A Fast Semi-implicit Method for Anisotropic Diffusion Simple finite differencing of the anisotropic diffusion equation, where diffusion is only along a given direction, does not ensure that the numerically calculated heat fluxes are in the correct direction. This can lead to negative temperatures for the anisotropic thermal diffusion equation. In a previous paper we proposed a monotonicity-preserving explicit method which uses limiters (analogous to those used in the solution of hyperbolic equations) to interpolate the temperature gradients at cell faces. However, being explicit, this method was limited by a restrictive Courant-Friedrichs-Lewy (CFL) stability timestep. Here we propose a fast, conservative, directionally-split, semi-implicit method which is second order accurate in space, is stable for large timesteps, and is easy to implement in parallel. Although not strictly monotonicitypreserving, our method gives only small amplitude temperature oscillations at large temperature gradients, and the oscillations are damped in time. With numerical experiments we show that our semi-implicit method can achieve large speed-ups compared to the explicit method, without seriously violating the monotonicity constraint. This method can also be applied to isotropic diffusion, both on regular and distorted meshes. Anisotropic diffusion equation arises frequently in diverse applications: microscopic transport in magnetized plasmas [7] ; image processing [17] ; diffusion-tensor magnetic resonance imaging [4] ; thermal properties of crystals [8] ; transport in geological formations [5] , etc. In [22] we showed that simple finite-differencing of the anisotropic diffusion equation resulted in unphysical numerical heat fluxes, which lead to negative temperatures at large temperature gradients. Negative temperatures, in addition to being unphysical, result in an imaginary sound speed and associated numerical instabilities.. Anisotropic diffusion equation satisfies important mathematical properties such as monotonicity preservation along the direction of diffusion, so that no new extrema are created in that direction and any existing extrema are not accentuated, i.e., maxima must drop or be unchanged, and minima must increase or be unchanged (e.g. see [11, 14] and references therein). For simplicity we will refer to this as an extrema reducing property. In two or three dimensions the anisotropic diffusion equation of the form we are considering here can always be transformed to the form ∂T (α, β, t)/∂t = ∂/∂α[D(α, β)∂T /∂α], where α is a coordinate that varies along field lines and β are field-line label coordinates that are constant along a magnetic field line, so the monotonicity properties of 1-D diffusion should be preserved.. In [22] we proved that the use of slope limiters (e.g., see [12] ) to interpolate temperature gradients at cell faces guarantees that temperature extrema are not accentuated, as required physically. The extremareducing property ensures that the temperature is positive for a CFL stable timestep. Since the temperature is positive, numerical instabilities that plague simple finite differencing of anisotropic diffusion (because of an imaginary sound speed!), do not arise with the use of limiters. Because of this desirable property our method has been used in astrophysical magnetohydrodynamic (MHD) simulations where thermal conduction is anisotropic, and large temperature gradients can arise [24, 16, 20] . In addition to thermal conduction, limiters have proved useful for anisotropic viscosity with large gradients [9] , and may help in various problems with anisotropic transport; e.g., cosmic ray streaming [23] . There is another explicit method that has been applied to astrophysical problems and avoids the problem of negative temperature in presence of large temperature gradients, though it is non-conservative and somewhat more complex [19] . We also discuss some other recent work below.. A key limitation of all explicit methods is that the timestep is limited by the usual CFL condition, ∆t < ∆x 2 /2χ , where ∆x is the grid spacing and χ is the anisotropic diffusion coefficient. For some applications, this timestep constraint is rather severe and the conduction timestep can be much smaller than the MHD CFL time limit. In such cases an implicit method, where there is no stability limit on the diffusion timestep, is desirable. Although it is straightforward to difference a linear anisotropic diffusion equation implicitly, the resulting scheme is still not monotonicity-preserving. E.g., see Table 1 in [3] , which shows that temperature oscillations remain till late times even with an implicit method, just as with the explicit schemes without limiters. One can try to solve the anisotropic diffusion equation fully implicitly, using limiters to prevent temperature oscillations, but the nonlinearities in the limiters will require a careful iterative treatment (some studies have found that these kinds of nonlinear limiters make iterative solvers more difficult).. We have experimented with a Jacobian-free nonlinear iterative implicit method (a two-stage Richardson iteration extension of the LGMRES(1,1) version of Loose GMRES [2] , a variant of the Generalized Minimal RESidual method [21] with restarting) but found that it requires a fairly large number of iterations per time step, because the Jacobian matrix is not strongly diagonally dominant for large timesteps and has a large condition number.. 2 The fast method proposed here might be able to serve as an effective preconditioner to further accelerate an unsplit iterative method.. We have also experimented with an explicit method which is stable for timesteps longer than the CFL limit, and where the internal iteration time-steps are chosen based on the properties of Chebyshev polynomials [1] . We were not able to obtain a speed-up of more than ∼10 compared to the CFL-limited scheme, irrespective of resolution, for any of the parameters that we varied. Moreover, the parameters for which maximum speed-up is obtained, without becoming numerically unstable, are difficult to choose (this is true even for isotropic diffusion!).. Here we present a conservative, directionally-split, semi-implicit method which is numerically stable for any choice of timestep, and is easy to implement. The method is based on directional splitting where the heat fluxes in each direction are updated sequentially. The heat flux in each direction, e.g.,. 2), consists of two terms: −χ b 2 x ∂T /∂x, the 'normal' term where temperature gradient need not be interpolated, and −χ b x b y ∂T /∂y, the 'transverse' term which involves temperature interpolation with limiters. In our directionally-split method the 'normal' terms are treated implicitly, and the transverse terms are treated explicitly. The directional splitting of 'normal' implicit terms results in a tridiagonal matrix which can be solved very quickly. The explicit treatment of 'transverse' terms with limiters ensures that extrema are not accentuated. The resulting scheme, while not strictly monotonicity-preserving for large timesteps, results in only small amplitude temperature oscillations which are damped in time. Speed-up of order 100-1000, compared to the explicit scheme, is easily achieved for our test problems.. There has been some interesting recent work on another approach to the problem of preserving positivity in presence of anisotropic diffusion tensors (or diffusion on distorted meshes), based on expressing the flux at cell faces in terms of the advected quantity at the cell centers on either side of the face (a "two-point flux expression"), but where the coefficients of this flux depend on the transverse gradients and so is nonlinear (for example see [13, 25] ). Future work could compare the nonlinear limiters and implicit solvers that are used in our algorithm with these other algorithms on the types of test problems considered here and in [22] .. The paper is organized as follows: Section 2 presents the method in detail and shows that the scheme is linearly stable for large timesteps. Section 3 presents results from three test problems which show the practical utility of our method. We conclude and discuss applications of our method in Section 4. We present a simple, directionally-split, conservative, semi-implicit method for anisotropic diffusion which is linearly stable for large timesteps. Directional splitting results in a tridiagonal matrix equation for each direction, which can be solved exactly and efficiently. For problem on a N × N grid our scheme (Eqs. 3 & 4) requires two independent tridiagonal solves. In comparison, the fastest unsplit methods like LGMRES will require O(N ) iterations to converge! Similarly, compared to the explicit method our scheme is ncfl (∼ 10-1000) times faster. Our method should be easily implemented in parallel using standard parallel linear algebra libraries like ScaLAPACK [6] .. Although our method is not monotonicity-preserving for arbitrarily large timesteps, the temperature oscillations are of quite small amplitude and are damped with time. Using test problems we show that large speedups (up to ∼100-1000 for our test problems) are achieved compared to the explicit method, without seriously violating the monotonicity constraint. A similar directional splitting may also prove effective for isotropic diffusion. Although ADI is numerically stable, fast, and second order accurate in time for isotropic diffusion, it does not give strong damping in the large timestep limit (just like the Crank Nicolson scheme; e.g., [18] ).. We also tried unsplit methods, both fully implicit using limiters for the 'transverse' terms, and semiimplicit where only 'transverse' terms with limiters are treated explicitly. The limiters lead to nonlinearities that require some care with an iterative solver, and these unsplit methods result in a large sparse matrix equation which is much more expensive to solve than the tri-diagonal systems of the split method, even with an iterative solver like conjugate gradients or Loose GMRES. Although both of these unsplit methods with limiters appear to be monotonicity-preserving for arbitrary ∆t, it takes many iterations to obtain a converged solution in both cases, and we generally find that the split algorithm is quite efficient by comparison. Our method might be able to serve as an effective preconditioner to further accelerate unsplit iterative methods, and works quite well as it is for our present purposes.. Thermal conduction is primarily along the magnetic field direction for hot plasmas. For astrophysical plasmas large temperature gradients exist, and it is important for the numerical scheme implementing anisotropic conduction to yield positive temperatures in regimes of interest. Another practical requirement is that the scheme be fast so that the conduction timestep is not much smaller than the MHD timestep. Here we present a simple, directionally-split method that gets close enough to monotonicity that the temperature remains positive in presence of relatively large temperature gradients, and results in a large speedup. Our method should find applications in modeling of hot astrophysical plasmas with large temperature gradients (e.g., the multiphase interstellar/intracluster medium, transition from chromosphere to corona in the Sun).

