, timed automata, multi-rate timed systems, n-rate time systems, integrator systems....

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  • ...Some undecidability results are known [1, 2, 10, 12], but only a small number of papers have been devoted to the study of hybrid systems as computational models: the work of Asarin, Maler and Pnueli [3, 4, 5] about Piecewise Constant Derivative systems and the work of Branicky [9] about simulation capabilities of Ordinary Differential Equations can however be mentioned....

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