Noise tolerant spatiotemporal chaos computing Behnam Kia, Sarvenaz Kia, John F. Lindner, Sudeshna Sinha, and William L. Ditto Citation: Chaos: An Interdisciplinary Journal of Nonlinear Science 24, 043110 (2014); doi: 10.1063/1.4897168 View online: http://dx.doi.org/10.1063/1.4897168 View Table of Contents: http://scitation.aip.org/content/aip/journal/chaos/24/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Spatiotemporal dynamics of a digital phase-locked loop based coupled map lattice system Chaos 24, 013116 (2014); 10.1063/1.4863859 Hyperlabyrinth chaos: From chaotic walks to spatiotemporal chaos Chaos 17, 023110 (2007); 10.1063/1.2721237 Spatiotemporal patterns driven by autocatalytic internal reaction noise J. Chem. Phys. 122, 214701 (2005); 10.1063/1.1900092 Localized error bursts in estimating the state of spatiotemporal chaos Chaos 14, 1042 (2004); 10.1063/1.1788091 Statistics of defect trajectories in spatio-temporal chaos in inclined layer convection and the complex Ginzburg–Landau equation Chaos 14, 864 (2004); 10.1063/1.1778495
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Noise tolerant spatiotemporal chaos computing Behnam Kia,1 Sarvenaz Kia,1 John F. Lindner,2 Sudeshna Sinha,3 and William L. Ditto1 1
Department of Physics and Astronomy, University of Hawaii at Manoa, Honolulu, Hawaii 96822, USA Physics Department, The College of Wooster, Wooster, Ohio 44691, USA 3 Indian Institute of Science Education and Research (IISER), Mohali, Punjab 140306, India 2
(Received 8 July 2014; accepted 23 September 2014; published online 20 October 2014) We introduce and design a noise tolerant chaos computing system based on a coupled map lattice (CML) and the noise reduction capabilities inherent in coupled dynamical systems. The resulting spatiotemporal chaos computing system is more robust to noise than a single map chaos computing system. In this CML based approach to computing, under the coupled dynamics, the local noise from different nodes of the lattice diffuses across the lattice, and it attenuates each other’s effects, resulting in a system with less noise content and a more robust chaos computing architecture. C 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4897168] V
Noise is everywhere, and in many engineering applications it is noise that puts an upper limit on the performance of a system. Specifically, noise issues are a main concern and limiting factor for conventional deep submicron digital integrated circuits. The field of chaos computing is an alternative approach for computing, in which the computations are implemented at the dynamics level. In this paper, we encapsulate computation and noise robustness at the dynamics level, and introduce a new approach for robust, reconfigurable computation. We show how by using coupled chaotic dynamics for computation we can make chaos computing systems more robust to local noise by exploiting the inherent noise reduction properties in coupled systems.
I. INTRODUCTION
A dynamical system can perform a computation by implementing functions with its orbits. The dynamical system maps an initial condition, which represents the input, to future states, which represent the outputs. Future states of a chaotic dynamical system are highly sensitive to the initial conditions.1 A very small change in the initial condition can change the future states of the chaotic system, and therefore the type of computations that the chaotic system can implement. This is the main idea behind chaos computing.2 In many chaos computing methods that have been introduced so far, a single chaotic system is initialized based on the inputs, and after evolution of the chaotic map for a period of time, the output of the computation is decoded from the final state.2 A generic schematic of chaos computing paradigm is depicted in Fig. 1. Specifically consider the m binary data inputs, d1,d2,…,dm and the n digital control inputs, c1,c2,…,cn, which are mapped by an encoding map E, to a point on the unstable manifold of the chaotic system. If L is a binary set {0,1}, then LðnþmÞ represents the domain of the encoding map E, which consists of all the possible combinations of digital data and control inputs. Let b be the unstable manifold of the chaotic system, Rs the general state space of the chaotic 1054-1500/2014/24(4)/043110/7/$30.00
system, and Y the output of E on the unstable manifold. In this case, the general form of the encoding map is as follows: E : LðnþmÞ ! b;
b Rs ; L ¼ f0; 1g;
x0 ¼ Eðd1 ; d2 ; :::; dm ; c1 ; c2 ; :::; cn Þ:
(1a) (1b)
Then starting from the initial conditions produced by the encoding map, the chaotic map evolves for a fixed time (or for a fixed iteration number, if the chaotic system is discrete). After this evolution time, the final state of the map is decoded to the outputs using a decoding map. The decoding of the binary outputs from the final state can be as simple as a threshold mechanism.2 An example will be presented and explained in Sec. III. The encoding map maps different sets of the inputs to different initial conditions on the unstable manifold of the chaotic system. The chaotic orbits and therefore the final state values and the decoded binary outputs will be very sensitive to the inputs. As a result, a single one-bit change in the inputs can change the outputs. Thus control inputs can be used to select different digital functions. A digital function maps binary inputs to binary outputs. To evaluate which digital function is selected with a particular control input, the association of a control input with a logic function is noted and then all possible combinations of data inputs are enumerated to construct the truth table of the function. A truth table of a digital function is a table that shows what outputs are produced when different data inputs are given to the function. By changing the control input and repeating this procedure (of constructing the truth table), a second digital function different (with high probability) from the first one is observed. This is the meaning of the reconfigurability of chaos computing. By using all possible control inputs and finding the type of function that the chaotic system
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FIG. 1. Block diagram of chaos computing. C 2014 AIP Publishing LLC V
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implements, the full instruction set of the chaotic system is obtained.2 Variations of this approach have been central to various theoretical, applied, and commercial versions of chaotic computing applications.3,4 But is chaotic computing robust to noise? Does sensitivity to initial conditions, which makes the system flexible in computation, make the computing system unreliable and overly sensitive to noise? Chaotic computing is robust to noise if the chaotic system evolves long enough so the control inputs change the outputs (via changing the orbits) but not long enough so the noise changes the outputs. There is a fine line between iterating long enough to get different functions, and iterating too long to allow the noise to randomly change the final states. We have previously investigated5 this issue and have examined the robustness levels of chaos computing. However, one of the main difficulties with chaos computing remains making it more robust to noise. While there has been significant work on exploiting noise to make computation based on dynamical systems more robust,6 there is also a need to understand how to reduce rather than exploit the overall noise sensitivity of chaotic computation architectures. Thus if a way were possible to incorporate noise reduction into the design of the dynamical system(s) that actually perform the computation then one would, in principle, have an additional tool to build the most robust and configurable chaotic computing architectures. This will become particularly impactful for constrained architectures, such as submicron VLSI,7 where the noise is problematic. Scaling the transistors has been the main method to increase the performance of the digital computing systems.8 With the continuous scaling of CMOS digital technologies, signal integrity and noise issues have become a main concern and limiting factor for deep submicron digital integrated circuits.7 As a result, semiconductor chip makers are conducting research on radical, but extremely expensive and complicated device architectures, such as FinFETS, to address the issue of noise in computing systems.9 In this paper, we encapsulate computation and noise robustness at the dynamics level, and introduce an alternative approach for robust, reconfigurable computation. In particular, we demonstrate that by using coupled chaotic systems for computation, we can make chaos computing systems more robust to noise by exploiting the inherent noise reduction properties in coupled systems to exceed the performance of a single isolated chaotic computing system. Coupled dynamics reduces the noise content of the coupled systems and can inherently function as an averaging filter for itself. The incoherent, independent noise terms added to different nodes of the coupled map lattice (CML) are diffused and averaged by the coupled dynamics, whereas the coherent input signals to the nodes remain preserved under the coupled dynamics. We demonstrate how a CML can be utilized to implement computation and how the application of coupled redundant maps increases the noise robustness of computations. Notice that an ensemble averaging of N independent chaotic maps can result in a similar level of noise robustness that we can obtain by dynamically coupling N maps together.
