## The complexity of analog computation

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Venue: | in Math. and Computers in Simulation 28(1986 |

Citations: | 38 - 0 self |

### BibTeX

@INPROCEEDINGS{Vergis_thecomplexity,

author = {Anastasios Vergis and Kenneth Steiglitz},

title = {The complexity of analog computation},

booktitle = {in Math. and Computers in Simulation 28(1986},

year = {},

pages = {91--113}

}

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### Abstract

We ask if analog computers can solve NP-complete problems efficiently. Regarding this as unlikely, we formulate a strong version of Church’s Thesis: that any analog computer can be simulated efficiently (in polynomial time) by a digital computer. From this assumption and the assumption that P ≠ NP we can draw conclusions about the operation of physical devices used for computation. An NP-complete problem, 3-SAT, is reduced to the problem of checking whether a feasible point is a local optimum of an optimization problem. A mechanical device is proposed for the solution of this problem. It encodes variables as shaft angles and uses gears and smooth cams. If we grant Strong Church’s Thesis, that P ≠ NP, and a certain ‘‘Downhill Principle’ ’ governing the physical behavior of the machine, we conclude that it cannot operate successfully while using only polynomial resources. We next prove Strong Church’s Thesis for a class of analog computers described by well-behaved ordinary differential equations, which we can take as representing part of classical mechanics. We conclude with a comment on the recently discovered connection between spin glasses and combinatorial optimization. 1.

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and J.J.Hopfield, "Simple neural optimization network
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Citation Context ...g combinatorial problem, and a detailed examination of proposed analog networks for the solution of other combinatorial problems, including ones in Digital P-time. Examples of the latter are found in =-=[13,27]-=-.s9. The Spin Glass Computer - 20 - Recent work on the properties of certain magnetic alloys called spin glasses has led to an interesting connection between physics and combinatorial optimization [1,... |

66 |
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Citation Context ..., scalar multipliers, and integrators — constrained by some natural conditions to ensure well-posedness, generates functions solving ordinary differential equations of a particular form [26]. Pour-El =-=[22]-=- added some necessary elaborations concerning existence of unique solutions. This work derived the following form for the differential equations corresponding to a (Bush) analog computer: A(Y(t)) dY(t... |

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The wave equation with computable initial data such that its unique solution is not computable
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Citation Context ...asure for the resources used. In particular it is the second derivative of the analog variables that appears as the natural measure. It is interesting to note that in the work of Pour-El and Richards =-=[23,24]-=-, where it was shown that the three-dimensional wave equation can transform computable initial data into noncomputable solution values, imposition of continuity conditions on the second derivative wil... |

51 |
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19 |
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Citation Context ...and try again. Obviously we cannot assert that this procedure requires polynomial time. (It is interesting to note that an electrical network for solving quadratic programming problems is proposed in =-=[5]-=-, but it is based on sufficiency of the Kuhn-Tucker conditions, and depends for its operation on positive definiteness of the quadratic form. At first this might seem to be another candidate for an an... |

16 |
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Citation Context ...blems are inherently difficult (i.e. not polynomial) for analog computers, even though they are easy for digital computers. Something like our Strong Church’s Thesis was discussed recently by Feynman =-=[8]-=- in connection with the problem of building a (digital) computer that simulates physics. He says: ‘‘The rule of simulation that I would like to have is that the number of computer elements required to... |

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12 |
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Citation Context ...alog computation considered by Shannon and Pour-El does not account for the limitations that are inherent in the general model of analog computer developed in previous sections. From work of Plaisted =-=[21]-=- it is known that solution of the NPcomplete PARTITION problem is equivalent to the evaluation of a particular definite integral. The integral may be computed as the solution to a differential equatio... |

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Citation Context ...g combinatorial problem, and a detailed examination of proposed analog networks for the solution of other combinatorial problems, including ones in Digital P-time. Examples of the latter are found in =-=[13,27]-=-.s9. The Spin Glass Computer - 20 - Recent work on the properties of certain magnetic alloys called spin glasses has led to an interesting connection between physics and combinatorial optimization [1,... |

6 |
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Citation Context ... over the years, to solve a variety of problems. Perhaps most widely known is the Differential Analyzer [4,26], which has been used to solve differential equations. To mention some other examples, in =-=[25]-=- an electronic analog computer is proposed to implement the gradient projection method for linear programming. In [18] the problem of finding a minimum-length interconnection network between given poi... |

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Citation Context ... obtained by pulling the strings. Another method is proposed there for this problem, based on the fact that soap films form minimal-tension surfaces. Many other examples can be found in books such as =-=[14]-=- and [16], including electrical and mechanical machines for solving simultaneous linear equations and differential equations. Given the large body of work on the complexity of Turing-machine computati... |

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Citation Context ...asure for the resources used. In particular it is the second derivative of the analog variables that appears as the natural measure. It is interesting to note that in the work of Pour-El and Richards =-=[23,24]-=-, where it was shown that the three-dimensional wave equation can transform computable initial data into noncomputable solution values, imposition of continuity conditions on the second derivative wil... |

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Citation Context ...ween spin glasses and combinatorial optimization. 1. Introduction Analog devices have been used, over the years, to solve a variety of problems. Perhaps most widely known is the Differential Analyzer =-=[4,26]-=-, which has been used to solve differential equations. To mention some other examples, in [25] an electronic analog computer is proposed to implement the gradient projection method for linear programm... |

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Citation Context ...otheses ares- 21 - important open questions, but we have been able to prove a restricted form of Strong Church’s Thesis and perhaps more general results will be forthcoming. In recent work of Bennett =-=[2,3]-=-, there have been discussions that are germane to analog computation and efficient simulation. He has suggested that efficient simulation of physical systems up to the errors induced by uncontrollable... |

2 |
On the logical ‘depth’ of sequences and their reducibilities to random sequences,’’ preprint
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Citation Context ...otheses ares- 21 - important open questions, but we have been able to prove a restricted form of Strong Church’s Thesis and perhaps more general results will be forthcoming. In recent work of Bennett =-=[2,3]-=-, there have been discussions that are germane to analog computation and efficient simulation. He has suggested that efficient simulation of physical systems up to the errors induced by uncontrollable... |

1 |
Analog Methods, 2nd
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- 1959
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Citation Context ... by pulling the strings. Another method is proposed there for this problem, based on the fact that soap films form minimal-tension surfaces. Many other examples can be found in books such as [14] and =-=[16]-=-, including electrical and mechanical machines for solving simultaneous linear equations and differential equations. Given the large body of work on the complexity of Turing-machine computation, and t... |

1 |
Fundamentals of Mechanical Design, 3rd
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Citation Context ...o shaft positions can be subtracted with a differential gear (see Fig. 4). The differential forces the angles p, q, r tos- 8 - satisfy the equation p − q = r. A full description of it can be found in =-=[20]-=-. (Differentials are used in automobile transmissions.) To preserve symmetry, we shall make the assumption that the differential adds the angles p and q; this can be accomplished easily by incorporati... |