- USA - Analog computation - Google Patents
- systron donner :: Handbook Of Analog Computation Jun67
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The new dynamical system is therefore:.
USA - Analog computation - Google Patents
The escape rate is an invariant measure of the dynamics introduced for characterizing transiently chaotic systems 55 , In a transiently chaotic system the asymptotic dynamics is not chaotic, but, for example, settles onto a simple attractor, or escapes to infinity in open systems , however, the nonasymptotic dynamics is chaotic, usually governed by a chaotic repeller. It is well known that for hyperbolic, transiently chaotic dynamical systems the probability of a randomly started trajectory not converging to an attractor by time t i. For permanently chaotic systems, such as our MaxSAT solver, however, this definition does not work, as there is no simple asymptotic attractor in the dynamics and the system is closed.
To be able to use a similar notion also for MaxSAT, we use a thresholding on the energy of the visited states. More precisely, we monitor the probability p E , t that a trajectory has not yet found an orthant of energy smaller than E by analog time t. Here, E acts as a parameter of the distribution. This can be measured by starting many trajectories from random initial conditions and monitoring the fraction of those that have not yet found a state with an energy less than E by analog time t. In Fig. For large E , all trajectories almost immediately find orthants with fewer unsatisfied clauses, but for lower E values the distributions decay exponentially.
systron donner :: Handbook Of Analog Computation Jun67
Naturally, if an energy level does not exist in the system e. From extensive simulations, we observe a power-law behavior with an intercept E 0 :. This observation is at the basis of our method to predict the global energy minimum for MaxSAT. Energy dependent escape rate. This estimation is convenient, as it is easier to automate in the algorithm than the fitting procedure see Methods. The dashed lines show the fitting of Eq. Here, we describe the algorithm along with the halting criterion for system 4 with details presented in the Methods section along with a flowchart shown in Supplementary Fig.
The exponentially decaying nature of the p E , t distributions implies that sooner or later every trajectory will visit the orthant with the lowest energy. Nevertheless, instead of leaving one trajectory to run for a very long time, it is more efficient starting many shorter trajectories from random initial conditions and tracking the lowest energy reached by each trajectory see Supplementary Fig. This also generates good statistics for p E , t and for obtaining the properties of the chaotic dynamics that are then exploited along with 5 to predict the value of the global minimum and to decide on the additional number of trajectories needed to find a lower energy state with high probability.
If this energy value has already been attained found at least one assignment for it , the algorithm outputs the corresponding assignment s. In the latter case it outputs the lowest energy value attained and the corresponding assignment s and the consistency status of the predicted value. Thus, one expects that the performance of the algorithm decreases as N increases, e.
Nevertheless, the results show that all three distributions have a large peak at 0. Positive errors are much smaller and shifted toward harder problems. Negative errors mean that the algorithm consistently predicts a slightly lower energy value than the optimum, which is good as this gives an increased assurance that we have found the optimum state. In Supplementary Fig.
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Algorithm statistics over random Max 3-SAT problems. Next, we present the performance of our solver on MaxSAT competition problems, from We are foremost interested if Max-CTDS is capable of predicting the global minimum energy value and finding assignments corresponding to that energy value for hard problems, within a reasonable time.
This problem was also used in Fig. No complete competition algorithm could solve this problem. We ran our algorithm on a regular iMac The problem can be downloaded from the competition site Similar figures for other hard problems such as for a 4-SAT problem and a spin-glass problem are shown in Supplementary Figs.
Algorithm performance on a hard benchmark problem. We use the same problem as in Fig. See Supplementary Fig. These deviations are presented in detail in Fig. This causes the effects of stiffness to appear earlier in the simulations than for the other problems, slowing them down; see Supplementary Note 1 and Supplementary Fig.
Algorithm performance on competition MaxSAT problems. Max-CTDS solves all problems within 0. In terms of computation time wall-clock time on digital machines using standard computers , Max-CTDS typically took on the order of hours to find an assignment for the minimum energy value. The average time over all the problems was 4. The lowest search time was 1. These numbers certainly depend on the digital hardware used. Note that in an analog circuit implementation, the current flow or voltage behavior would correspond to the equations of the solver, eliminating numerical integration issues and thus the algorithm should run much faster 46 shows a possible 10 4 speedup.
Ramsey theory deals with the unavoidable appearance of order in large sets of objects partitioned into few classes, with deep implications in many areas of mathematics 51 , 61 but also with practical applications Although it has several variants, in the standard, two-color Ramsey number problem we have to find the order for the smallest complete graph for which no matter how we color its edges with two colors red and blue , we cannot avoid creating a monochromatic m -clique. The number of nodes for the smallest such complete graph is denoted by R m , m.
The best lower bound of 43 was first found in by Exoo 64 , and the upper bound was only recently reduced from 49 53 to 48 by Angeltveit and McKay Using various heuristic methods, researchers have found in total solutions graphs and their complements for the complete graph on 42 nodes Starting from these solutions they searched for a 5-clique-free coloring in We are satisfied with a coloring a solution when no m -clique is monochromatic, i.