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But the main purpose of this paper is to show that as we utilize dynamics to implement computation, we can also utilize coupled dynamics to implement an averaging filter in addition to the implemented computation to improve the noise robustness of the computing system. Here, we are not explicitly designing and engineering an averaging filter, rather we demonstrate that the coupled dynamics itself can function as an averaging filter, and therefore it can be used as a physical realization of the averaging filter. This results in a fully dynamics based approach for computation, where dynamics and coupled dynamics are used for computation and improving noise robustness of the computation, respectively. Utilizing redundancy has been one of the main techniques to reduce the probability of error in a system and it dates back to the first days of the modern computers. John von Neumann discusses that10 “instead of running the incoming data into a single machine, the same information is simultaneously fed into a number of identical machines, and the result that comes out of a majority of these machines is assumed to be true.” However, when the error is caused by noise, averaging the noisy states of the redundant systems can reduce and control the errors better than using a majority wins technique among the redundant systems. In this paper, we use coupled dynamics as a technique to organize the redundant systems, and to implement a dynamics level and dynamics based averaging filter. Therefore, it reduces noise better than using the majority wins technique. But if the error is caused by catastrophic failure of an element, averaging would not work and majority wins technique is the effective method to recover the system from such failures. The organization of the paper is as follows: in Sec. II, we study how coupled dynamics can reduce the noise content of the coupled lattice. In Sec. III, we introduce coupled map lattice computing and investigate noise robustness of the resulting system. In Sec. IV, we study noise robustness of a globally coupled map (GCM) lattice based computing of size N. The paper is concluded in Sec. V. II. NOISE ROBUSTNESS THROUGH DYNAMICALLY COUPLING IDENTICAL SYSTEMS
The Kaneko coupling scheme,11 which we use in our study as a model of coupled dynamics, has the recurrence relation e ¼ ð1 eÞf ðxij Þ þ ½f ðxij1 Þ þ f ðxijþ1 Þ; xiþ1 j 2
(2)
where xj represents the state of the jth node, i is the iteration number, f is a one-dimensional, discrete map, e is the coupling parameter, and e 2 ½0; 1. For simplicity, we study a CML of size 3, which means j ¼ 1,2,3, and also assume the first and the last maps of the lattice, j ¼ 1 and j ¼ 3, are connected to each other. All maps in the CML receive the same initial condition x0. But the maps are subjected to local noise. This means that the noise terms that are added to different maps of the CML are statistically independent from each other. However, the noise terms added to different maps of the CML are all identically distributed. More specifically, the
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initial conditions of the maps in the CML are set to x0j ¼ x0 þ rd0j , j ¼ 1,2,3, where rd0j is the additive noise term at step 0 to map number j. The additive noise is added to each map at each future iteration like ¼ f ðxij Þ þ rdiþ1 xiþ1 j ; j
(3)
Nð0; 1Þ and d0j Nð0; 1Þ are normal Gaussian where diþ1 j noise with zero mean and unit variance. The variance r2 of the additive noise to any map is the same. Equation (3) presents the noise effects on a single node. In a coupled map lattice, f will be replaced by the coupled map dynamics of Eq. (2). The future noisy state of a map in the CML will be x1j
e ¼ ð1 þ ½f ðx0j1 Þ þ f ðx0jþ1 Þ þ rd1j 2 e ¼ ð1 eÞf ðx0 þ rd0j Þ þ ½f ðx0 þ rd0j1 Þ 2 þ f ðx0 þ rd0jþ1 Þ þ rd1j : eÞf ðx0j Þ
(4)
After linearizing the function f around initial condition x0, we obtain e 0 e 0 0 1 0Þ ð ð Þ xj ¼ f x þ rk1 1 e dj þ dj1 þ djþ1 þ rd1j ; (5) 2 2 where k1 ¼ df =dx is evaluated at x0. The first term of Eq. (5) is the noise free evolution of the map starting from the noise free initial condition, x0; the second term, which we are interested in, is evolution of previous noise terms under the coupled dynamics, rk1 ½ð1 eÞd0j þ 2e d0j1 þ 2e d0jþ1 ; and the last term is the noise term that is added to the map at the current iteration, rd1j . The noise terms added to different maps are independent and identically distributed. Therefore by using the law of summation of independent random variables, the variance of the second term, which is the one-step noise variance, becomes e2 e2 2 r21 ¼ r2 k21 ð1 eÞ r2d þ r2d þ r2d 4 4 2 e 2 2 : ¼ ðrk1 Þ ð1 eÞ þ 2
(6)
The one-step noise variance in a single isolated map can be easily obtained by setting e ¼ 0 in Eq. (6) to get r21 ¼ ðrk1 Þ2 :
(7)
By comparing Eqs. (6) and (7), we observe that the one-step variance of noise deviations in the coupled map lattice is rescaled by r ¼ (1 e)2 þ e2/2, where this factor is less than 1, indicating that the coupled dynamics reduces noise deviations. In Fig. 2, this rescaling factor is plotted for different possible values of coupling parameters. By taking the first derivative of this rescaling factor and setting it to zero, dr/dt ¼ 0, we find e ¼ 2=3 as the optimal coupling parameter, and by setting e ¼ 2=3 in Eq. (6), the one-step noise variance becomes
FIG. 2. The rescaling factor r versus different values of coupling parameter e.
1 2 r21 ¼ ðrk1 Þ : 3
(8)
Similarly, the further iterations of the CML can be linearized around the noise free evolution, and after applying the law of summation of independent random variables on the independent noise terms from different nodes, eventually we can estimate the variance of noise deviation in the coupledmap lattice after m iterations as 1 2 2 r2m ¼ r2 ½ðk1 k2 :::km1 km Þ þ ðk2 :::km1 km Þ þ ::: þ k2m ; (9) 3 where kk ¼ df/dx is evaluated at f m1(x0), and f m1 means m 1 iterations of function f, whereas the variance of noise deviations in the single map after m iterations is r2m ¼ r2 ½ðk1 k2 :::km1 km Þ2 þ ðk2 :::km1 km Þ2 þ ::: þ k2m : (10) We observe that the coupled dynamics of the identical maps reduces the variance of noise deviation to 1/3 of the variance of noise deviation in a single map when they are subjected to the same amount of noise. In a GCM lattice of size N, all nodes are connected to each other as12 e X i i ð Þ xiþ1 ¼ 1 e f ðx Þ þ f xk : (11) j j N 1 k6¼j By linearizing GCM lattice dynamics along the noise free orbit, it is easy to show that the coupled dynamics shrinks the variance of the noise deviation. The dynamic coupling e ¼ (N 1)/N converts the GCM into an inherent averaging filter and provides the maximum noise reduction, where the variance of noise deviation is reduced to 1/N of the variance of noise deviation in a single map. For theoretical simplicity and computational speed, we have focused this paper on coupled map lattices and Kaneko model, but the phenomenon of coupling reducing noise is more general and applies to many other kinds of coupling and other topologies for coupling. As an example, we have observed that the global coupling is not necessary to reduce noise effects. Rather a sparser coupled lattice, such as a CML with just local connectivity can reduce noise effects almost as much as a GCM even for large N values. Even though in a locally coupled CML not all nodes are coupled together, different noises from different nodes can still
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FIG. 3. Block diagram of a CML based chaos computing.