O O O O O B I O B I O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Continuous interaction with a virtual human Abstract This paper presents our progress in developing a Virtual Human capable of being an attentive speaker. Such a Virtual Human should be able to attend to its interaction partner while it is speaking-and modify its communicative behavior on-the-fly based on what it observes in the behavior of its partner. We report new developments concerning a number of aspects, such as scheduling and interrupting multimodal behavior, automatic classification of listener responses, generation of response eliciting behavior, and strategies for generating appropriate reactions to listener responses. On the basis of this progress, a task-based setup for a responsive Virtual Human was implemented to carry out two user studies, the results of which are presented and discussed in this paper. Continuous interaction is one of the fundaments underlying Attentive Speaking and Active Listening for Virtual Humans (VHs). Attentive Speaking and Active Listening require that a Virtual Human be capable of simultaneous perception/interpretation and generation of communicative behavior. A Virtual Human should be able to signal its attitude and attention while it is listening to its interaction partner, and be able to attend to its interaction partner while it is speaking-and modify its communicative behavior on-thefly based on what it observes in the behavior of its partner.. This paper presents our progress in developing a Virtual Human that supports continuous interaction. We discuss our work on perception capabilities, involving development and evaluation of automatic classifiers for vocal listener responses. We also present our work on multimodal generation capabilities: flexible and adaptive scheduling and planning including graceful interruption, generation of response eliciting behavior, and models for appropriate reactions to listener responses. Finally, we worked on a task-based setup in which a Virtual Human explains a route to a user, combining the above mentioned capabilities with a Wizard of Oz in order to have the Virtual Human act as an Attentive Speaker. Using this setup, two user studies were carried out, the results of which are presented and discussed at the end of this paper.. In addition to the results presented in this paper, the project yielded a number of deliverables that are released for public access, among which a public release of Elckerlyc 1 (a new platform for building Virtual Humans), and a database of motion capture animations containing over 100 direction-giving-task related gestures in the route giving domain. 2 So far we have not been able to manipulate when a listener responds using subtle cues. Replicating cues discussed in literature and seen in corpora, like the use of rising pitch near the end of a sentence, a head nod accompanying the rise in pitch, looking for mutual gaze and manipulating pause lengths, we could not measure a significant effect on the amount of responses elicited by the VH.. The fact that participants did respond at most of the response opportunities created by the system (74% in Experiment 1, 67% in Experiment 2) suggests that another cue, present in all variations, is much more important. Most probably this is the syntax of the sentences and the nature of the task. Almost every sentence ended with a response opportunity. Each of these sentences carry a piece of information about the next step in the route. Given the task-oriented nature of the interaction, the user is inclined to acknowledge or express misunderstanding about the piece of information just given, like we do in real life when we are explained a route. The few variations we add with the vocal and nonverbal cues are dominated by this fact.. Another reason why some of our elicitation cues did not elicit more responses, is the fact that the task was difficult for the participants. This is illustrated by the fact that in the second experiment, only 9 out of 48 routes were drawn correctly afterwards. Because of this some of the participants did not always look at the virtual human, but at a neutral point, to reduce cognitive load. Therefore, they would not see the nonverbal elicitation cues we have implemented. We have worked on a virtual human that, in the long term, should be able to interact with a human subject in an continuous manner. That is: it should be capable of the human-like interaction in which all partners perceive each other, express themselves, and coordinate their behavior to each other, continually and in parallel. The work reported in this paper resulted in progress on several aspects of continuous interaction, such as flexible and adaptive scheduling and planning of multimodal behavior (speech, gestures, facial expressions) including graceful interruption, automatic online classification of listener responses, and models for appropriate reactions to listener responses. We have set up two experiments in which a virtual human interacts with a subject. The aim of the experiment was to elicit Listener Response behavior, to provide us with more information on what user responses occur, and to serve as inspiration for further interaction models.. In the first experiment, we have observed Listener Responses given by our subjects that are much shorter than the waiting time between steps; other Listener Responses are much longer. Furthermore, Listener Responses are not given at every response opportunity. Starting to speak through a repetition or waiting for a Listener Response that is already finished confused some of our subjects. In the second experiment we used dynamic pauses. We used a Wizard-of-Oz setup to put the virtual human on hold while the subject was speaking. The resulting behavior was that, if no Listener Response was forthcoming, the virtual human would continue after the planned waiting time. If a response was given, the virtual human waited until it finished. None of the subjects in the second experiment experienced confusion because the virtual human reacted too soon. In an autonomous version, the Wizard-of-Oz setup could be replaced by speech detection in combination with the classifiers presented in Sect. 5 to achieve similar results.. We have observed several "teach back" responses from the listener, in which (s)he repeated part of the utterance from the speaker. Detecting such repetitions is still an open issue. Since the repetitions often repeat the landmarks used in the route, perhaps the occurrence of landmarks (as detected by a keyword spotter) could be used as one of the cues for the identification of repetions. Assuming that we can automatically assess whether a response is a "teach back" repetition, the new preplanning mechanisms we have developed can be used to generate an acknowledgment of the repetition (see Interaction Example 3). The set of 64 face-to-face dialogs from the HCRC MapTask are officially divided into 8 subsets called quads. For all experiments, the training set consists of so-called quads 1-4, the development set holds quads 5-6 and the evaluation set holds quads 7-8. Based on the analysis of overlap durations in Sect. 4.3, a maximum latency threshold, τ , of 100 ms, 300 ms or 500 ms is desirable for this task.. For this experiment, the set-up diverges from the set-up described in Sect. 5.2. For training and testing the classifier, we used the COMPETITIVE and COOPERATIVE annotations that were obtained with two human annotators (see Sect. 4.1). Only those talkspurts which had labels agreed upon by both annotators were used, in total 88 and 319 talkspurts for the COMPETITIVE and COOPERATIVE class respectively. Since we have relatively little data, an N -fold cross-validation scheme was applied for training and testing the classifier. There were 4 quads available. To ensure strict separation of training, development and testing sets, in each fold, 2 quads were used for training, 1 quad for optimization and 1 quad for evaluation. All possible combinations of quads with strict separation of training, development, and testing sets were made which yielded a total of 12 folds. When the optimal parameters were found, the training and optimization set were merged and used for training. This procedure allowed better use of the data, especially the sparse occurrence of the COMPETITIVE class. As pointed out earlier, the desirable choice for the maximum latency thresholds starts at 300 ms, adding 200 ms in steps until 1100 ms. We invited 9 participants (8 male, 1 female, aged between 25 and 54, all non-native English speakers) to interact with our route giving VH. Participants were told that the VH is able to perceive and react to short vocal and nonverbal listener responses (like nodding, saying "Uh-huh", or "Yes").. Before each interaction the user was presented with the map with the starting point of the route. This map was taken away before the interaction started. During the interaction, the route giving VH gave a route description to the user. It was the task of the user to remember the route and reproduce it on the map afterwards.. Each participant interacted three times with the route giving VH. During each interaction the VH explained a different route. Each route description was given with a different elicitation strategy. Every participant interacted with the Default and Combined VH and either the Vocal or the Nonverbal VH. Permutations of routes and elicitation strategies were varied among participants.. We invited 24 participants (14 male, 10 female, aged between 23 and 54; all, except 1, non-native English speakers) to interact with our route giving VH. Participants are told that the VH is able to perceive and react to short vocal and nonverbal listener responses (like nodding, saying "Uh-huh", or "Yes").. Before each interaction the user was presented the map with the starting point of the route. This map was taken away before the interaction started. During the interaction the route giving VH explained a route to the user. It was the task of the user to remember the route and reproduce it on the map, afterwards.. Each participant interacted two times with the route giving VH. The VH presented a different route each time. At the various response opportunities in the explanation of one route (i.e., between the steps) different combinations of elicitation strategy and pause length were offered. Each combination was offered exactly once per route per participant. The order of the conditions was varied between subjects.. In the previous experiment the participants sometimes repeated part of the sentence spoken by the VH as a acknowledgment (see Interaction Example 2). The VH started during this repetition with its next sentence. To prevent this from happening again we build a Wizard-of-Oz setup emulating the COMPETITIVE/COOPERATIVE detector. We did this using the anticipaters presented in Sect. 6.4. The wizard would press a button when the participant was speaking and release it when the participant stopped. The next step would not start as long as the button was pressed.

