Abstract
The increasing demands on information processing require novel computational concepts and true parallelism. Nevertheless, hardware realizations of unconventional computing approaches never exceeded a marginal existence. While the application of optics in supercomputing receives reawakened interest, new concepts, partly neuroinspired, are being considered and developed. Here we experimentally demonstrate the potential of a simple photonic architecture to process information at unprecedented data rates, implementing a learningbased approach. A semiconductor laser subject to delayed selffeedback and optical data injection is employed to solve computationally hard tasks. We demonstrate simultaneous spoken digit and speaker recognition and chaotic timeseries prediction at data rates beyond 1 Gbyte/s. We identify all digits with very low classification errors and perform chaotic timeseries prediction with 10% error. Our approach bridges the areas of photonic information processing, cognitive and information science.
Introduction
The past century has seen an unprecedented increase in information processing abilities. However, certain tasks are not efficiently solved by standard computers. Unconventional computation schemes and the inclusion of photonic hardware could provide new opportunities^{1,2,3,4,5}. Photonics might extend and complement the repertoire of techniques for certain information processing tasks, with far reaching consequences^{3}. Realizing learningbased computation by utilizing the nonlinear transients of standard photonic components could lead to a paradigm shift in the field of optical computing, away from binarylogic based methods, towards machine learningbased approaches using offtheshelf, fast and power efficient devices. In particular, semiconductor lasers exhibit nonlinear interaction between the laser field and semiconductor medium, resulting in complex behaviour when subjected to feedback, electrical modulation or optical injection^{6}. They are offtheshelf, high bandwidth, power efficient components, and form the backbone of modern fibre communication.
In this article, we implement the computational concept of reservoir computing (RC)^{4} in photonics, in particular realizing the reservoir and information injection alloptically. At the same time, we boost the bandwidth compared with previous approaches by more than two orders of magnitude. We achieve this by utilizing the analogue transient dynamics generated by a semiconductor laser coupled to a fibreoptic feedback loop. Following the RC concept, we generate nonlinear transient states in the context of previous input responses. Consequently, the system is capable of processing temporal sequences of information.
Results
Computational concept
Our computational concept is schematically illustrated in Fig. 1a (ref. 7). Following ideas introduced in RC, we generate transient states for processing the information. These transient states are generated by a central nonlinear element, here a semiconductor laser, which is subjected to optical feedback with delay τ_{D}, and to injection of the input information. The input information is sampled in time, resulting in a discrete sequence a_{n}, n={0,1,…}. In the case of realtime processing, the sampling rate should be chosen as T_{2}^{−1}=τ_{D}^{−1}. For successful information processing, a significant number of different nonlinear transients is required. In the case of a scalar input, we multiply a_{n} by a temporal mask sequence M(t), the duration of which is equal to the sampling time T_{2}. Each computational cycle T_{2} is divided into N subintervals of duration T_{1}=T_{2}/N. With this procedure, we obtain N different, timemultiplexed transient states x_{m}(t), with m=1…N and N typically chosen as 100≤N≤1,000. The resulting sequence injected into the laser during one delay time is therefore given by u_{n}(t)=a_{n}·M(t). The induced nonlinear transient states have to contain a maximum amount of information about the injected input^{7}. Therefore, we keep the system sufficiently far from reaching a steady state during its dynamical response. We achieve this by choosing T_{1}=Θ·T_{0}, with Θ=0.2, where T_{0} is the characteristic time scale of the laser’s relaxation oscillations (ROs). In the case of multiple component input a_{n}, the information injected each T_{1} consists of a random, linear combination of input components.
The final step in our computational process is based on the summation of these induced transient states x_{m}(t). This is done by calculating a linear combination of the laser output intensities, detected at a rate of T_{1}^{−1}. The coefficients of this linear combination, called weights, are determined by a standard machine learning training procedure, in which the difference between the resulting readout value and its target is minimized. Owing to the delay loop, the induced transients are the system’s response to the currently injected information in the context of the response to previous inputs. Theoretical^{8,9} and experimental^{7,10,11,12} investigations show that our scheme can indeed be interpreted as RC.