diffuse through the local coupling and average each other’s effects. Homogeneous, global coupling gives the highest amount of noise reduction in the lattice. However, other coupling topologies can result in noise reduction levels that are not very different from the amount one can get from this best case, the global coupling. Further details and discussions about noise reduction in locally coupled maps can be found in Ref. 13. III. ROBUST TO NOISE CML BASED CHAOS COMPUTING
The schematic of our proposed CML based chaos computing approach to improve robustness to noise is depicted in Fig. 3 for a CML of size 3. Circles indexed by 1, 2, and 3 represent three maps, and the solid lines connecting these circles together represent the coupling. The encoding map maps the data and controls inputs to an initial condition x0 and initializes all three maps to this value. After n iterations of the coupled dynamics, the output of the computation is decoded from the final state of one of the maps. Notice that it does not matter which node we choose to decode the output from. All three maps are identical (except for noise realizations) and initialized to the same value. Therefore, noise free evolution of these nodes under the coupled dynamics results in the same output no matter from which node we select to read the output. Under noisy condition, these maps might result in different output values, however, the error rate, the probability of error, is the same for all nodes. Despite the fact that a single map can be enough to implement the desired digital functions,2 here we use redundant maps coupled together based on Eq. (2) to implement the same digital functions, but now with a higher level of
noise tolerance. This higher noise tolerance will be achieved at the expense of redundancy of the maps. The redundant maps receive the same, coherent inputs, and the maps perform the computation in the presence of incoherent, local noise. As demonstrated in Sec. II, the coherent inputs are preserved by the coupled dynamics, whereas the incoherent local noise is averaged by the coupled dynamics. Therefore, these dynamically coupled redundant maps can perform computations at noise levels that a single map cannot perform at the required error rate. As a numerical example, consider the one-humped map xiþ1 ¼ 1 – 2ðxi Þ2
(12)
on the interval [ 1, 1] which nests like f(2)[x] ¼ f[f[x]], and so on, as in Fig. 4. Assume the aim is to implement different two-input, one-output digital functions. Additionally assume there are six binary control inputs to program the CML to implement the different types of the function. The encoding map can be as simple as a digital-to-analog converter. Include the linear scaling s½y ¼ 2y 1
(13)
from the unit interval [0,1] with the binary encoding 7 2 d1 þ26 d2 þ25 c1 þ24 c2 þ23 c3 þ22 c4 þ21 c5 þc6 ; Ec ½d ¼ s 28 (14) where the two high order bits encode the data d ¼ {dn} and the six low order bits encode the controls c ¼ {cn}. Using the binary decoding ( 0; y 0 (15) D½y ¼ 1; y > 0; and six iterations of the CML to find the output Dc ½d ¼ D½f ð6Þ ½Ec ½d
(16)
as summarized by Table I and Fig. 5. Notice that different controls can build the same function. Therefore, in Table I, in some rows there are multiple controls for a same function. However, these different realizations of the same function
FIG. 4. Iterates of the quadratic map increasingly oscillate causing chaos.
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TABLE I. Truth tables and instruction set for the Eq. (16) chaotic map. Output Dc ½d1 ; d2
N
Dc[0,0]
Dc[0,1]
Dc[1,0]
Dc[1,1]
Controls c1 c2 c3 c4 c5 c6 25c1 þ 24c2þ23c3þ 22c4þ21c5þc6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1
0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
(15), 28, 36, 49 25, (26), 27, 50, 61, 62 48, (58), 59 (35) 5, (6), 16 (17), 41, 42, 54 9, (55) (7), 8, 18, 19, 33, 34, 43, 44 2, 3, 14, 37, (38), 39 1, 13, 24, 40, (51), 63 10, 22, 23, (47) 11, (12), 60 (29) (4), 52, 53 20, 21, 30, 31, 45, 46, 56, (57) (32)
noise under the chaotic dynamics needs to be kept far from this threshold value. If the noise cloud crosses the threshold that means that there is a possibility that noise can change the output of the computation, which results in an error. The probability of this error in chaos computing can be determined by calculating the cumulative probability of the portion of the noise cloud that has passed the threshold value in Eq. (15). The theoretical models that we developed in Sec. II estimate that the variance of noise deviations, in a CML of size 3 is three times smaller than the variance of noise deviations in a single map. This suggests that a CML based chaos computing system can tolerate up to three times higher noise levels than a single map based chaos computing system can tolerate. This subject can be easily verified by examining Eq. (9). The three-times increase in additive noise variance can be compensated by the rescale factor 1/3 in Eq. (9). More precisely, if we assume r2Cm ¼ r2Sm ;
(17)
h i 1 2 2 rC ðk1 k2 :::km1 km Þ þ ::: þ k2m 3 h i 2 ¼ r2S ðk1 k2 :::km1 km Þ þ ::: þ k2m ;
(18)
r2C ¼ 3r2S ;
(19)
then we get might, and usually will, have different levels of robustness to noise. Please see Fig. 5. In this research, we pick up the most robust to noise realization. Monte Carlo simulation can be used the estimate noise robustness of these different realizations. In Table I, for each function, the more robust control input is distinguished by parenthesis. Now imagine the CML is subjected to additive noise. We can consider the additive noise around the initial condition of a map as a noise “cloud.” Assuming the mean value of noise is zero, the center of this cloud would be the noise free initial condition, and the size of the cloud would be proportional to the variance of the noise. We cannot be sure where exactly the noisy initial condition is; it can be anywhere in this cloud. But we can define a probability distribution, which gives us the probability that the noisy initial condition can be at each region of the cloud. After each iteration of the coupled map, this cloud of noise is rescaled and reshaped based on the eigenvectors and eigenvalues of the map. We know that in a chaotic system, this kind of uncertainty grows larger under the chaotic dynamics. Even though at some specific locations of the attractor, the eigenvalues can be less than 1, but still we can say that on average, the cloud size grows bigger over time because the chaotic map has a positive Lyapunov exponent. In one implementation of chaos computing, the output of computation is determined from the final state of the chaotic system using a simple threshold mechanism (Eq. (16)). An error in chaos computing happens when noise moves an orbit and its final state passes the threshold value. Therefore, the evolved cloud of
where r2Cm is the variance of noise deviations in a CML of size 3 after m iterations, r2Sm is the variance of noise deviations in a single map after m iterations, rC is the variance of the additive noise to the CML, and rS is the variance of additive noise to the single map. This means that the CML can be subjected to three times higher noise level than the noise level in a single map, and the variance of noise deviations for both CML and the single map would be the same. We define relative noise tolerance s ¼ r2C =r2S ;
(20)
where r2s is the maximum variance of additive noise that a single-map based chaos computing system can tolerate without exceeding a specified error rate, and r2C is the maximum variance of additive noise that an improved, more robust to noise CML based chaos computing, can tolerate without exceeding the same specified error rate. Therefore, based on Eq. (19), relative noise tolerance for a CML based chaos computing of size 3 is s ¼ 3. To investigate the accuracy of these theoretical estimates, we set up the following simulations. First, we numerically calculate the relative noise tolerance of a CML based
FIG. 5. Black lines superimposed on the 5th iterate of the quadratic map indicate the digital functions for two binary inputs and three binary controls. Slight changes in initial conditions flip the outputs from 1 (blue) to zero (red), especially at the interval’s ends.