B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O B I O O O O B I O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O B I O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Capacity of Ad Hoc wireless networks Early simulation experience with wireless ad hoc networks suggests that their capacity can be surprisingly low, due to the requirement that nodes forward each others' packets. The achievable capacity depends on network size, traffic patterns, and detailed local radio interactions. This paper examines these factors alone and in combination, using simulation and analysis from first principles. Our results include both specific constants and general scaling relationships helpful in understanding the limitations of wireless ad hoc networks.We examine interactions of the 802.11 MAC and ad hoc forwarding and the effect on capacity for several simple configurations and traffic patterns. While 802.11 discovers reasonably good schedules, we nonetheless observe capacities markedly less than optimal for very simple chain and lattice networks with very regular traffic patterns. We validate some simulation results with experiments.We also show that the traffic pattern determines whether an ad hoc network's per node capacity will scale to large networks. In particular, we show that for total capacity to scale up with network size the average distance between source and destination nodes must remain small as the network grows. Non-local traffic patterns in which this average distance grows with the network size result in a rapid decrease of per node capacity. Thus the question "Are large ad hoc networks feasible?" reduces to a question about the likely locality of communication in such networks. Ad hoc wireless networks promise convenient infrastructure-free communication. We expect the total capacity of such networks to grow with the area they cover, due to spatial re-use of the spectrum: nodes sufficiently far apart can transmit concurrently. However, ad hoc routing requires that nodes cooperate to forward each others' packets through the network. This means that the throughput This research was supported by grants from NTT Corporation under the NTT-MIT collaboration.. available to each single node's applications is limited not only by the raw channel capacity, but also by the forwarding load imposed by distant nodes. This effect could seriously limit the usefulness of ad hoc routing.. In this paper, we focus our analysis and simulations on static ad hoc networks. Note that in most mobility scenarios, nodes do not move significant distances during packet transit times. Thus, for capacity analysis, we can view mobile networks as effectively static.. The following simplification of an analysis by Gupta and Kumar [8] estimates the per node capacity to be expected in an ad hoc network. Radios that are sufficiently distant can transmit concurrently; the total amount of data that can be simultaneously transmitted for one hop increases linearly with the total area of the ad hoc network. If node density is constant, this means that the total one-hop capacity is O´nµ, where n is the total number of nodes. However, as the network grows larger, the number of hops between each source and destination may also grow larger, depending on communication patterns. One might expect the average path length to grow with the spatial diameter of the network, or equivalently the square root of the area, or O´Ônµ. With this assumption, the total end-toend capacity is roughly O´n Ô nµ, and the end-to-end throughput available to each node is. Gupta and Kumar also demonstrated the existence of a global scheduling scheme achieving ª´1 Ô n log nµ for a uniform random network with random traffic pattern.. It is not encouraging that the throughput available to each node approaches zero as the number of nodes increases. Furthermore, this simple analysis omits the constant factors which determine whether any particular networks will have a useful per node throughput.. A common observation in analyses of ad hoc routing protocols [2, 10, 4] is that capacity is the limiting factor; that is, the symptom of failure under stress is congestion losses. A high volume of routing queries or updates, caused by mobility or a large number of nodes, causes congestion; the result is not just dropped data packets, but also lost routing information and consequent mis-routing of data. Evaluations of ad hoc protocols tend to use very low data rates in order to avoid running out of capacity. For example, Das et al. [4] observe that in a simulated network of 100 nodes, each with a 2 Mbps radio, the throughput available to each node is on the order of a few kilobits per second. They report that their network has an area large enough that 7 transmissions may proceed concurrently without interfering; this means that the per node throughput actually available was about 50 times smaller than the apparent capacity. The loads used in other ad hoc routing studies are consonant with this; for example, both Karp and Kung [9] and Broch et al. [2] limit the total offered load to about 60 Kbps despite using 2 Mbps radios. The interaction of ad hoc routing and capacity suggests that any evaluation of an ad hoc network requires an understanding of network capacity.. While the above discussion suggests that ad hoc networks are fundamentally non-scalable, it may not reflect reality. The studies cited above assume a random communication pattern: each pair of nodes is equally likely to communicate, so that packet path lengths grow along with the physical diameter of the network. This assumption is probably reasonable for small networks. However, users in large networks may communicate mostly with physically nearby nodes: their neighbors in the same lecture hall of a university, or on the same floor of a building, or in the same company in a city. If local communication predominates, path lengths could remain nearly constant as the network grows, leading to constant per node available throughput. This paper makes two contributions to the understanding of practical ad hoc network scalability. At a detailed level, it examines the interaction between ad hoc forwarding and the 802.11 medium access protocol in order to estimate the constants in Equation 1. At a system level, it examines the impact of communication patterns on the form of Equation 1, and determines some conditions under which per node capacity is likely to scale to large networks. These results are likely to be useful both in understanding simulation studies of ad hoc network performance and in the deployment of real ad hoc networks. This paper examines the capacity of wireless ad hoc networks via simulations and analysis from first principles. In particular, it studies 802.11 MAC interactions with ad hoc forwarding, their effect on network capacity, and the scaling behavior of per node capacity as networks grow bigger.. The ideal capacity of a long chain of nodes in isolation is 1 4 of the raw channel bandwidth obtainable from the radio. The simulated chain capacity that the 802.11 MAC achieves with a greedy sender is about 1 7 , because nodes early in the chain starve later nodes.. We find that, in general, 802.11 does a reasonable job of scheduling packet transmissions in ad hoc networks. 802.11 is more efficient for orderly local traffic patterns, such as a lattice network with only horizontal flows. 802.11 is also able to approach the theoretical maximum capacity of O´1 Ô nµ per node in a large random network of n nodes with random traffic.. We argue that the key factor deciding whether large ad hoc networks are feasible is the locality of traffic. We present specific criteria to distinguish traffic patterns that allow scalable capacity from those that do not. Gupta and Kumar [8] show that the per node capacity in an n-node random ad hoc network is ¢´1 Ô n log nµ, using a geometric analysis. They also show a global scheduling scheme which achieves that bound. In their work, a random communication pattern is assumed. Our work extends theirs by further considering the effects of different traffic patterns on the scalability of per node capacity. We also examine how ad hoc forwarding interacts with the 802.11 MAC and show that the use of 802.11 instead of a global scheduling scheme does not seem to affect the asymptotic bound on per node capacity.. Shepard [13] considers limits on capacity imposed by aggregate interference from many senders spread over a large area, concluding that such networks are scalable. He points out that capacity can be increased with minimum-energy routing, and proposes an efficient distributed channel-access technique. Our work, in contrast, focuses on the capacity likely to be available with the existing 802.11 channel access algorithm, which cannot easily support minimumenergy routing. We also focus on capacity limits imposed by multihop traffic patterns rather than by aggregate interference.. We assume that nodes are stationary. Grossglauser and Tse [6] consider ad hoc networks of mobile nodes, showing that long term per node throughput can stay constant in a network where node movement process is ergodic with a stationary distribution uniform over the network. The basic idea is for a source node to distribute packets to as many different nodes as possible; these nodes relay the packets to the final destination whenever they get close to the destination. Therefore, the expected path length remains constant.. However, this result depends critically on the movement model. Furthermore, the fixed throughput guarantee is achieved only over very long time frames. This result, nevertheless, suggests a way to take advantage of node movement when sending packets from applications that can tolerate long delays.. Some existing studies have focused on the fairness of 802.11 in the context of ad hoc forwarding. Nandagopal et al. [12] propose an algorithm that gives each flow in the network a fair allocation of capacity no matter how much more contention it perceives in comparison to other flows. Luo et al. [11] propose an algorithm that, in addition to giving each flow its fair share, maximizes the total network capacity by giving more chances to flows whose transmissions cause less interference. The proposed algorithms might improve 802.11's efficiency in ad hoc forwarding.