This is seen from the plot of E vs.
baannernnam.com/images/out/1286.php This is simply due to the fact that 5 is a statistical average behavior characteristic of the chaotic trajectory, from the neighborhood of the chaotic repeller of the dynamics and away from the region in which the solution resides. However, once the trajectory enters the basin of attraction and nears the solution, the dynamics becomes simple, nonchaotic, and runs into the solution, reflected by the sudden drop in energy.
This is not due to statistical errors, because the curve remains consistent when plotting it using 10 3 , 10 4 , or 10 5 initial conditions the figure shows 10 5 initial conditions. Finding the Ramsey number R 4, 4. E 0 is the extrapolated value based on the fit from Eq. The long vertical bars indicate the lower end of the fitting range. Colorings for the R 5, 5 Ramsey number problem. The thicker red blue edges from c are represented with darker red blue cells. Supplementary Fig. In summary, we presented a continuous-time dynamical system approach to solve a quintessential discrete optimization problem, MaxSAT.
The solver is based on a deterministic set of ordinary differential equations and a heuristic method that is used to predict the likelihood that the optimal solution has been found by analog time t. The prediction part of the algorithm exploits the statistics of the ensemble of trajectories started from random initial conditions, by introducing the notion of energy-dependent escape rate and extrapolating this dependence to predict both the minimum energy value lowest number of unsatisfied clauses and the expected time needed by the algorithm to reach that value.
This statistical analysis is very simple; it is quite possible that more sophisticated methods can be used to better predict minima values and time lengths.
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Due to its general character, the presented approach can be extended to other optimization problems as well, to be presented in forthcoming publications. Instead of a numerical implementation on a digital computer, one would ideally like to use a direct implementation by analog circuits, the feasibility of which has been shown in ref.
One reason for this is that in such analog circuits the von Neumann bottleneck is eliminated, with the circuit itself serving its own processor and memory, see ref. Implementation on a digital computer, however, as it was done here requires the use of an ODE integrator algorithm, which discretizes the continuous-time equations and evolves them step by step, while controlling for errors. The time-cost of the dynamics in this case is the wall-clock time not t , which also depends on the computer hardware and the numerical integration method used.
In digitized form, the solver is not performing better than current MaxSAT competition solvers simply because the dynamics evolves many several thousands or more coupled ODEs, and this integration is time consuming on digital machines.
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Additionally, to manage the occasional stiffness of the differential equations, one needs to use implicit or higher-order integration methods, also contributing to the slowing down of the simulations. Note that this would not be an issue for analog circuit implementations, as there are no discretization schemes or numerical integration methods; the physical system evolves its currents and voltages according to the ODEs, flowing toward a halting condition, solving the problem. Nevertheless, even when simulated on a digital machine, the solver finds very good solutions to hard problems in reasonable time.
This is because continuous-time analog dynamical systems represent an entirely novel family of solvers and search dynamics, and for this reason they behave differently and thus may perform better, than existing algorithms on certain classes of hard problems. It is also important to note that the system 4 is not unique, other ODEs can be designed with similar or even better properties.
This is useful, because the form given in 4 is not readily amenable to simple hardware implementations, due to the constantly growing auxiliary variable dynamics all variables represent a physical characteristic such as a voltage or a current and thus they will have to have an upper limit value for a given device. However, the auxiliary variables do not need to grow always exponentially, one can devise other variants in which they grow exponentially as needed, otherwise they can decay to be presented in a future publication , allowing for better hardware implementations.
To illustrate the effectiveness of our solver, we applied it to the famous two-color Ramsey problem and in particular for R 5, 5 , which is still open. Note that after posting our paper to arxiv, Geoffrey Exoo in a private communication mentioned that he also found the same, smallest energy coloring as presented here. If all clauses contain exactly k literals, the problem is k -SAT. Max 2-SAT i. Due to the exponentially growing weights, the changes in V are dominated by the clause that was unsatisfied the longest. Keeping only that term in V and inserting it into 2 , it is easily seen that the dynamics drives the corresponding clause function toward zero exponentially fast, until another clause function takes over.
This is repeated until all clauses are satisfied, for solvable SAT problems.
The properties and performance of this solver have been discussed in previous publications 42 , 43 , Here, we give a simple, nonoptimized variant of the algorithm see flowchart in Supplementary Fig. Better implementations can be devised, for example with better fitting routines, however the description below is easier to follow and works well.
Given a SAT problem, we first determine the b parameter as described previously. Step 6: we check the consistency of the prediction defined here as saturation of the predicted values. If it is not consistent yet, we continue running new trajectories Step 4. If it is large enough e. As seen in Fig. Theoretically the escape rate can be obtained by fitting the exponential on that last part of the curves Fig.
However, while running the algorithm it would be difficult to automatically estimate the region where the exponential should be fitted.