For successful and reproducible computation, the system has to be capable of generating consistent transients, that is, for identical or highly alike inputs the induced transients have to be equally alike. This requirement corresponds to the approximation property in machine learning^{4,7}. Therefore, the rest state (reservoir state in the absence of input) and the dynamical properties have to be chosen appropriately. Hence, computational performance benefits from a reservoir rest state corresponding to steady state emission of the laser. In the experiment, this can be achieved by, for example, biasing the laser close to the threshold current of the free running laser.
Our scheme combines photonics with the true parallel computation properties of RC. First, the transients are induced by injecting information from multiple sources simultaneously into the laser. Second, as the transient states are not affected by the training, in principle, an arbitrary amount of k scalar outputs can be defined and trained, performing different computations. Parallel data injection can be realized using a single optical fibre without crosstalk between the individual input channels. Although, at this stage, data preprocessing and readout are carried out offline, hardware implementations for a realtime preprocessing are being discussed^{2}. Additional information about data injection and data readout can be found in the Methods section.
Experimental scheme
A schematic representation of the experimental setup is given in Fig. 1b. The emission of the semiconductor laser diode (λ=1,542 nm) is collected by a standard single mode fibre. Feedback (with delay τ_{D}=77.6 ns) is realized via a fibre loop, comprising an optical circulator, two fibre splitters, an optical attenuator and a polarization controller. The optical attenuator and polarization controller facilitate the control of the optical feedback conditions. Using T_{2}=77.6 ns and N=388, we obtain T_{1}=200 ps.
In our experiment, information can either be injected electrically, via modulating the laser diode current, or optically by injecting the modulated intensity of a tunable laser. Diode current modulation was facilitated by directly connecting the diode to a radio frequency cable using a 40 Ω resistor. Combined with the 10 Ω impedance of the laser diode, the system therefore was impedance matched to the radio frequency electronics used for electrical modulation. Following this approach, we achieved a diode current modulation bandwidth exceeding 10 GHz. For electrical modulation, the laser’s RO frequency acts as bandwidth limiting factor. In our laser, the RO frequency changed from 1.4 to 5 GHz when changing the bias current from I_{b}=9 mA to I_{b}=20 mA. The optical signal was injected via the optical fibres forming the delayed feedback loop. In principle, efficient nonlinear interactions of optical signals in semiconductor lasers can be achieved for frequencies up to several hundreds of gigahertz^{13}. So far, however, the signal source used for generating the input information remains the bandwidth limiting factor in our experiments.
We evaluated the performance of our system focusing on two different, general classes of information processing tasks. Various informationprocessing problems require classification of information, associating different inputs to different classes. As such, these tasks equally require discrete classes as classifier targets, and a system response sufficiently diverse to allow for clear separation. A second typical task in information processing is based on nonlinear processing of dynamical information. Accordingly, the classifier target values can be continuous. Furthermore, the system has to provide memory to capture the dynamical nature of the injected information. We selected two computationally challenging tasks, each representing one of the two introduced classes of information processing tasks, as benchmark tests to evaluate the information processing capability of our scheme: parallel spoken digit/speaker recognition and chaotic timeseries prediction. Both tasks are standard to the machine learning community and allow for a direct comparison between different approaches to information processing^{7,10,11,14}.
Spoken digit recognition
In Fig. 2a we show the experimental results of spoken digit classification in terms of the classification error as a function of the laser bias current (I_{b}). Results of optical (electrical) information injection are given by the blue (red) data set. Optical information was injected using a tunable laser, the polarization of which was parallel to the laser emission, with the optical injection power modulated between 15 nW ≤u^{o}(t)≤15 μW. In the absence of information, that is, the rest state, the injected power corresponded to 15 nW. Electrical information was injected by modulating the laser current between 0 mA ≤u^{e}(t)≤12 mA. The optical feedback was attenuated by 20 dB and rotated in polarization. Under these conditions, we achieved a very low classification error of (0.014+0.051/−0.014)% for I_{b}=7.7 mA ((0.64±0.17)%, I_{b}=7.6 mA) for optical (electrical) injection at a laser bias current close to the laser threshold. A classification error of 0.014% corresponds to one misclassification per ∼7,000 digits, with an uncertainty that was limited by the size of the database. Our results present an improvement in classification error by a factor of ∼3 and in speed by a factor 260 when compared with previous, optoelectronic implementations of this information processing scheme^{10,11}.