O O O O B I O B B O B I I I O O O O O O O O O O O O O O B O O O O O O O O B B B O O O O O O O B O O O O O O O O B O B O O O O O O O O O O B O B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O B O O O O O O O O O O B O O O O O O O O O B O O O O B O O O O B O O O O O O O B B O O O O O O O O O O B O O O O O O O O O O O O B O O B B O O B O O O O O O O O O O B O O B O O O O O B O O O B B O O O O O O O O O O O O O O B I I O O O O O O O O B O B O O O O O B I O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O B O O O B O O O O B I I O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O B O O B B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O B O O O O B O O O O O O O B I I O O O O O O O O O O O O O O O B B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O B O O O O O O O O O B O O O O O O O O O O B O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O B O O O O B O O O O O O O B I I O B O O O O B I I I O O O B O O O O O O O O O O O O O O O B O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O B O O O B I I O O O O O O O B O O O O B B O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O B O O O O O B O O O B I I O O O O B O B O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O B O O O O O B O O O O O O O O O O B B O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O B O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O


Good Reduction of Puiseux Series and Applications We have designed a new symbolic-numeric strategy to compute efficiently and accurately floating point Puiseux series defined by a bivariate polynomial over an algebraic number field. In essence, computations modulo a well chosen prime number p are used to obtain the exact information needed to guide floating point computations. In this paper, we detail the symbolic part of our algorithm: First of all, we study modular reduction of Puiseux series and give a good reduction criterion to ensure that the information required by the numerical part is preserved. To establish our results, we introduce a simple modification of classical Newton polygons, that we call "generic Newton polygons", which turns out to be very convenient. Finally, we estimate the size of good primes obtained with deterministic and probabilistic strategies. Some of these results were announced without proof at ISSAC '08. Let K be a number field (finite extension of Q, the field of rational numbers), and F (X, Y ) be a bivariate polynomial in K[X, Y ] such that:. • F is squarefree and primitive with respect to Y . If R F (X) denotes the resultant of F and F Y , its derivative with respect to Y , then a root of R F is called a critical point. Critical points can also be defined as the set of numbers x 0 such that F (x 0 , Y ) has less than d Y roots. Non-critical points will be called regular.. Email addresses: adrien.poteaux@unilim.fr (Adrien Poteaux), marc.rybowicz@xlim.fr (Marc Rybowicz).. At X = x 0 , the d Y roots of F , viewed as a univariate polynomial in Y , can be represented by fractional Laurent series in (X − x 0 ) called Puiseux series; see Section 3. If x 0 is regular, Puiseux series reduce to classical Laurent series.. Puiseux series are fundamental objects of the theory of algebraic curves (Walker, 1950; Brieskorn and Knörrer, 1986) and provide important information: They give ramification indices of the X-plane covering defined by F , they can be used to compute the genus of the curve defined by F using Riemann-Hurwitz formula, or to compute integral bases and linear spaces associated to divisors on the curve (Bliss, 1933; Duval, 1987; van Hoeij, 1994) , which in turn have many applications, such as the determination of parametrizations of genus 0 curves (van Hoeij, 1997) , the integration of algebraic functions (Trager, 1984; Bronstein, 1990) , or the absolute factorization of polynomials (Duval, 1991) .. Moreover, the equation F (X, Y ) = 0 defines d Y algebraic functions of the variable X, which are analytic in any simply connected domain D ⊂ C free of critical points. If D is included in a sufficiently small disc centered at a critical point x 0 , it is well-known that numerical approximations of these functions in D can be obtained directly via truncated Puiseux series at X = x 0 .. We have used this fact to devise an algorithm to compute the monodromy of the Xplane covering defined by the curve F (X, Y ) = 0 (Poteaux, 2007) . The algorithm follows paths along a minimal spanning tree for the set of critical points; expansions above critical point are used to bypass them. Our ultimate goal was to build an effective version of the celebrated Abel-Jacobi Theorem (Miranda, 1995; Forster, 1981) , which requires the integration of algebraic functions along paths on the Riemann Surface defined by F (see Deconinck and van Hoeij (2001) for instance). Again, in this context, Puiseux series are definitely useful (Deconinck and Patterson, 2008) .. We know of three methods to compute Puiseux series: Differential Equation. It has been known for a long time (Comtet, 1964) that Puiseux series can be efficiently computed using the differential equation satisfied by the algebraic functions. More recently, Chudnovsky, 1986, 1987; van der Hoeven, 1999 van der Hoeven, , 2005 have advocated this approach and designed asymptotically fast algorithms, in terms of truncation order and precision required. In our monodromy context, though, we do not need a high precision since we just need enough information to separate functions. Moreover, no differential equation is known a priori; the minimal order differential equation may have high degree coefficients and its determination may be a bottleneck. Bostan et al. (2007) have recenlty proposed a method to reduce the degrees of the coefficients, but this leads to a higher order differential equation. Hence, it is not clear that these asymptotically fast methods are relevant. Generalized Hensel constructions. Cohn (1984) has proposed a matrix analogue of Hensel's Lemma that gives an alternative proof of Puiseux's Theorem (Theorem 1 below). Diaz-Toca and Gonzalez-Vega (2002) deduced an algorithm that can be viewed as an iterative method to compute the Jordan normal form of a matrix whose eigenvalues are the roots of F , i.e. Puiseux series. But computations