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chaos computing, where CML is composed of three maps of Eq. (12), coupled together based on Eq. (2). For illustrative purposes, we assume that the computing system is specified to have an error rate of no more than 0.4%, four errors in each 103 operations. Notice that this choice of error rate does not change the results and conclusions. We have tried other error rates and we obtained the same results. We run a Monte Carlo simulation to numerically calculate the maximum variance of additive noise to a CML of size 3 and to a single map, both performing chaos computing, without exceeding the given error rate. We then calculate the ratio of these variances to obtain the relative noise tolerance. More specifically, we gradually increase the variance of additive noise, and for each variance value, we perform the chaoscomputing algorithm in the noisy environment 103 times. By observing the fifth error, we record the previous value of the noise variance value, which meets the specified error rate. This Monte Carlo simulation yields approximate results. To reduce the error of simulation and to estimate the size of simulation errors, we repeat this Monte Carlo simulation 25 times. The average value of these 25 relative noise tolerance values for each function is depicted in Fig. 6 by green circles. Notice that each digital function has a different level of noise robustness. This is why there are separate results for each function. The error bars represent a measure of error in Monte Carlo simulation results and are defined as the variance of 25 observed values for each function. We observe that the average value of relative noise tolerance s for each function is very close to 3, which is what the theory predicts. Notice that some functions, such as functions 7, 10, and 11, have a slightly lower noise tolerance than we expected. The reason is that the averaging filter is a first order approximation for the coupled dynamics. The effects of the second order terms that we drop during linearization manifest themselves as deviations from the perfect result that we expect, here three. Sometimes the actual result is greater than three and sometimes less than three. Four data inputs to each function are encoded as four different initial conditions. To calculate Fig. 7, we calculate the relative noise robustness for different orbits starting from these initial conditions. Some of these four relative noise robustness values would be slightly greater than three and some less than three. We take the lowest number as the relative noise robustness for that function. As a result, this technique tends to take the worst values, which are normally less than three. In some
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FIG. 7. Relative noise tolerance for GCM based chaos computing of size N.
functions, such as 7, 10, and 11, the deviations from the expected relative noise tolerance are higher due to their greater second order effects. IV. GLOBALLY COUPLED MAP LATTICE BASED CHAOS COMPUTING
Based on discussions in Sec. II, the variance of noise deviations in a GCM lattice of size N where e ¼ (N 1)/N, is 1/N of variance of noise deviations in a single map, when they both are subjected to the same amount of noise. As a result, a GCM based chaos computing of size N can tolerate up to N times more noise level than a single-map based chaos computing can tolerate. Therefore, the relative noise tolerance for a GCM based chaos computing of size N with e ¼ (N 1)/N is s ¼ N. Next, we numerically investigate the effect of lattice size, N, on relative noise tolerance, s, of a GCM based chaos computing. The GCM lattice is composed of N maps of Eq. (12) and they are coupled based on Eq. (11). We assume that the computing system is specified to have an error rate of no more than 0.4%, four errors in each 103 operations. The results from Monte Carlo simulation are presented in Fig. 7. The horizontal axis is the lattice size, N, and the vertical axis is the average relative noise tolerance of 16 different functions. In the Monte Carlo simulation, we numerically calculate the relative noise tolerance for each function, and then we average these values and plot these average values for each lattice size in Fig. 7. Notice that the Monte Carlo simulation produces approximate results. However, because we use the average values obtained for 16 different functions, the amount of error in results reported in Fig. 7 is reduced. The error bars represent the variance of the relative noise tolerance values for 16 different functions. We observe that in a GCM based chaos computing of size N, noise tolerance is s ¼ N. V. CONCLUSION
FIG. 6. Relative noise tolerance s for a size-3 CML based chaos computing.
In this paper, we have introduced CML based chaos computing approach, and we demonstrated that it is more robust to noise than a single-map based chaos computing approach. All the maps in the coupled map lattice receive the same inputs, therefore these coherent inputs remain unchanged under the coupled dynamics. But incoherent, local noise is averaged by the coupled dynamics. As a result, the CML based chaos computing is less sensitive to noise.
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This enables us to perform chaos computing at noise levels where a single map cannot reliably perform the same computation. Coupling dynamical systems, such as electronic circuits, cost very little or no extra structural complexity and computational complexity. We have discovered that a very straightforward coupling mechanism for coupling electronic circuits is sufficient for noise reduction purposes, and we have shown that these coupled electronic circuits are more robust to local noise than single, independent circuits. Results will be published in a future paper. We have shown that the coupled dynamics can be considered as a dynamics based realization of an averaging filter and by utilizing the inherent noise reduction properties in the coupled dynamics, we achieve noise reduction similar to that achieved by applying an averaging filter. This provides us with a unified, dynamics-based approach for computation, where both computation and robustness (to noise) are realized and implemented using intrinsic dynamics of the systems, with no need to explicitly design and engineer computation or noise filtering. ACKNOWLEDGMENTS
We gratefully acknowledge support from the Office of Naval Research under Grant No. N000141-21-0026 and STTR Grant No. N00014-14-C-0033.
1
S. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (Perseus Publishing, 2006).