We speculate that the improved error, as compared with earlier experiments^{7,10,11,14}, is related to the influence of the nonlinearity on the classification performance. Such an influence has been empirically identified in Larger et al.^{10} A thorough understanding of the interplay of nonlinearity and specific task performances is, however, still lacking. Even when compared with software emulations of neural networks, our performance is highly competitive. There, for the spoken digit recognition without the influence of noise, classification errors reported for a system based on hidden Markovian models were 0.168% (ref. 15), and 0.5% for a reservoir computing system^{16}. Further above threshold, our system exhibits complex dynamics even in the absence of injected information^{6}, yielding a worse performance for the case of electrical modulation (Fig. 2a). For the case of optical information injection the performance dependence above threshold is less intuitive. This might be due to the complex influence of the injected light on the stability of the rest state^{17}.
As an illustration of the parallel computation capabilities we utilized the same transient responses to identify the speaker of a digit. The results are shown in Fig. 2b. A (0.88±0.18)% speaker classification error was obtained for a bias current of I_{b}=7.91 mA. At this point it is worth emphasizing the high speed of our setup, which allows us to process about one word in 3.3 μs, corresponding to ∼300,000 spoken digits per second. Our results represent the lowest spoken digit classification error as well as the highest speed reported in the literature.
Chaotic timeseries prediction
The timeseries prediction task requires a single point prediction of an experimentally obtained chaotic time series, which was also used in the Santa Fe competition^{18}. The performance, using optical data injection, is depicted in Fig. 3. For timeseries prediction, the classifier target value is a nonlinear transformation of several, continuously distributed, previous data points, and therefore is (quasi) continuous. Consequently, noise has a direct impact on the accuracy of the classifier. In order to reduce noise, the range of optical modulation was the same as before, however, an injection power of 7.5 μW was chosen, defining the rest state. The external laser therefore additionally served as an injection locking source, increasing the performance significantly by reducing the rest state noise. In our experiment, we performed a onetime step prediction test, that is, predicting a datapoint onetime step ahead in the future. It is worth mentioning that we processed one datapoint per delay time, and consequently no explicit memory was added at the data injection stage, a preprocessing technique commonly used to artificially introduce memory in machine learning techniques^{19}. The quasicontinuous nature of our classifier is caused by the analoguedigital conversion in our experiment, which, in possible future applications, could be avoided.
The prediction performance, depending on the bias current, is depicted in Fig. 3a. The best performance is again obtained for I_{b} close to the laser threshold, with a prediction error of 10.6% (I_{b}=7.62 mA, feedback attenuation 10 dB) at a prediction rate of 1.3 × 10^{7} points per second. When repeating the same input five times and averaging the responses (keeping the 8 bit resolution), the signaltonoise ratio was improved, and an error of only 5.5% was obtained. The signal averaging was executed directly at the acquisition stage. Experimentally, this procedure could be implemented by running five systems in parallel or by making a tradeoff between speed and performance. Timeseries prediction using machine learning techniques and realized in software, achieved errors below 1% (refs 20,21)^{20,21}. These approaches, however, neglect the influence of noise and finite experimental precision and, more importantly, add external memory in the data injection procedure. Although these implementations, for now, offer superior accuracy, hardware systems could offer advantages in speed, integrability and energy efficiency. These points will be discussed later in the manuscript.
The importance of the memory introduced by the feedback loop can be seen in Fig. 3b. Owing to experimental constraints, the lowest feedback attenuation that we achieved was 10 dB, for which the best prediction performance was obtained. An increase by only 2 dB resulted in a prediction error exceeding 40%. We would like to emphasize that the data point with the lowest error in Fig. 3b does not stem from an individual measurement, but was confirmed by several independent measurements. The s.d. (±1.9%) of these measurements is given by the red error bars in Fig. 3. Additional deviations arise from the crossvalidation procedure. The entire training and testing procedure was carried out five times, using each data point for training and testing according to a random selection. Prediction results showed small deviations (∼0.3%) for different partitions. This error is displayed by the blue error bars in Fig. 3.
The direct influence of I_{b} on the transient responses can be seen in Fig. 4. Panel (a) shows a part of the Santa Fe time series used in the experiment, illustrating the sudden changes in amplitude, which are characteristic to this data set. Timeseries prediction in the vicinity of these points is deemed particularly hard. Transients, induced by data containing such a rapid amplitude change, are displayed for three values of I_{b} in panel (b). Although for I_{b}=7.62 mA the dynamics is largely dominated by the system’s response to the injected information, this correlation between input and transient response reduces for larger I_{b}. For a bias current of I_{b}=10.78 mA, the characteristic amplitude change within the injected data is no longer vivid in the induced transients. The high quality of our timeseries prediction is illustrated in Fig. 4c, where we compare the target (black) with the predicted (red) time series. The predicted time trace includes data points in the vicinity to a rapid amplitude transition. Here, top and bottom horizontal axes display the time step number in the original target trace, and the temporal duration of the actual experiment, respectively.
Discussion
Apart from the demonstrated highly competitive figures of merit, our novel computational concept might offer significant energy reduction for certain tasks. At this stage we can only provide a conservative estimate. The calculated energy consumption for spoken digit recognition using our alloptical transient computing scheme, including alloptical data input and readout hardware, would be of the order of 10 mJ per digit, compared with 2 J per digit required by a standard desktop computer. These numbers demonstrate the great potential of our computational scheme realized in hardware. Additional information about the energy consumption can be found in the Methods sections.
Other hardware implementations of machine learning concepts are based on field programmable gate arrays. There, the states utilized for computation are realized based on digital electronics. The speed of these implementations is measured in connections per second, with the fastest systems typically achieving rates from 1 × 10^{9} to 5 × 10^{9} connections per second for a single field programmable gate arrays^{22}. It is worth noting that this number refers to connections and not to the number of nodes of the neural network, hence a more complex network structure will inevitably decrease the global processing rates. Our scheme, with the injection rate limited by our signal source, therefore allows for identical or even superior processing speed.
In conclusion, we have experimentally demonstrated the computational power of a single photonic device using nonlinear transient laser responses. The reported results open new perspectives for tackling certain tasks with analogue optical computation. Instead of relying on complicated, power and spaceconsuming architectures, one can realize simple, yet powerful implementations of optical information processing with inherent memory and the possibility of true parallelism. Of major consequence might be the large computational power achieved with the nonlinearity of a laser diode. Similar computational schemes could in the future be implemented, benefiting from the vast variety of optical nonlinearities already available. As such, implementations could span numerous areas, including ultrafast optical nonlinearities, some even on the femtosecond time scales, as well as utilizing nanoand metamaterials.
We achieved excellent figures of merit in spoken digit recognition, with the lowest reported error rate (0.014%) at simultaneously highest data rate (1.1 Gbyte/s), and timeseries prediction with an error of only 10.6% with a prediction rate of 1.3 × 10^{7} data points per second. Especially for optical information injection, the factor ultimately limiting the speed of this scheme might not arise from the maximum bandwidth of the photonic nonlinear element^{13}, but from a decreasing signaltonoise ratio when going to higher bandwidths. Using a slightly modified injection scheme, demonstrated in the timeseries prediction task, we have shown how these effects can, to some degree, be compensated for.
We would like to point out that for all tasks, digit and speaker recognition as well as timeseries prediction, best performance was achieved for laser bias currents close to threshold. Consequently, the main parameter determining the performance does not require a fine adjustment for executing different tasks. For tasks requiring extended or multiple time scale memory, our scheme can be expanded by utilizing several delay lines of different length^{14} or by adding additional nonlinear nodes (lasers).
Our results, reported for the two different tasks, indicate the potential of our information processing schemes and open new perspectives for future applications in photonics. Spoken digit recognition is a classification task, and as such our excellent figures of merit might be transferable to technological challenges like alloptical routing^{23}. However, tackling alloptical routing with the implementation as presented would currently still be out of reach becaus of the employed time multiplexing. A possible extension includes the use of a multiple node reservoir, based on semiconductor lasers. With such extended schemes, data rates of tens of gigabit per second should be achievable. Furthermore, based on the alloptical nonlinearity of laser diodes^{13}, such systems might even reach processing speeds of hundreds of gigahertz. These numbers demonstrate the potential of RC based on offtheshelf photonic components for telecommunication applications. However, many questions still need to be investigated on the way. The fast speed and good performance in timeseries prediction equally demonstrate possible applications in ultrafast control system. Here, the increase to tens of megahertz bandwidth could offer great opportunities for fast control of photonic systems. In addition, the simplicity of our scheme could allow technological implementations in distributed networks and smart systems, enabling closed loop control of individual elements.
Methods
Information injection and readout
In our experiment, the information to be processed can be injected either electrically, via directly modulating the laser bias current (u(t)^{(e)}), or optically via modulating the optical intensity of an external injection laser using a Mach–Zehnder modulator (u(t)^{(o)}). In both cases, the modulation rate is given by T_{1}^{−1}=(Θ·T_{0})^{−1}. We chose Θ such that a high sensitivity of the induced transient states on the injected information is achieved, which we find optimized for Θ∼0.2. Larger Θ values would allow the induced transient to approach their steady state, while for too small Θ values the system’s inertia would result in a neglegible response to the injected information.
Input information for the nth input interval is described by the Ldimensional data vector a_{n}. The data vector is created via sampling the input information in discrete time steps. For realizing parallel injection of the Ldimensional input, we construct vector ũ with N dimensions according to
where is a connectivity matrix, defining connections between input dimensions and the individual transients.
A large diversity of transient states is required in our computational scheme. This is achieved by an additional multiplication of vector ũ with mask M(t). According to this, mask M defines the temporal position m·T_{1} of vector components u_{n,m} inside one clock cycle T_{2}. The mask is defined as a sequence of N values, randomly selected from discrete values. For the temporal interval, defined by ([n·T_{2}+(m−1)·T_{1}]≤t<[n·T_{2}+m·T_{1}]), the information injected into the reservoir is given by
The resulting, u(t) is the information directly injected into the nonlinear node. Following these data injection procedure, it is possible to inject L data values in parallel into the nonlinearity during one computational time step T_{2}.
For the training, standard methods from machine learning can be employed^{4}. In our computational scheme, the processed information is represented by scalar values y_{k} of the individual classifiers, indicated by k. Combined, these values (dimensions) form the output vector y. Per time step T_{2} one computation is executed, hence we normalize time according to s=t/T_{2}. The output vector is given by
where the matrices contain a linear combination of the transient states, generated by the information injected into the laser and a constant bias term, respectively. For performing a certain information processing operation on readout vector y, the linear weights are determined in a training procedure. Information with known target values is injected into the system to optimize . Consequently, the s.d. between y(s) and its target is minimized. During this procedure, referred to as training, the readout weights are adjusted in an offline procedure.
For the case of an input with s time steps, we build a transient state matrix of dimension N × s, containing all transient states at each time s. To account for a constant offset in the target value, we additionally include a constant bias, resulting in a state matrix of dimension (N+1) × s. Furthermore, we concatenate all readout weights defining matrix . For a number of k individual readouts y, the dimensionality of is k × (N+1). We define a target output matrix , given the optimal output values at each time step (that is, dimension k × s). Using these three matrices, the root mean square can be minimized according to
Here, † denotes the Moore–Penrose pseudoinverse, which allows to avoid problems with illconditioned matrices.
Overfitting of the training data is avoided by the noise inherently present in the experiment, and by the Tikhonov regularization method (ridge regression). Tikhonov regularization is included in the training procedure by minimizing
where the factor λ acts as a penalizing factor, forcing the components of towards smaller values.
Spoken digit recognition
The spoken digit data set consists of five female speakers, uttering the numbers zero to nine with a tenfold repetition for statistics^{24}. Before injecting the information into the laser, we performed standard preprocessing creating a cochleagram of each digit via the Lyon ear model^{25}. For simplicity, mask M and connectivity matrix were merged into a single matrix . In this matrix, 98% of the elements were set to zero in order to realize a sparse connectivity between cochleagram channels and transients. The remaining values were randomly selected among (0.41,0.59). The preprocessing following the digit’s chocleagram is illustrated in Fig. 5. After preprocessing, each cycle of length T_{2} data of the cochleagram’s 86 frequency channels were injected into the system with 8 bit resolution, corresponding to a rate of 1.1 Gbyte/s. One Byte here refers a unit of injected information, which can comprise different numbers of bits. Out of the entire body of spoken digits, we chose 20 random partitions of 25 samples each, using 475 samples for training the readout weights, keeping the remaining 25 for testing. Each random partition and each sample are used exactly once for testing (20fold crossvalidation). Statistical information is provided by repeating this procedure five times, creating different random combinations of testing and training. At the two data points, where the system approaches 0% error, the crossvalidation procedure was repeated 100 times.
Timeseries prediction
In the timeseries prediction task the system was injected with one data point per T_{2}. The target of the classifier was to predict the value of the respective next time step. By injecting information at a rate of 1 Byte per clock cycle T_{2} (1 Byte=8 bits), we could predict 1.3 × 10^{7} data points per second. We evaluated the performance of our scheme using the chaotic timeseries of the Santa Fe timeseries competition, data set A (ref. 18). The data set, created by a farinfraredlaser operating in a chaotic state^{26}, consists of 4,000 data points, from which we used 80% for training and 20% for testing and fivefold crossvalidation.
The input information for the timeseries prediction task consisted of scalar values only. Accordingly, input and transient states are fully connected. Timeseries prediction is very sensitive to the presence of noise. To evaluate the performance of our system, we carried out numerical simulations, that showed that for an 8bit digital to analogue converter digitization noise would already limit the performance of our system. A mask with a larger variety of values can reduce the performance limiting influence of this factor to some extent^{27}. Hence, for this task we employed a sixvalued mask, with a random selection from the values {−1, −0.6, −0.2, 0.2, 0.6, 1}.
An additional positive effect of this injection scheme is that a single value of the injected information is the origin of all N transient states during one delay. The impact of noise is therefore reduced by averaging over the large number of states, hence increasing prediction accuracy and robustness. It is worth noting that in our experiments we aim at demonstrating timeseries prediction utilizing only the memory inherently present in the system. For this reason, we inject a single data point per delay T_{2}. Injecting multiple points within one delay time would result in an artificial enlargement of the memory.
Energy efficiency estimation
The energy consumption for a system, entirely realized in hardware, was calculated using offtheshelf components. Masking of the input and realizing the scalar multiplication factors in the readout classifier can, due to the time multiplexing, be done via temporal amplitude modulation (for example, with a Mach–Zehnder modulator). The readout additionally requires integration for the duration of T_{2}. This can be realized electronically with a gated integrator^{28} or optically with an alloptical integrator^{29}. Our calculations are based on the gated integrator, for which we estimate a power consumption of 200 mW. The temporal modulation sequence for the input and readout is provided by an arbitrary waveform generator (AWG, Euvis AWG452), with a power consumption of 13 W. One AWG for each input and output were used in our estimation which gives an upper limit for the power consumption, with large potential for further reductions. We assumed the transformation by the Cochleagram to be carried out in advance. As such, a system with 10 classifiers would consume roughly ∼150 W. We included an additional 50 W for all other components like the laser diode current source and temperature stabilizer, as well as the driver of the Mach–Zehnder modulator. This number does not rely on any specifically designed electronic hardware, and hence represents a highly conservative guess.
Additional information
How to cite this article: Brunner, D. et al. Parallel photonic information processing at gigabyte per second data rates using transient states. Nat. Commun. 4:1364 doi: 10.1038/ncomms2368 (2013).
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Acknowledgements
This work was supported by MICINN (Spain), Comunitat Autònoma de les Illes Balears, FEDER, and the European Commission under Projects TEC200914101 (DeCoDicA), Grups Competitius and EC FP7 Projects PHOCUS (Grant No.240763) and NOVALIS (Grant no. 275840).
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Instituto de Física Interdisciplinar y Sistemas Complejos, IFISC (UIBCSIC), Campus Universitat de les Illes Balears, Palma de Mallorca E07122, Spain
 Daniel Brunner
 , Miguel C. Soriano
 , Claudio R. Mirasso
 & Ingo Fischer
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Contributions
All authors have contributed to development and/or implementation of the concept. D.B. and I.F. designed the experiment, D.B. performed the experiment, D.B. and M.C.S. analysed the data, I.F. and C.M. supervised the research. All authors contributed to the discussion of the results and to the writing of the manuscript.
Competing interests
The authors declare no competing financial interests.
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Correspondence to Daniel Brunner.
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