are performed in algebraic extensions of K larger than residue fields (see Section 3.3) and there is no evidence that, in its current setting, this approach is competitive.. Hensel lifting has also been extended by several authors to compute factorizations of F in K[ [X]] [Y ] when X = 0 is a critical point; see Sasaki and Inaba (2000) , who consider the case where X is a multi-variable, and references therein. This method could be used to compute Puiseux series, if there is no ramification or if ramification indices are known in advance, which is not our case. Moreover, we have found no information on the efficiency and complexity of the method, nor any implementation. Newton-Puiseux Algorithm. This method is based on Newton polygons and is wellestablished (Walker, 1950; Brieskorn and Knörrer, 1986) . A variant that allows to perform all computations in the residue fields, called "Rational Newton-Puiseux Algorithm", was introduced by Duval (1989) . An implementation due to Mark Van Hoeij is available in the Maple library since release V; see also the Magma implementation (Bosma et al., 1997) . Our approach is based on the Newton-Puiseux algorithm and its rational version; we shall give details in Section 4.. Unfortunately, applying a floating point Newton-Puiseux algorithm above a critical point is doomed to failure. Indeed, if the critical point x 0 is replaced with an approximation, expansion algorithms return approximate series with very small convergence discs and do not retain important information, such as ramification indices. Therefore, the output is not aidful.. On the other hand, coefficient growth considerably slows down symbolic methods. Since the degree of R F may be equal to d X (2d Y − 1), a critical point x 0 may be an algebraic number with large degree. Furthermore, Puiseux series coefficients above x 0 may belong to a finite extension of degree d Y over K(x 0 ): For d Y = d X = 10, the degree over K may already be excessively large for practical computations. Moreover, when these coefficients are expressed as linear combinations over Q, the size of the rational numbers involved may also be overwhelming. Floating point evaluation of such coefficients must, in some cases, be performed with a high number of digits because spectacular numerical cancellations occur; see examples in (Poteaux, 2007) . For instance, the degree 6 polynomial F (X, Y ) = (Y 3 − X) ((Y − 1) 2 − X) (Y − 2 − X 2 ) + X 2 Y 5 has a resultant R F (X) = X 3 P (X), where P (X) is an irreducible polynomial of degree 23 over Q. Rational Puiseux series (see Section 3.3) above roots of P (X) have coefficients in a degree 23 extension of Q. Rational numbers with 136 digits appear in the first term of the expansions. Walsh (2000) has shown that, for any ǫ > 0, the singular part of Puiseux series can be computed using O(d. X log h 2+ǫ ) bit operations, where h is the height of F . Although this bound is probably not sharp, it is not encouraging and tends to confirm fast coefficient growth.. To alleviate these problems, we have introduced a symbolic-numeric approach: exact relevant information is first obtained by means of computation modulo a well chosen prime number p, then this information is used to guide floating point computations. The coefficient size is therefore kept under control while numerical instability is reduced. Experimental evidences reported in Poteaux (2007) seem to validate this approach. Exact important data, such as ramification indices, Puiseux pairs and intersection multiplicities of branches, are preserved by our reduction criterion; as a byproduct, we also obtain a modular method to compute the genus of a plane curve and the topological type of its singularities (Zariski, 1981; Campillo, 1980) . This paper presents several contributions: • We introduce "generic Newton polygons" and "polygon trees" (Section 4). The latter captures precisely the symbolic information needed for floating point computations and other aplications.. • We study modular reduction of Puiseux series and rational Puiseux expansions. This leads to a fully proved and easy to check criteria for the choice of a "good prime" p such that polygon trees can be obtained using modular arithmetic (Section 5). We rely on technical results that are proven or recalled in Section 3 and 4.. • Finally, we study deterministic and probabilistic methods to obtain such a prime and give estimates for the size of p (Section 6). It turns out that probabilistic methods yield primes with logarithmic size, with respect to the size of F . Many of these results were announced without proofs by Poteaux and Rybowicz (2008) at ISSAC '08, and only for F monic. This paper is an extended and (almost) self-contained version that includes all proofs and additional material such as the non monic case, which requires some care, as well as a global good reduction criterion.. In (Poteaux and Rybowicz, 2009) , we study how to efficiently implement the rational Newton-Puiseux algorithm over finite fields and deduce improved arithmetic complexity bounds. Combined with our results herein about the size of a good prime, they give estimates for the bit-complexity of the symbolic part of our symbolic-numeric method, as well as bit-complexity estimates for the computation of the genus and similar problems.. Obtaining floating point Puiseux series from polygon trees is not a trivial task. A first method was briefly described in (Poteaux, 2007) and experimental results were provided. A more elegant and more appropriate approach based on Singular Value Decomposition is given in (Poteaux, 2008) ; this will be the topic of a forthcoming article.. Finally, we remark that modular methods are extensively used in Computer Algebra to avoid intermediate coefficient swell, via Hensel lifting or the Chinese Remainder Theorem; see for instance von zur Gathen and Gerhard (1999) . However, a "reduce mod p and lift" or "reduce mod p i and combine" method would not help much in this case since we are not facing an intermediate coefficient growth problem, but an intrinsically large symbolic output. Modular methods are much less common when it comes to directly obtaining numerical results. We therefore claim some originality with this approach.

