Vergis A, Steiglitz K and Dickinson B, The complexity of analog computation, Math. Comp. Sim. 28 91--113 (1986)  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... that cannot be simulated by Turing machine stretches back over at least four decades (for example Da Costa and Doria 1991; Doyle 1982; Geroch and Hartle 1986; Komar 1964; Kreisel 1967, 1974; Penrose 1989, 1994; Pour El 1974; Pour El and Richards 1979, 1981; Scarpellini 1963; Stannett 1990; Vergis et al. 1986). If such processes do exist then perhaps future engineers will use them to implement the non classical part of some O machine. Science fiction or not, this theorizing suffices to illustrate why it is an empirical matter whether or not the disjunction of hypotheses (1) and (2) is true. III Not so ....

Vergis, A., Steiglitz, K., Dickinson, B. 1986. The complexity of analog computation. Mathematics and Computers in Simulation 28: 91-113.

.... Speculation as to whether there may actually be Uncomputable physical processes stretches back over at least four decades (see, for example, Da Costa and Doria 1991, Doyle 1982, Kreisel 1967, 1974, Pour El 1974, Pour El and Richards 1979, 1981, Penrose 1989, 1994, Scarpellini 1963, Stannett 1990, Vergis et al. 1986; Copeland and Sylvan 1997 is a survey) A notional oracle can certainly be implemented by means of a machine similar to M1. For example, relative to some given ordering of the arguments of the halting function, the successive values of the function form a certain infinite sequence of binary ....

Vergis, A., Steiglitz, K., Dickinson, B. 1986. 'The Complexity of Analog Computation'.

....updates these real numbers by performing elementary operations, such as addition, upon them. As was pointed out above, models such as this, with infinite precision (or even exponential precision) arithmetic are capable of carrying out tasks such as factoring or satisfiability in polynomial time [Sha1, VSD]. What distinguishes quantum computation from classical computation is our ability to prepare the system to solve a computational problem of our choice. If we try to build a classical device to carry out a desired computation, the imprecision in the realization of the system, effectively leaves us ....

Vergis, A., Steiglitz, K., Dickinson, B., "The complexity of analog computation", Mathematics and Computers in Simulation 28 (1986) 91-113. 16

....of the computational power of continuoustime processes are at the moment relatively rare in the literature, but we expect their number to grow along with the current increase of interest in analog computation. So far only a few papers have explicitly addressed computational complexity issues [5, 29, 33]. Computability aspects have been studied more often; see, e.g. 2, 6, 7, 20, 21, 22, 27, 28] and the surveys [23, 26] 2 A Continuous Time Neural Network Model An electrical model of Hopfield s continuous time neuron is shown in Fig. 1. Here oe denotes the characteristic of the nonlinear ....

Vergis, A., Steiglitz, K., Dickinson, B. The complexity of analog computation. Math. and Computers in Simulation 28 (1986), 91--113. 20

....by the recent papers [2, 49] which we review. We also discuss Shor s [38] paper, which describes the quantum polynomial time factoring algorithm that has provided a major motivation for much of the recent activity. 1 Introduction Quantitative Church s Thesis. The Quantitative Church s Thesis [50, 48] claims that Turing machines are as efficient as any realistic computer, within a polynomial factor. However, Feynman [25] has pointed out that Turing machines seem to be unable to efficiently simulate quantum physics; that is, they seem to require an exponential slowdown to simulate it (although ....

A. Vergis, K. Steiglitz, and B. Dickinson. The complexity of analog computation. Math. and Computers in Simulation, 28:91--113, 1986. .

....power of continuous time processes are at the moment relatively rare in the literature, but we expect their number to grow along with the current increase of interest in analog computation. The only other analysis that explicitly addresses computational complexity issues seems to be that in [22]. Computability aspects have been studied more often; see, e.g. 2, 5, 6, 15, 16, 19, 20] 2 A continuous time neural network model An electrical model of Hopfield s continuous time neuron is shown in Figure 1. Here oe denotes the characteristic of the nonlinear amplifier, and ae i and C i are ....

Vergis, A., Steiglitz, K., Dickinson, B. The complexity of analog computation. Math. and Computers in Simulation 28 (1986), 91--113. 22

....exist, which were invented to solve one complicated task. Such are the differential analyzer invented by Lord Kelvin in 1870[120] which uses friction, wheels, and pressure to draw the solution of an input differential equations. The spaghetti sort is another example, and there are many more[194]. Are these systems computers We do not want to construct and build a completely different machine for each task that we have to compute. We would rather have a general purpose machine, which is universal . A mathematical model for a universal computer was defined long before the invention ....

....DNA computer which enables a solution of NP complete problems (these are hard problems to be defined later) in polynomial time[4, 140] However, the cost of the solution is exponential because the number of molecules in the system grows exponentially with the size of the computation. Vergis et al.[194] suggested a machine which seems to be able to solve instantaneously an NP complete problem using a construction of rods and balls, which is designed such that the structure moves according to the solution to the problem. A careful consideration[178] reveals that though we tend to think of rigid ....