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H. R. Pourshaghaghi, B. Kia, W. L. Ditto, and M. R. Jahed-Motlagh, “Reconfigurable logic blocks based on a chaotic Chua circuit,” Chaos Solitons Fractals 41, 233–244 (2008). 3 See http://www.chaologix.com/ for ChaoLogix; last accessed July 4, 2014. 4 D. Boothroyd, “Harnessing the butterfly effect for electronics,” New Electronics, April 28, 13–14 (2009); D. Graham-Rowe, “Engines of Chaos,” New Scientist, November 1st, 40–44 (2008); L. Dailey Paulson “Chip Takes a New Reconfigurability Approach,” Computer, IEEE, September, 22–24 (2006); C. McCarty, A. Cohen, S. Rupley, and R. Peterson “The 10 Coolest Technologies You’ve Never Heard Of – Chaos Computing,” PC Magazine 25, 66 (2006). 5 B. Kia, M. Spano, and W. Ditto, “Chaos computing in terms of periodic orbits,” Phys. Rev. E 84, 036207–036214 (2011); B. Kia et al., “Unstable periodic orbits and noise in chaos computing,” Chaos 21, 047520 (2011). 6 K. Murali, A. Miliotis, W. L. Ditto, and S. Sinha, Phys. Lett. A 373, 1346 (2009); K. Murali, I. Raja Mohamed, S. Sinha, W. L. Ditto, and A. R. Bulsara, “Realization of reliable and flexible logic gates using noisy nonlinear circuits,” Appl. Phys. Lett. 95, 194102 (2009). 7 Z. Liang, M. Ikeda, and K. Asada, “Analysis of noise margins due to device parameter variations in sub-100 nm CMOS technology,” in IEEE Conference on Design and Diagnostics of Electronic Circuits and Systems, Krak ow, Poland, April 11–13, 2007; E. N. Shauly, “CMOS leakage and power reduction in transistors and circuits: Process and layout considerations,” J. Low Power Electron. Appl. 2(1), 1–29 (2012). 8 N. H. E. Weste and D. M. Harris, CMOS VLSI Design, A Circuits and Systems Perspective (Addison-Wesley, Boston, 2011). 9 See http://phys.org/news/2011-05-radical-intel-transistor-based-uc.html for “Radical New Intel Transistor Based on UC Berkeley’s FinFET” (last accessed June 26, 2014). 10 J. V. Neumann, “Probabilistic logics and the synthesis of reliable organisms from unreliable components,” in John von Neumann Collected Works, edited by A. H. Taub (A Program Press Book, New York, 1963), Vol. V, pp. 329–378. 11 K. Kaneko, “Overview of coupled map lattices,” Chaos 2, 279–282 (1992). 12 K. Kaneko, “Clustering, coding, switching, hierarchical ordering, and control in network of chaotic elements,” Physica D 41, 137–172 (1990). 13 B. Kia, S. Kia, J. Lindner, S. Sinha, and W. Ditto, “Coupling reduces noise,” Int. J. Bifurcation Chaos arXiv:1409.5186v1 (to be published).
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CHAOS 24, 043110 (2014)
Noise tolerant spatiotemporal chaos computing Behnam Kia,1 Sarvenaz Kia,1 John F. Lindner,2 Sudeshna Sinha,3 and William L. Ditto1 1
Department of Physics and Astronomy, University of Hawaii at Manoa, Honolulu, Hawaii 96822, USA Physics Department, The College of Wooster, Wooster, Ohio 44691, USA 3 Indian Institute of Science Education and Research (IISER), Mohali, Punjab 140306, India 2
(Received 8 July 2014; accepted 23 September 2014; published online 20 October 2014) We introduce and design a noise tolerant chaos computing system based on a coupled map lattice (CML) and the noise reduction capabilities inherent in coupled dynamical systems. The resulting spatiotemporal chaos computing system is more robust to noise than a single map chaos computing system. In this CML based approach to computing, under the coupled dynamics, the local noise from different nodes of the lattice diffuses across the lattice, and it attenuates each other’s effects, resulting in a system with less noise content and a more robust chaos computing architecture. C 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4897168] V
Noise is everywhere, and in many engineering applications it is noise that puts an upper limit on the performance of a system. Specifically, noise issues are a main concern and limiting factor for conventional deep submicron digital integrated circuits. The field of chaos computing is an alternative approach for computing, in which the computations are implemented at the dynamics level. In this paper, we encapsulate computation and noise robustness at the dynamics level, and introduce a new approach for robust, reconfigurable computation. We show how by using coupled chaotic dynamics for computation we can make chaos computing systems more robust to local noise by exploiting the inherent noise reduction properties in coupled systems.
I. INTRODUCTION
A dynamical system can perform a computation by implementing functions with its orbits. The dynamical system maps an initial condition, which represents the input, to future states, which represent the outputs. Future states of a chaotic dynamical system are highly sensitive to the initial conditions.1 A very small change in the initial condition can change the future states of the chaotic system, and therefore the type of computations that the chaotic system can implement. This is the main idea behind chaos computing.2 In many chaos computing methods that have been introduced so far, a single chaotic system is initialized based on the inputs, and after evolution of the chaotic map for a period of time, the output of the computation is decoded from the final state.2 A generic schematic of chaos computing paradigm is depicted in Fig. 1. Specifically consider the m binary data inputs, d1,d2,…,dm and the n digital control inputs, c1,c2,…,cn, which are mapped by an encoding map E, to a point on the unstable manifold of the chaotic system. If L is a binary set {0,1}, then LðnþmÞ represents the domain of the encoding map E, which consists of all the possible combinations of digital data and control inputs. Let b be the unstable manifold of the chaotic system, Rs the general state space of the chaotic 1054-1500/2014/24(4)/043110/7/$30.00
system, and Y the output of E on the unstable manifold. In this case, the general form of the encoding map is as follows: E : LðnþmÞ ! b;
b Rs ; L ¼ f0; 1g;
x0 ¼ Eðd1 ; d2 ; :::; dm ; c1 ; c2 ; :::; cn Þ:
(1a) (1b)
Then starting from the initial conditions produced by the encoding map, the chaotic map evolves for a fixed time (or for a fixed iteration number, if the chaotic system is discrete). After this evolution time, the final state of the map is decoded to the outputs using a decoding map. The decoding of the binary outputs from the final state can be as simple as a threshold mechanism.2 An example will be presented and explained in Sec. III. The encoding map maps different sets of the inputs to different initial conditions on the unstable manifold of the chaotic system. The chaotic orbits and therefore the final state values and the decoded binary outputs will be very sensitive to the inputs. As a result, a single one-bit change in the inputs can change the outputs. Thus control inputs can be used to select different digital functions. A digital function maps binary inputs to binary outputs. To evaluate which digital function is selected with a particular control input, the association of a control input with a logic function is noted and then all possible combinations of data inputs are enumerated to construct the truth table of the function. A truth table of a digital function is a table that shows what outputs are produced when different data inputs are given to the function. By changing the control input and repeating this procedure (of constructing the truth table), a second digital function different (with high probability) from the first one is observed. This is the meaning of the reconfigurability of chaos computing. By using all possible control inputs and finding the type of function that the chaotic system
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FIG. 1. Block diagram of chaos computing. C 2014 AIP Publishing LLC V
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implements, the full instruction set of the chaotic system is obtained.2 Variations of this approach have been central to various theoretical, applied, and commercial versions of chaotic computing applications.3,4 But is chaotic computing robust to noise? Does sensitivity to initial conditions, which makes the system flexible in computation, make the computing system unreliable and overly sensitive to noise? Chaotic computing is robust to noise if the chaotic system evolves long enough so the control inputs change the outputs (via changing the orbits) but not long enough so the noise changes the outputs. There is a fine line between iterating long enough to get different functions, and iterating too long to allow the noise to randomly change the final states. We have previously investigated5 this issue and have examined the robustness levels of chaos computing. However, one of the main difficulties with chaos computing remains making it more robust to noise. While there has been significant work on exploiting noise to make computation based on dynamical systems more robust,6 there is also a need to understand how to reduce rather than exploit the overall noise sensitivity of chaotic computation architectures. Thus if a way were possible to incorporate noise reduction into the design of the dynamical system(s) that actually perform the computation then one would, in principle, have an additional tool to build the most robust and configurable chaotic computing architectures. This will become particularly impactful for constrained architectures, such as submicron VLSI,7 where the noise is problematic. Scaling the transistors has been the main method to increase the performance of the digital computing systems.8 With the continuous scaling of CMOS digital technologies, signal integrity and noise issues have become a main concern and limiting factor for deep submicron digital integrated circuits.7 As a result, semiconductor chip makers are conducting research on radical, but extremely expensive and complicated device architectures, such as FinFETS, to address the issue of noise in computing systems.9 In this paper, we encapsulate computation and noise robustness at the dynamics level, and introduce an alternative approach for robust, reconfigurable computation. In particular, we demonstrate that by using coupled chaotic systems for computation, we can make chaos computing systems more robust to noise by exploiting the inherent noise reduction properties in coupled systems to exceed the performance of a single isolated chaotic computing system. Coupled dynamics reduces the noise content of the coupled systems and can inherently function as an averaging filter for itself. The incoherent, independent noise terms added to different nodes of the coupled map lattice (CML) are diffused and averaged by the coupled dynamics, whereas the coherent input signals to the nodes remain preserved under the coupled dynamics. We demonstrate how a CML can be utilized to implement computation and how the application of coupled redundant maps increases the noise robustness of computations. Notice that an ensemble averaging of N independent chaotic maps can result in a similar level of noise robustness that we can obtain by dynamically coupling N maps together.
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But the main purpose of this paper is to show that as we utilize dynamics to implement computation, we can also utilize coupled dynamics to implement an averaging filter in addition to the implemented computation to improve the noise robustness of the computing system. Here, we are not explicitly designing and engineering an averaging filter, rather we demonstrate that the coupled dynamics itself can function as an averaging filter, and therefore it can be used as a physical realization of the averaging filter. This results in a fully dynamics based approach for computation, where dynamics and coupled dynamics are used for computation and improving noise robustness of the computation, respectively. Utilizing redundancy has been one of the main techniques to reduce the probability of error in a system and it dates back to the first days of the modern computers. John von Neumann discusses that10 “instead of running the incoming data into a single machine, the same information is simultaneously fed into a number of identical machines, and the result that comes out of a majority of these machines is assumed to be true.” However, when the error is caused by noise, averaging the noisy states of the redundant systems can reduce and control the errors better than using a majority wins technique among the redundant systems. In this paper, we use coupled dynamics as a technique to organize the redundant systems, and to implement a dynamics level and dynamics based averaging filter. Therefore, it reduces noise better than using the majority wins technique. But if the error is caused by catastrophic failure of an element, averaging would not work and majority wins technique is the effective method to recover the system from such failures. The organization of the paper is as follows: in Sec. II, we study how coupled dynamics can reduce the noise content of the coupled lattice. In Sec. III, we introduce coupled map lattice computing and investigate noise robustness of the resulting system. In Sec. IV, we study noise robustness of a globally coupled map (GCM) lattice based computing of size N. The paper is concluded in Sec. V. II. NOISE ROBUSTNESS THROUGH DYNAMICALLY COUPLING IDENTICAL SYSTEMS
The Kaneko coupling scheme,11 which we use in our study as a model of coupled dynamics, has the recurrence relation e ¼ ð1 eÞf ðxij Þ þ ½f ðxij1 Þ þ f ðxijþ1 Þ; xiþ1 j 2
(2)
where xj represents the state of the jth node, i is the iteration number, f is a one-dimensional, discrete map, e is the coupling parameter, and e 2 ½0; 1. For simplicity, we study a CML of size 3, which means j ¼ 1,2,3, and also assume the first and the last maps of the lattice, j ¼ 1 and j ¼ 3, are connected to each other. All maps in the CML receive the same initial condition x0. But the maps are subjected to local noise. This means that the noise terms that are added to different maps of the CML are statistically independent from each other. However, the noise terms added to different maps of the CML are all identically distributed. More specifically, the
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initial conditions of the maps in the CML are set to x0j ¼ x0 þ rd0j , j ¼ 1,2,3, where rd0j is the additive noise term at step 0 to map number j. The additive noise is added to each map at each future iteration like ¼ f ðxij Þ þ rdiþ1 xiþ1 j ; j
(3)
Nð0; 1Þ and d0j Nð0; 1Þ are normal Gaussian where diþ1 j noise with zero mean and unit variance. The variance r2 of the additive noise to any map is the same. Equation (3) presents the noise effects on a single node. In a coupled map lattice, f will be replaced by the coupled map dynamics of Eq. (2). The future noisy state of a map in the CML will be x1j
e ¼ ð1 þ ½f ðx0j1 Þ þ f ðx0jþ1 Þ þ rd1j 2 e ¼ ð1 eÞf ðx0 þ rd0j Þ þ ½f ðx0 þ rd0j1 Þ 2 þ f ðx0 þ rd0jþ1 Þ þ rd1j : eÞf ðx0j Þ
(4)
After linearizing the function f around initial condition x0, we obtain e 0 e 0 0 1 0Þ ð ð Þ xj ¼ f x þ rk1 1 e dj þ dj1 þ djþ1 þ rd1j ; (5) 2 2 where k1 ¼ df =dx is evaluated at x0. The first term of Eq. (5) is the noise free evolution of the map starting from the noise free initial condition, x0; the second term, which we are interested in, is evolution of previous noise terms under the coupled dynamics, rk1 ½ð1 eÞd0j þ 2e d0j1 þ 2e d0jþ1 ; and the last term is the noise term that is added to the map at the current iteration, rd1j . The noise terms added to different maps are independent and identically distributed. Therefore by using the law of summation of independent random variables, the variance of the second term, which is the one-step noise variance, becomes e2 e2 2 r21 ¼ r2 k21 ð1 eÞ r2d þ r2d þ r2d 4 4 2 e 2 2 : ¼ ðrk1 Þ ð1 eÞ þ 2
(6)
The one-step noise variance in a single isolated map can be easily obtained by setting e ¼ 0 in Eq. (6) to get r21 ¼ ðrk1 Þ2 :
(7)
By comparing Eqs. (6) and (7), we observe that the one-step variance of noise deviations in the coupled map lattice is rescaled by r ¼ (1 e)2 þ e2/2, where this factor is less than 1, indicating that the coupled dynamics reduces noise deviations. In Fig. 2, this rescaling factor is plotted for different possible values of coupling parameters. By taking the first derivative of this rescaling factor and setting it to zero, dr/dt ¼ 0, we find e ¼ 2=3 as the optimal coupling parameter, and by setting e ¼ 2=3 in Eq. (6), the one-step noise variance becomes
FIG. 2. The rescaling factor r versus different values of coupling parameter e.
1 2 r21 ¼ ðrk1 Þ : 3
(8)
Similarly, the further iterations of the CML can be linearized around the noise free evolution, and after applying the law of summation of independent random variables on the independent noise terms from different nodes, eventually we can estimate the variance of noise deviation in the coupledmap lattice after m iterations as 1 2 2 r2m ¼ r2 ½ðk1 k2 :::km1 km Þ þ ðk2 :::km1 km Þ þ ::: þ k2m ; (9) 3 where kk ¼ df/dx is evaluated at f m1(x0), and f m1 means m 1 iterations of function f, whereas the variance of noise deviations in the single map after m iterations is r2m ¼ r2 ½ðk1 k2 :::km1 km Þ2 þ ðk2 :::km1 km Þ2 þ ::: þ k2m : (10) We observe that the coupled dynamics of the identical maps reduces the variance of noise deviation to 1/3 of the variance of noise deviation in a single map when they are subjected to the same amount of noise. In a GCM lattice of size N, all nodes are connected to each other as12 e X i i ð Þ xiþ1 ¼ 1 e f ðx Þ þ f xk : (11) j j N 1 k6¼j By linearizing GCM lattice dynamics along the noise free orbit, it is easy to show that the coupled dynamics shrinks the variance of the noise deviation. The dynamic coupling e ¼ (N 1)/N converts the GCM into an inherent averaging filter and provides the maximum noise reduction, where the variance of noise deviation is reduced to 1/N of the variance of noise deviation in a single map. For theoretical simplicity and computational speed, we have focused this paper on coupled map lattices and Kaneko model, but the phenomenon of coupling reducing noise is more general and applies to many other kinds of coupling and other topologies for coupling. As an example, we have observed that the global coupling is not necessary to reduce noise effects. Rather a sparser coupled lattice, such as a CML with just local connectivity can reduce noise effects almost as much as a GCM even for large N values. Even though in a locally coupled CML not all nodes are coupled together, different noises from different nodes can still
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FIG. 3. Block diagram of a CML based chaos computing.
diffuse through the local coupling and average each other’s effects. Homogeneous, global coupling gives the highest amount of noise reduction in the lattice. However, other coupling topologies can result in noise reduction levels that are not very different from the amount one can get from this best case, the global coupling. Further details and discussions about noise reduction in locally coupled maps can be found in Ref. 13. III. ROBUST TO NOISE CML BASED CHAOS COMPUTING
The schematic of our proposed CML based chaos computing approach to improve robustness to noise is depicted in Fig. 3 for a CML of size 3. Circles indexed by 1, 2, and 3 represent three maps, and the solid lines connecting these circles together represent the coupling. The encoding map maps the data and controls inputs to an initial condition x0 and initializes all three maps to this value. After n iterations of the coupled dynamics, the output of the computation is decoded from the final state of one of the maps. Notice that it does not matter which node we choose to decode the output from. All three maps are identical (except for noise realizations) and initialized to the same value. Therefore, noise free evolution of these nodes under the coupled dynamics results in the same output no matter from which node we select to read the output. Under noisy condition, these maps might result in different output values, however, the error rate, the probability of error, is the same for all nodes. Despite the fact that a single map can be enough to implement the desired digital functions,2 here we use redundant maps coupled together based on Eq. (2) to implement the same digital functions, but now with a higher level of
noise tolerance. This higher noise tolerance will be achieved at the expense of redundancy of the maps. The redundant maps receive the same, coherent inputs, and the maps perform the computation in the presence of incoherent, local noise. As demonstrated in Sec. II, the coherent inputs are preserved by the coupled dynamics, whereas the incoherent local noise is averaged by the coupled dynamics. Therefore, these dynamically coupled redundant maps can perform computations at noise levels that a single map cannot perform at the required error rate. As a numerical example, consider the one-humped map xiþ1 ¼ 1 – 2ðxi Þ2
(12)
on the interval [ 1, 1] which nests like f(2)[x] ¼ f[f[x]], and so on, as in Fig. 4. Assume the aim is to implement different two-input, one-output digital functions. Additionally assume there are six binary control inputs to program the CML to implement the different types of the function. The encoding map can be as simple as a digital-to-analog converter. Include the linear scaling s½y ¼ 2y 1
(13)
from the unit interval [0,1] with the binary encoding 7 2 d1 þ26 d2 þ25 c1 þ24 c2 þ23 c3 þ22 c4 þ21 c5 þc6 ; Ec ½d ¼ s 28 (14) where the two high order bits encode the data d ¼ {dn} and the six low order bits encode the controls c ¼ {cn}. Using the binary decoding ( 0; y 0 (15) D½y ¼ 1; y > 0; and six iterations of the CML to find the output Dc ½d ¼ D½f ð6Þ ½Ec ½d
(16)
as summarized by Table I and Fig. 5. Notice that different controls can build the same function. Therefore, in Table I, in some rows there are multiple controls for a same function. However, these different realizations of the same function
FIG. 4. Iterates of the quadratic map increasingly oscillate causing chaos.
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TABLE I. Truth tables and instruction set for the Eq. (16) chaotic map. Output Dc ½d1 ; d2
N
Dc[0,0]
Dc[0,1]
Dc[1,0]
Dc[1,1]
Controls c1 c2 c3 c4 c5 c6 25c1 þ 24c2þ23c3þ 22c4þ21c5þc6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1
0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
(15), 28, 36, 49 25, (26), 27, 50, 61, 62 48, (58), 59 (35) 5, (6), 16 (17), 41, 42, 54 9, (55) (7), 8, 18, 19, 33, 34, 43, 44 2, 3, 14, 37, (38), 39 1, 13, 24, 40, (51), 63 10, 22, 23, (47) 11, (12), 60 (29) (4), 52, 53 20, 21, 30, 31, 45, 46, 56, (57) (32)
noise under the chaotic dynamics needs to be kept far from this threshold value. If the noise cloud crosses the threshold that means that there is a possibility that noise can change the output of the computation, which results in an error. The probability of this error in chaos computing can be determined by calculating the cumulative probability of the portion of the noise cloud that has passed the threshold value in Eq. (15). The theoretical models that we developed in Sec. II estimate that the variance of noise deviations, in a CML of size 3 is three times smaller than the variance of noise deviations in a single map. This suggests that a CML based chaos computing system can tolerate up to three times higher noise levels than a single map based chaos computing system can tolerate. This subject can be easily verified by examining Eq. (9). The three-times increase in additive noise variance can be compensated by the rescale factor 1/3 in Eq. (9). More precisely, if we assume r2Cm ¼ r2Sm ;
(17)
h i 1 2 2 rC ðk1 k2 :::km1 km Þ þ ::: þ k2m 3 h i 2 ¼ r2S ðk1 k2 :::km1 km Þ þ ::: þ k2m ;
(18)
r2C ¼ 3r2S ;
(19)
then we get might, and usually will, have different levels of robustness to noise. Please see Fig. 5. In this research, we pick up the most robust to noise realization. Monte Carlo simulation can be used the estimate noise robustness of these different realizations. In Table I, for each function, the more robust control input is distinguished by parenthesis. Now imagine the CML is subjected to additive noise. We can consider the additive noise around the initial condition of a map as a noise “cloud.” Assuming the mean value of noise is zero, the center of this cloud would be the noise free initial condition, and the size of the cloud would be proportional to the variance of the noise. We cannot be sure where exactly the noisy initial condition is; it can be anywhere in this cloud. But we can define a probability distribution, which gives us the probability that the noisy initial condition can be at each region of the cloud. After each iteration of the coupled map, this cloud of noise is rescaled and reshaped based on the eigenvectors and eigenvalues of the map. We know that in a chaotic system, this kind of uncertainty grows larger under the chaotic dynamics. Even though at some specific locations of the attractor, the eigenvalues can be less than 1, but still we can say that on average, the cloud size grows bigger over time because the chaotic map has a positive Lyapunov exponent. In one implementation of chaos computing, the output of computation is determined from the final state of the chaotic system using a simple threshold mechanism (Eq. (16)). An error in chaos computing happens when noise moves an orbit and its final state passes the threshold value. Therefore, the evolved cloud of
where r2Cm is the variance of noise deviations in a CML of size 3 after m iterations, r2Sm is the variance of noise deviations in a single map after m iterations, rC is the variance of the additive noise to the CML, and rS is the variance of additive noise to the single map. This means that the CML can be subjected to three times higher noise level than the noise level in a single map, and the variance of noise deviations for both CML and the single map would be the same. We define relative noise tolerance s ¼ r2C =r2S ;
(20)
where r2s is the maximum variance of additive noise that a single-map based chaos computing system can tolerate without exceeding a specified error rate, and r2C is the maximum variance of additive noise that an improved, more robust to noise CML based chaos computing, can tolerate without exceeding the same specified error rate. Therefore, based on Eq. (19), relative noise tolerance for a CML based chaos computing of size 3 is s ¼ 3. To investigate the accuracy of these theoretical estimates, we set up the following simulations. First, we numerically calculate the relative noise tolerance of a CML based
FIG. 5. Black lines superimposed on the 5th iterate of the quadratic map indicate the digital functions for two binary inputs and three binary controls. Slight changes in initial conditions flip the outputs from 1 (blue) to zero (red), especially at the interval’s ends.