O O O B I O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O B I O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O B I O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Strong qualitative independence The subsets A, B of the n-element X are said to be s-strongly separating if the two sets divide X into 4 sets of size at least s. The maximum number h(n, s) of pairwise s-strongly separating subsets was asymptotically determined by Frankl [8] for fixed s and large n. A new proof is given. Also, estimates for h(n, cn) are found where c is a small constant. Let X be a finite set of n elements. The following notion was introduced by Marczewski [17] . The subsets A, B ⊂ X are called qualitatively independent if they divide X into four non-empty parts that is A ∩ B, A ∩ B, A ∩ B, A ∩ B are all non-empty. The significance of this notion in search theory lies in the consequence that after knowing if an unknown x is in A or not we cannot decide the same question for B, independenly on the answer for the first question. The family F ⊂ 2 X is called independent if their members are pairwise qualitatively independent.. Rényi asked [23] the question what is the maximum size of an independent family in an n-element set. The answer was found by the present author ( [12] , see also [23] ), and independently by Brace and Daykin [5] , Bollobás [3] and Kleitman and Spencer [16] (see also Schönheim [24] ).

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Learning DFA from simple examples Abstract. We p r e s e n t a framework for learning DFA f r o m simple examples. We s h o w that e cient P AC learning of DFA is possible if the class of distributions is restricted to simple distributions where a teacher might choose examples based on the knowledge of the target concept. This answers an interesting variant of an open research question posed in Pitt's seminal paper: Are D F A's PAC-identi able if examples are d r awn from the uniform distribution, or some other known simple distribution? Our approach uses the RPNI algorithm for learning DFA from labeled examples. In particular, we describe an e cient learning algorithm for exact learning of the target DFA with high probability when a bound on the number of states (N) of the target DFA i s k n o wn in advance. When N is not known, we s h o w h o w this algorithm can be used for e cient P AC learning of DFAs. The problem of learning the minimum state DFA that is consistent w i t h a g i v en sample has been actively studied for over two decades. DFAs are recognizers for regular languages which constitute one of the simplest class of languages in the formal language hierarchy Chomsky, 1956 Hopcroft & Ullman, 1979 ]. An understanding of the issues and problems encountered in learning regular languages (or equivalently, identi cation of the corresponding DFA) are therefore likely to provide insights into the problem of learning more general classes of languages.. Exact learning of the target DFA from an arbitrary presentation of labeled examples is a hard problem Gold, 1978] . Gold showed that the problem of identifying the minimum state DFA consistent with a presentation S comprising of a nite non-empty set of positive examples S + and possibly a nite non-empty set of negative examples S ; is NP-hard. Under the standard complexity theoretic assumption P 6 = NP, Pitt and Warmuth showed that no polynomial time algorithm can be guaranteed to produce a DFA with at most N (1; )loglog(N) states from a set of labeled examples corresponding to a DFA with N states Pitt & Warmuth, 1988] .. E cient learning algorithms for identi cation of DFA assume that additional information is provided to the learner. Trakhtenbrot and Barzdin described a polynomial time algorithm for constructing the smallest DFA consistent with a complete labeled sample i.e., a sample that includes all strings up to a particular length and the corresponding label that states whether the string is accepted by the target DFA o r n o t T rakhtenbrot & Barzdin, 1973] . Angluin showed that given a live-complete set of examples (that contains a representative string for each live state of the target DFA) and a knowledgeable teacher to answer membership queries it is possible to exactly learn the target DFA Angluin, 1981] . In a later paper, Angluin relaxed the requirement of a live-complete set and has designed a polynomial time inference algorithm using both membership and equivalence queries Angluin, 1987] . The RPNI algorithm is a framework for identifying a DFA consistent with a given sample S in polynomial time Oncina & Garc a, 1992] . Further, if S is a superset of a characteristic set (see section 2.1) for the target DFA then the DFA output by the RPNI algorithm is guaranteed to be equivalent to the target Oncina & Garc a, 1992 Dupont, 1996b .. Pitt surveyed several approaches for approximate identi cation of DFA Pitt, 1989 ]. Valiant's distribution-independent model of learning (also called the PAC model) Valiant, 1984] is widely used for learning several di erent concept classes approximately. When adapted to the problem of learning DFA, the goal of a PAC learning algorithm is to obtain in polynomial time, with high probability, a D F A that is a good approximation of the target DFA. Even approximate learnability o f D F A w as proven to be a hard problem. Pitt and Warmuth showed that the problem of polynomially approximate predictability of the class of DFA i s hard Pitt & Warmuth, 1989] . They made use of prediction preserving reductions to show t h a t i f D F As are polynomially approximately predictable then so are other known hard to predict concept classes such a s boolean formulas. F urther, under certain cryptographic assumptions, Kearns and Valiant s h o wed that an e cient algorithm for learning DFA w ould entail e cient algorithms for solving several known hard problems such as breaking the RSA cryptosystem, factoring Blum integers, and detecting quadratic residues Kearns & Valiant, 1989] .. The PAC model's requirement of learnability under all conceivable distributions is often too stringent for practical learning scenarios. Pitt's paper identi ed the following open research problem: Are D F A's PAC-identi able if examples are d r awn from the uniform distribution, or some other known simple distribution? Pitt, 1989] . Using a variant o f T rakhtenbrot and Barzdin's algorithm, Lang empirically demonstrated that random DFAs are approximately learnable from a sparse uniform sample Lang, 1992] . However, exact identi cation of the target DFA w as not possible even in the average case with a randomly drawn training sample. Several concept classes are e ciently PAC learnable under restricted classes of distributions while their learnability under the distribution free model is not known Li & Vit anyi, 1991] . Li and Vit anyi proposed a model for PAC learning with simple examples wherein the examples are drawn according to the Solomono -Levin universal distribution. T h e y s h o wed that learnability under the universal distribution implies learnability under a broad class of simple distributions. Thus, this model is su ciently general. Recently, this learning model has been extended to a framework where a teacher might c hoose examples based on the knowledge of the target concept Denis et al., 1996] . We s h o w that under this extended framework of learning from simple examples (called the PACS model) it is possible to e ciently learn DFA thereby answering an interesting variant of Pitt's open research question in the a rmative.. The rest of this paper is organized as follows: Section 2 introduces the necessary de nitions and notation. Section 3 summarizes the RPNI algorithm. Section 4 describes the learning of DFA with simple examples and section 5 concludes with a discussion of several interesting avenues that merit further investigation. The problem of exactly learning the target DFA from an arbitrary set of labeled examples and the problem of approximating the target DFA from labeled examples under Valiant's PAC learning framework are both known to be hard problems. Thus, the question as to whether DFA are e ciently learnable under some restricted yet fairly general and practically useful classes of distributions was clearly of interest. In this paper, we h a ve answered an important v ariant of this question in the a rmative b y p r o viding a framework for e cient P AC learning of DFA from simple examples. In particular, we h a ve s h o wn that if a benign teacher (or nature) provides labeled examples of a target DFA d r a wn according to the universal distribution m r , then with probability at least 1 ; , this sample is guaranteed to include a c haracteristic set of examples for the target DFA when the sample size is polynomial in the number of states of the target DFA and lg( 1 ). The RPNI algorithm is guaranteed to return a DFA t h a t i s e q u i v alent to the target DFA provided the set of examples given to the algorithm includes a characteristic set. Thus, by casting the RPNI algorithm in the framework of learning from simple examples we demonstrate probably exact leanability o f D F A from labeled examples. Further, we s h o w that when an upper bound on the number of states of the target DFA is unknown, we can use this algorithm iteratively, to e ciently PAC learn the concept class of DFAs with any desired error and con dence parameters.. The class of simple distributions includes a large variety of probability distributions (including all computable distributions that are characterized by nite precision parameters). Li and Vitany i showed that a concept class is e ciently learnable under the universal distribution if and only if it is e ciently learnable under each simple distribution provided the sampling is done according to the universal distribution Li & Vit anyi, 1991] . This raises the possibility o f using sampling under the universal distribution to learn under all computable distributions. However, the universal distribution is not computable. Whether one can instead get by with a polynomially computable approximation of the universal distribution remains an open question. It is known that the universal distribution for the class of polynomially-time bounded simple distributions is computable in exponential time Li & Vit anyi, 1991] . This raises a number of interesting possibilities for learning under simple distributions.. A related question of interest has to do with the nature of environments that can be modeled by simple distributions. Learning scenarios that involve biological intelligent systems (e.g., language acquisition) appear to bene t from simple examples. It is not inconceivable that these systems rst process simpler examples and then as they mature, take on the more complex examples. In particular, if Kolmogorov complexity is an appropriate measure of the intrinsic complexity of objects in nature and if nature (or the teacher) has a propensity for simplicity, then it stands to reason that the examples presented to the learner by the environment are likely to be generated by a simple distribution. Against this background, empirical evaluation of the performance of the proposed algorithms using examples that come from natural domains is clearly of interest.. In the incremental version of the RPNI algorithm Dupont, 1996a ] the learner maintains a hypothesis that is consistent with all labeled examples seen thus far and modi es it whenever a new inconsistent example is observed. The convergence of this algorithm relies on the fact that sooner or later, the set of labeled examples seen by the learner will include a characteristic set. If in fact the stream of examples provided to the learner is drawn according to a simple distribution, our results show that in an incremental setting the characteristic set would be made available relatively early (during learning) with a su ciently high probability and hence the algorithm will converge quickly to the desired target.. Some of the negative results in approximate identi cation of DFA a vailable in the literature were derived by showing that an e cient algorithm for learning DFA w ould entail algorithms for solving known hard problems such a s learning boolean formulae Pitt & Warmuth, 1988] and breaking the RSA cryptosystem Kearns & Valiant, 1989] . It would be interesting to explore the implications (if any) of our results on e cient learning of DFA from simple examples on these problems.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