Vergis A, Steiglitz K and Dickinson B , "The Complexity of Analog Computation", Math. Comput. Simulation 28, pp. 91-113. 1986

....a number of steps polynomial in the input size is known as P. For this classification to make sense, we need to know that whether a function is computable in polynomial time is independent of the kind of computing device used. This corresponds to a quantitative version of Church s thesis, which Vergis et al. 1986] have called the Strong Church s Thesis and which makes up half of the Invariance Thesis of van Emde Boas [1990] This quantitative Church s thesis is: Any physical computing device can be simulated by a Turing machine in a number of steps polynomial in the resources used by the computing ....

....time (computation steps) and space (memory) There are more resources pertinent to analog computation; some proposed analog machines that seem able to solve NP complete problems in polynomial time have required the machining of exponentially precise parts, or an exponential amount of energy. See Vergis et al. 1986] and Steiglitz [1988] this issue is also implicit in the papers of Canny and Reif [1987] and Choi et al. 1995] on three dimensional shortest paths. For quantum computation, in addition to space and time, there is also a third potential resource, accuracy. For a quantum computer to work, ....

[Article contains additional citation context not shown here]

A. Vergis, K. Steiglitz, B. Dickinson (1986) "The complexity of analog computation," Math. and Computers in Simulation 28, 91--113.

....the ultimate capabilities of such devices from a theoretical standpoint. Part of the problem is that, much interesting work notwithstanding, analog computation is hard to model, as difficult questions about precision of data and readout of results are immediately encountered see for instance [VSD86] and the many references there. We take the point of view that artificial neural nets provide an opportunity to reexamine some of the foundations of analog computation from the new perspective afforded by an extremely simple yet surprisingly rich model, in a context where techniques from ....

A. Vergis, K. Steiglitz, and B. Dickinson. The complexity of analog computation. Math. and Computers in Simulation, 28:91--113, 1986.

....the ultimate capabilities of such devices from a theoretical standpoint. Part of the problem is that, much interesting work notwithstanding, analog computation is hard to model, as difficult questions about precision of data and readout of results are immediately encountered see for instance [18], and the many references there. With the constraint of an unchanging structure, it is easy to see that classical McCullochPitts that is, binary neurons would have no more power than finite automata, which is not an interesting situation from a theoretical complexity point of view. Therefore, ....

Vergis A., K. Steiglitz, B. Dickinson, "The complexity of analog computation," in Math. and Computers in Simulation 28(1986): 91-113.

....rate limits of the circuit. At the low end of the bandwidth, there is a limit due to the number of electrons and the consequent number of transitions that can occur. Although we have stated that there is a dearth of work in the area of analog complexity theory, there is some work. Vergis et al. [Vergis 86] discusses the complexity of mechanical analog computation for a very restricted class of problems. Their analysis explicitly rules out quantum mechanics, other probabilistic behavior, and nonlinear devices as not applicable to their analysis. By eliminating nonlinearities, Vergis explicitly ....

Vergis, Anastasios, Kenneth Steiglitz, and Bradley Dickinson, "The Complexity of Analog Computation," Mathematics and Computers in Simulation 28 (1986) 91-113, NorthHolland.

....updates these real numbers by performing elementary operations, such as addition, upon them. As was pointed out above, models such as this, with infinite precision (or even exponential precision) arithmetic are capable of carrying out tasks such as factoring or satisfiability in polynomial time [Ad, VSD]. What distinguishes quantum computation from classical computation is our ability to prepare the system to solve a computational problem of our choice. If we try to build a classical device to carry out a desired computation, the imprecision in the realization of the system, effectively leaves us ....

Vergis, A., Steiglitz, K., Dickinson, B., "The complexity of analog computation", Mathematics and Computers in Simulation 28 (1986) 91-113.

....the ultimate capabilities of such devices from a theoretical standpoint. Part of the problem is that, much interesting work notwithstanding, analog computation is hard to model, as difficult questions about precision of data and readout of results are immediately encountered see for instance [21], and the many references there. With the constraint of an unchanging structure, it is easy to see that classical McCullochPitts that is, binary neurons would have no more power than finite automata, which is not an interesting situation from a theoretical complexity point of view. Therefore, ....

Vergis A., K. Steiglitz, B. Dickinson, "The complexity of analog computation," in Math. and Computers in Simulation 28(1986): 91-113.

....Very little work has been done on the potentially most fruitful field of computational complexity 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 possible by digital means. By a standard numerical integration argument, they come to the conclusion ....

....R, the maximum magnitude of the second derivative of the simulated system. The intended implication is then that Lipschitzian analog systems cannot be superpolynomially more efficient than digital computers for solving limited precision problems. However, looking more carefully at the argument in [52], one notices that the number of steps in the digital simulation is in fact exponential in the length of the analog time interval [0; t] which is assumed predetermined in the proof. Of course, an analog computation can be artificially sped up to occur within any given time interval, but then the ....

[Article contains additional citation context not shown here]

A. Vergis, K. Steiglitz, B. Dickinson, The complexity of analog computation. Math. and Computers in Simulation 28 (1986), 91--113.

No context found.

Vergis A, Steiglitz K and Dickinson B, The complexity of analog computation, Math. Comp. Sim. 28 91--113 (1986)

No context found.

Vergis A, Steiglitz K and Dickinson B , "The Complexity of Analog Computation", Math. Comput. Simulation 28, pp. 91-113. 1986

No context found.

A. Vergis, K. Steiglitz, and B. Dickinson. The complexity of analog computation. Math. and Computers in Simulation, 28:91-113, 1986. .

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC