## A Survey of Continuous-Time Computation Theory (1997)

Venue: | Advances in Algorithms, Languages, and Complexity |

Citations: | 29 - 6 self |

### BibTeX

@INPROCEEDINGS{Orponen97asurvey,

author = {Pekka Orponen},

title = {A Survey of Continuous-Time Computation Theory},

booktitle = {Advances in Algorithms, Languages, and Complexity},

year = {1997},

pages = {209--224},

publisher = {Kluwer Academic Publishers}

}

### Years of Citing Articles

### OpenURL

### Abstract

Motivated partly by the resurgence of neural computation research, and partly by advances in device technology, there has been a recent increase of interest in analog, continuous-time computation. However, while special-case algorithms and devices are being developed, relatively little work exists on the general theory of continuous-time models of computation. In this paper, we survey the existing models and results in this area, and point to some of the open research questions. 1 Introduction After a long period of oblivion, interest in analog computation is again on the rise. The immediate cause for this new wave of activity is surely the success of the neural networks "revolution", which has provided hardware designers with several new numerically based, computationally interesting models that are structurally sufficiently simple to be implemented directly in silicon. (For designs and actual implementations of neural models in VLSI, see e.g. [30, 45]). However, the more fundamental...

### Citations

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Citation Context ...r automata [17] and other "complex systems" models of computation (e.g. [36, 40]), molecular computing [1, 28, 42], optical computing [16, 41], and --- somewhat futuristically --- quantum co=-=mputation [6, 49]-=-. While the work on analog computation theory goes back at least to Claude Shannon's papers in the early 1940's [46, 47], the literature is not very extensive, and also does not answer many of the que... |

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Citation Context ...y based, computationally interesting models that are structurally sufficiently simple to be implemented directly in silicon. (For designs and actual implementations of neural models in VLSI, see e.g. =-=[30, 45]-=-). However, the more fundamental explanation for this development is that as hardware technology has advanced, it has become generally easier to experiment with and manufacture individualized, special... |

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Citation Context ...tical issues in analog computing, a similar situation exists, or can be foreseen, in e.g. cellular automata [17] and other "complex systems" models of computation (e.g. [36, 40]), molecular =-=computing [1, 28, 42]-=-, optical computing [16, 41], and --- somewhat futuristically --- quantum computation [6, 49]. While the work on analog computation theory goes back at least to Claude Shannon's papers in the early 19... |

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Citation Context ...ve to models that have arisen from some real or idealized physical implementations, such as mechanical or electrical differential analyzers [46, 38, 43], or electronically implemented neural networks =-=[20, 34]-=-. In Section 4 we discuss the few existing papers on computational complexity issues. We conclude in Section 5 by listing some of the main open research directions. 2 Unconstrained Models By a continu... |

483 | Neural computation of decisions in optimization problems - Hopfield, Tank - 1985 |

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Citation Context ...r automata [17] and other "complex systems" models of computation (e.g. [36, 40]), molecular computing [1, 28, 42], optical computing [16, 41], and --- somewhat futuristically --- quantum co=-=mputation [6, 49]-=-. While the work on analog computation theory goes back at least to Claude Shannon's papers in the early 1940's [46, 47], the literature is not very extensive, and also does not answer many of the que... |

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Differential Equations, Dynamical Systems and Linear Algebra
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Citation Context ...ld appear most interesting from a present-day perspective. For instance, while in recent years quite a number of papers have appeared on the computational aspects of discrete-time analog models (e.g. =-=[3, 7, 13, 24, 25, 29, 31, 36, 51]-=-), the number of papers on the implementationally more significant continuous-time models is far fewer. (And P. O. Box 35, FIN--40351 Jyvaskyla, Finland. E-mail: orponen@math.jyu.fi. the amount of wor... |

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Citation Context ...tical issues in analog computing, a similar situation exists, or can be foreseen, in e.g. cellular automata [17] and other "complex systems" models of computation (e.g. [36, 40]), molecular =-=computing [1, 28, 42]-=-, optical computing [16, 41], and --- somewhat futuristically --- quantum computation [6, 49]. While the work on analog computation theory goes back at least to Claude Shannon's papers in the early 19... |

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Citation Context ...ld appear most interesting from a present-day perspective. For instance, while in recent years quite a number of papers have appeared on the computational aspects of discrete-time analog models (e.g. =-=[3, 7, 13, 24, 25, 29, 31, 36, 51]-=-), the number of papers on the implementationally more significant continuous-time models is far fewer. (And P. O. Box 35, FIN--40351 Jyvaskyla, Finland. E-mail: orponen@math.jyu.fi. the amount of wor... |

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Citation Context ... existing research on the general computability and complexity aspects of continuous-time computation models. In Section 2 we outline some mathematicallybased, implementationwise unconstrained models =-=[3, 8, 9, 10, 11, 33, 44]-=-. (We are staying here at the level of individual, concretely specified models. In particular, an important collection of results we do not cover, although it properly would belong to this context, is... |