On Higher Arthur-Merlin Classes We study higher Arthur-Merlin classes de ned via several natural probabilistic operators BP; R and co-R. We investigate the complexity classes they de ne, and a number of interactions between these operators and the standard polynomial time hierarchy. We prove a hierarchy theorem for these higher Arthur-Merlin classes involving interleaving operators, and a theorem giving non-trivial upper bounds to the intersection of the complementary classes in the hierarchy. Arthur-Merlin Games were introduced by B abai B85] BM88] to study the power of randomization in interaction. Goldwasser and Sipser GS89] proved soon afterwards that these classes are equivalent in power to Interactive Proof Systems introduced by GMR85]. In the last 15 years, this study has proved to be exceedingly successful in complexity theory ZH86, BHZ87, ZF87, LFKN90, Sh90] . Eventually the study of these proof systems (and multi prover systems) led to perhaps the most spectacular achievement in the Theory of Computing in the last decade BGKW88, LFKN90, Sh90, BFL91, BFLS91, FGLSM91, AS92, ALMSS92]. It is well known that some traditional complexity classes can be characterized by operators.. For example NP = 9 P; 2 = 9 8 P, using the existential and universal operators 9 and 8 respectively. They have been used fruitfully to prove (or to give simpler proofs of known) relations between complexity classes ZF87, To91, WT92, VW93, VW95, VW97]. For example, Toda's Theorem was proved in this framework To91] . In this approach, Zachos and F urer ZF87] showed that AM can also be characterized by the operator BP, i.e. AM = BP NP. By successfully employing this operator machinery, they were able to give a simple and natural proof that the one-sided error version of AM coincides with the two-sided error version, and thus AM 2 . In this paper, we de ne higher Arthur-Merlin classes using several natural probabilistic operators BP, R and co-R. We consider the hierarchy of classes BP k , for k 1, and prove some intricate relations among these classes and `a nd ZPP. The class BP k can also be thought of as an Arthur-Merlin game where Arthur has access to k?1 oracle. By investigating the interactions between the BP operators and the polynomial-time hierarchy, we are able to obtain some non-trivial containments; for example, we show that AM k = k+1 for k 2, whereas the na ve argument gives AM k k+2 . Similarly, by interleaving n levels of k classes and n levels of BP `c lasses, we obtain results that improve trivial containments by n levels.. Recently classes ZPP NP and AM \ co-AM received much attention. Arvind We note that this containment is a signi cant improvement over the na ve bound.. We consider a generalization of the class AM \ co-AM. Let E k be BP k \ BP k . We prove NP E 1 ZPP NP using the following technique: the ZPP machine sends the random bits to the NP oracle, which simulates NP AM\co-AM using those random bits. Arvind and K obler's result follows as a simple corollary, and thereby, we obtain an alternative proof of their lowness theorem. We then prove a far reaching extension of this non-trivial upper bound. By interleaving n levels of E k and n levels of l classes, we obtain an upper bound that improves over the trivial bound by n levels. Our proofs illustrate the power of the operator machinery, when properly deployed. The apparent ease (even a tautness) with which these theorems on the hierarchy are revealed speaks volumes of its e ectiveness. While the formalism of this operator machinery may appear austere and less vivid intuitively, it should also be pointed out that only through such a formalism can we even begin to state some of the delicate relations in this hierarchy. We have considered natural extensions of the class AM using the BP operator. We have established several non-trivial relations involving these classes and the polynomial hierarchy. We note that by extending a result of Agrawal and Thierauf AT97] we get problems in BP k \ k . This can be achieved by considering Circuit Isomorphism problem with k?2 oracle gates. Observe that these languages can not be complete for k unless hierarchy collapses.. However, it would be interesting to give natural examples of languages in BP k \ k .. Another possible extension will be to obtain a similar operator characterization of MA, and to shed some light on their relationships with the hierarchy.

O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Integrating

learning

styles

and

adaptive

e-learning

system:

Current

developments,

problems

and

opportunities

Learning

styles

which

refer

to

students'

preferred

ways

to

learn

can

play

an

important

role

in

adaptive

e-learning

systems.

With

the

knowledge

of

different

styles,

the

system

can

offer

valuable

advice

and

instructions

to

students

and

teachers

to

optimise

students'

learning

process.

Moreover,

e-leaning

system

which

allows

computerised

and

statistical

algorithms

opens

the

opportunity

to

overcome

drawbacks

of

the

traditional

detection

method

that

uses

mainly

questionnaire.

These

appealing

reasons

have

led

to

a

growing

number

of

researches

looking

into

the

integration

of

learning

styles

and

adaptive

learning

system.

This

paper,

by

reviewing

studies,

delves

deeply

into

different

parts

of

the

integration

process.

It

captures

a

variety

of

aspects

from

learning

styles

theories

selection

in

e-learning

environment,

online

learning

styles

predictors,

automatic

learning

styles

classification

to

numerous

learning

styles

applications.