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chaos computing, where CML is composed of three maps of Eq. (12), coupled together based on Eq. (2). For illustrative purposes, we assume that the computing system is specified to have an error rate of no more than 0.4%, four errors in each 103 operations. Notice that this choice of error rate does not change the results and conclusions. We have tried other error rates and we obtained the same results. We run a Monte Carlo simulation to numerically calculate the maximum variance of additive noise to a CML of size 3 and to a single map, both performing chaos computing, without exceeding the given error rate. We then calculate the ratio of these variances to obtain the relative noise tolerance. More specifically, we gradually increase the variance of additive noise, and for each variance value, we perform the chaoscomputing algorithm in the noisy environment 103 times. By observing the fifth error, we record the previous value of the noise variance value, which meets the specified error rate. This Monte Carlo simulation yields approximate results. To reduce the error of simulation and to estimate the size of simulation errors, we repeat this Monte Carlo simulation 25 times. The average value of these 25 relative noise tolerance values for each function is depicted in Fig. 6 by green circles. Notice that each digital function has a different level of noise robustness. This is why there are separate results for each function. The error bars represent a measure of error in Monte Carlo simulation results and are defined as the variance of 25 observed values for each function. We observe that the average value of relative noise tolerance s for each function is very close to 3, which is what the theory predicts. Notice that some functions, such as functions 7, 10, and 11, have a slightly lower noise tolerance than we expected. The reason is that the averaging filter is a first order approximation for the coupled dynamics. The effects of the second order terms that we drop during linearization manifest themselves as deviations from the perfect result that we expect, here three. Sometimes the actual result is greater than three and sometimes less than three. Four data inputs to each function are encoded as four different initial conditions. To calculate Fig. 7, we calculate the relative noise robustness for different orbits starting from these initial conditions. Some of these four relative noise robustness values would be slightly greater than three and some less than three. We take the lowest number as the relative noise robustness for that function. As a result, this technique tends to take the worst values, which are normally less than three. In some
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FIG. 7. Relative noise tolerance for GCM based chaos computing of size N.
functions, such as 7, 10, and 11, the deviations from the expected relative noise tolerance are higher due to their greater second order effects. IV. GLOBALLY COUPLED MAP LATTICE BASED CHAOS COMPUTING
Based on discussions in Sec. II, the variance of noise deviations in a GCM lattice of size N where e ¼ (N 1)/N, is 1/N of variance of noise deviations in a single map, when they both are subjected to the same amount of noise. As a result, a GCM based chaos computing of size N can tolerate up to N times more noise level than a single-map based chaos computing can tolerate. Therefore, the relative noise tolerance for a GCM based chaos computing of size N with e ¼ (N 1)/N is s ¼ N. Next, we numerically investigate the effect of lattice size, N, on relative noise tolerance, s, of a GCM based chaos computing. The GCM lattice is composed of N maps of Eq. (12) and they are coupled based on Eq. (11). We assume that the computing system is specified to have an error rate of no more than 0.4%, four errors in each 103 operations. The results from Monte Carlo simulation are presented in Fig. 7. The horizontal axis is the lattice size, N, and the vertical axis is the average relative noise tolerance of 16 different functions. In the Monte Carlo simulation, we numerically calculate the relative noise tolerance for each function, and then we average these values and plot these average values for each lattice size in Fig. 7. Notice that the Monte Carlo simulation produces approximate results. However, because we use the average values obtained for 16 different functions, the amount of error in results reported in Fig. 7 is reduced. The error bars represent the variance of the relative noise tolerance values for 16 different functions. We observe that in a GCM based chaos computing of size N, noise tolerance is s ¼ N. V. CONCLUSION
FIG. 6. Relative noise tolerance s for a size-3 CML based chaos computing.
In this paper, we have introduced CML based chaos computing approach, and we demonstrated that it is more robust to noise than a single-map based chaos computing approach. All the maps in the coupled map lattice receive the same inputs, therefore these coherent inputs remain unchanged under the coupled dynamics. But incoherent, local noise is averaged by the coupled dynamics. As a result, the CML based chaos computing is less sensitive to noise.
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This enables us to perform chaos computing at noise levels where a single map cannot reliably perform the same computation. Coupling dynamical systems, such as electronic circuits, cost very little or no extra structural complexity and computational complexity. We have discovered that a very straightforward coupling mechanism for coupling electronic circuits is sufficient for noise reduction purposes, and we have shown that these coupled electronic circuits are more robust to local noise than single, independent circuits. Results will be published in a future paper. We have shown that the coupled dynamics can be considered as a dynamics based realization of an averaging filter and by utilizing the inherent noise reduction properties in the coupled dynamics, we achieve noise reduction similar to that achieved by applying an averaging filter. This provides us with a unified, dynamics-based approach for computation, where both computation and robustness (to noise) are realized and implemented using intrinsic dynamics of the systems, with no need to explicitly design and engineer computation or noise filtering. ACKNOWLEDGMENTS
We gratefully acknowledge support from the Office of Naval Research under Grant No. N000141-21-0026 and STTR Grant No. N00014-14-C-0033.
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