92 |
Time/space trade-offs for reversible computation
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Citation Context ...[25] can in principle be extended into smooth, locally three-dimensional systems. (Technically, one needs to require here that the Turing machines to be simulated are invertible, but it is well known =-=[4, 5]-=- that any Turing machine can be converted into an invertible one.) Moore in his paper [32] in fact discusses the issue of continuous-time embedding at some length, and in [31] even presents a quasi-ph... |

68 | Universal computation and other capabilities of hybrid and continuous dynamical systems
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Citation Context ...in the corresponding discrete-time model, asymmetric and symmetric networks are computationally equivalent.) Also, the simulation in [34] uses a similar twophasing trick as Branicky's construction in =-=[8, 9]-=-, and is thus in the same way somewhat unsatisfactory. And finally, the result is only a lower bound on the computational power: the possibility still remains that polynomial-size Hopfield networks mi... |

66 |
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Citation Context ...ommon analytical and physical operators.) In Section 3 we move to models that have arisen from some real or idealized physical implementations, such as mechanical or electrical differential analyzers =-=[46, 38, 43]-=-, or electronically implemented neural networks [20, 34]. In Section 4 we discuss the few existing papers on computational complexity issues. We conclude in Section 5 by listing some of the main open ... |

56 |
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Citation Context ...art of computational complexity research. While the present paper concentrates on the theoretical issues in analog computing, a similar situation exists, or can be foreseen, in e.g. cellular automata =-=[17] and other-=- "complex systems" models of computation (e.g. [36, 40]), molecular computing [1, 28, 42], optical computing [16, 41], and --- somewhat futuristically --- quantum computation [6, 49]. While ... |

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Citation Context ...ld appear most interesting from a present-day perspective. For instance, while in recent years quite a number of papers have appeared on the computational aspects of discrete-time analog models (e.g. =-=[3, 7, 13, 24, 25, 29, 31, 36, 51]-=-), the number of papers on the implementationally more significant continuous-time models is far fewer. (And P. O. Box 35, FIN--40351 Jyvaskyla, Finland. E-mail: orponen@math.jyu.fi. the amount of wor... |

51 |
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51 |
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Citation Context ...ical computing [16, 41], and --- somewhat futuristically --- quantum computation [6, 49]. While the work on analog computation theory goes back at least to Claude Shannon's papers in the early 1940's =-=[46, 47]-=-, the literature is not very extensive, and also does not answer many of the questions that would appear most interesting from a present-day perspective. For instance, while in recent years quite a nu... |

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Citation Context ...hich was the central topic of interest in the study of differential analyzers.) Based on this convergence behavior, and the particular type of Liapunov function used in the proof, Hopfield and others =-=[20, 21, 14]-=- have proposed various special-case uses of such networks for associative memory and combinatorial optimization applications. The general computational power of Hopfield's network model was studied in... |

43 | et W.de Melo, Geometric theory of dynamical systems - Palis - 1982 |

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Citation Context ...ity analysis of continuous-time systems. Even the basic definitions have not yet been fixed in a universally acepted manner. Apparently, the only published paper in this area is that of Vergis et al. =-=[52]-=-, where the authors study the possibility of using GPAC-type systems (cf. equation (2)), or more generally Lipschitz-continuous systems of ODE's, for solving combinatorial problems faster than is poss... |

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Citation Context ...ysical hardware. The earliest theoretical study of the computational capabilities of analog devices seems to have been Shannon's 1941 [46] work on the generative power of Bush's Differential Analyzer =-=[12]-=-. Differential analyzers can be built either mechanically, out of rotating gears and shafts connecting them, or electronically from resistors and capacitors. Bush's original machine was (electro-)mech... |

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Citation Context ...ommon analytical and physical operators.) In Section 3 we move to models that have arisen from some real or idealized physical implementations, such as mechanical or electrical differential analyzers =-=[46, 38, 43]-=-, or electronically implemented neural networks [20, 34]. In Section 4 we discuss the few existing papers on computational complexity issues. We conclude in Section 5 by listing some of the main open ... |

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Citation Context ...ve to models that have arisen from some real or idealized physical implementations, such as mechanical or electrical differential analyzers [46, 38, 43], or electronically implemented neural networks =-=[20, 34]-=-. In Section 4 we discuss the few existing papers on computational complexity issues. We conclude in Section 5 by listing some of the main open research directions. 2 Unconstrained Models By a continu... |

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8 |
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Citation Context ...er in the result, and the argument in [52] is inconclusive. A promising approach to defining a general notion of analog computation time is suggested (based on discussions with the present author) in =-=[50]. Let-=- us assume that a system given by a field dx dt = f(x) relaxes from an initial state x(0) = x towards a stable equilibrium state x(1) = x . Define the computation time for input x and precision "... |

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Analog Computer Techniques, 2nd Ed
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1 |
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