The

results

offer

insights

into

different

developments,

achievements

and

open

problems

in

the

field.

Based

on

these

findings,

the

paper

also

provides

discussion,

recommendations

and

guidelines

for

future

researches.

Different

students

have

different

preferred

ways

to

learn.

Some

may

understand

quickly

through

images,

others

may

prefer

texts

and

readings.

Some

may

deal

well

with

theories,

others

may

learn

through

experiments

and

examples.

By

gaining

insights

into

different

learning

styles,

it

offers

means

to

design

and

provide

interventions

that

tailored

to

individual

needs.

Moreover,

several

valuable

advice

can

be

provided

to

a

wide

range

stakeholders.

For

example,

for

learners,

insights

into

their

own

styles

will

enable

them

to

be

more

confident

in

learning

and

optimize

their

learning

paths

(Herod,

For

teachers,

it

will

be

able

to

offer

valuable

feedback

on

how

to

match

suitable

instructions

and

learning

materials

to

different

groups

of

students

at

the

appropriate

stage

of

the

learning

process

(Stash,

For

instance,

under

FelderSilverman's

theory

(Felder

Silverman,

learning

styles

can

be

differentiated

between

the

way

students

process

information:

active

experimentation

or

reflective

observations.

For

"active"

students,

they

do

not

perform

very

well

in

a

standard

classroom

situation.

Conversely,

they

learn

effectively

through

interaction

with

other

students.

Thus,

it

is

advisable

for

teachers

to

provide

such

E-mail

address:

mtruong@corvinno.com

group

the

opportunity

to

cooperate

and

discuss

the

topic.

Furthermore,

there

is

evidence

in

previous

researches

(such

as

in

Kolb,

Boyatzis,

Mainemelis,

and

Plovnick,

which

shows

the

connection

between

learning

styles

and

career

choices.

Based

on

this,

recommendations

and

guidance

to

support

the

career

path

planning

can

be

developed.

With

this

strong

appeal,

learning

styles

have

been

gaining

significant

interest

from

researchers

and

educators.

Coffield,

Moseley,

Hall,

and

Ecclestone

in

their

review,

reported

over

theories

that

were

developed

over

the

past

years..

In

a

more

comprehensive

way,

learning

styles,

which

according

to

Keefe

can

be

defined

as:

"The

composite

of

characteristic

cognitive,

affective,

and

physiological

factors

that

serve

as

relatively

stable

indicators

of

how

a

learner

perceives,

interacts

with,

and

responds

to

the

learning

environment".

The

crowed

theories

area

can

be

divided

into

five

groups

depending

on

their

assumptions

on

how

flexible

the

learning

styles

can

change

over

the

lifetime

(Coffield

et

al.,

On

one

extreme,

there

are

theories

such

as

Dunn

and

Dunn's

models

and

instruments

of

learning

styles

(cited

in

Dunn

Griggs,

which

suggests

that

learning

styles

are

constitutionally

based,

that

is

fixed.

On

the

other

extreme,

researchers

consider

tactics

rather

than

learning

styles.

Learning

tactics

assume

that

learning

behaviours

can

change

depending

on

Elsevier

Ltd.

All

rights

reserved.

the

situation.

This

branch

includes

theories

such

as

Entwistle's

approaches

and

study

skills

inventory

for

students

(Entwistle,

and

Vermunt's

inventory

of

learning

styles

(Vermunt,

Other

prominent

theories

include

Kolb's

(Kolb,

Osland,

Rubin,

and

Felder-Silverman's

(Felder

Silverman,

which

consider

learning

styles

as

rather

stable

indicators

but

may

change

over

the

lifetime..

Traditionally,

learning

styles

are

mainly

measured

using

surveys

and

questionnaires,

asking

students

to

self-evaluate

their

own

behaviours.

This

development

is

suitable

with

the

traditional

classroom

where

it

is

difficult

to

observe

and

analyse

students'

preferences

over

the

whole

learning

process.

However,

as

for

every

qualitative

survey,

this

type

of

measurement

suffers

many

drawbacks.

Firstly,

it

can

be

biased

as

it

depends

on

students'

judgment.

Secondly,

it

is

done

only

at

a

point

in

time

while

the

learning

styles,

according

to

several

theories,

can

change

over

time.

Some

of

these

surveys

can

reach

over

long

(such

as

Vermunt's

(Vermunt,

and

Felder-Silverman's

(Felder

Silverman,

and

hence,

they

are

not

easy

to

update..

These

drawbacks

have

encouraged

a

growing

number

of

researchers

to

integrate

the

framework

of

learning

styles

into

elearning

system.

On

one

hand,

e-learning

system,

which

allows

researchers

to

observe

students'

behaviours

throughout

the

learning

process

and

with

the

use

of

data

mining

and

computerised

algorithms,

to

quickly

identify

and

analyse

trends

in

big

dataset,

opens

opportunities

to

develop

new

framework

to

observe

and

measure

learning

styles

through

online

behaviours.

On

the

other

hand,

learning

styles

are

also

useful

sources

to

develop

an

adaptive

e-learning

system

that

effectively

personalises

learning

resources

to

individuals'

learning

needs.

With

strong

appeal,

the

integration

of

the

IT-related

and

psychology

and

pedagogy-related

area

have

gained

significant

interest

over

the

past

years.

A

recent

paper

surveying

e-learning

system

developers

by

Thalmann

even

suggested

that

learning

styles

models

were

the

most

useful

frameworks

for

adaptive

system

development

among

other

sources

such

as

previous

knowledge

and

student

background..

The

constructs

and

applications

of

learning

styles

into

adaptive

e-learning

system

have

observed

several

positive

results

in

both

learning

styles

detection

(such

as

in

García,

Amandi,

Schiaffino,

Campo,

Graf,

Kinshuk,

Liu,

Scott,

Rodríguez,

Soria,

Campo,

Özpolat

Akar,

and

applications

(such

as

personalising

learning

materials

and

learning

contents

as

in

Kurilovas,

Kubilinskiene,

Dagiene,

and

developing

educational

games

as

in

Lin,

Yeh,

Hung,

Chang,

This

paper

provides

an

update

and

a

systematic

review

on

this

integration

of

learning

styles

into

adaptive

e-learning

system.

Through

the

literature

review,

it

offers

insights

into

different

methodologies,

constructs,

developments

and

applications

that

have

been

studied

in

the

research

field.

Moreover,

broadening

from

previous

reviews

such

as

papers

by

Vandewaetere,

Desmet,

and

Clarebout

and

Akbulut

and

Cardak

which

either

only

briefly

looked

at

learning

styles

as

part

of

many

other

personalisation

characteristics

or

only

presented

parts

of

the

whole

applications,

this

paper,

focusing

on

learning

styles,

delves

deeply

into

the

integration

process.

It

captures

several

aspects

from

learning

styles

measurement

in

e-learning

system,

to

the

usage

of

learning

styles

in

e-learning

application.

Based

on

the

findings,

it

also

provides

new

insights

into

current

developments,

issues

and

challenges

and

as

the

result,

valuable

recommendations

can

be

offered

for

future

studies.

O B I O O B O O O O O O B I O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O B I I O O O O O O O O O O O O O O O O O O O O O O B I O O O B O O B I O O B I O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O B I O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O B I O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O B I O O B O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O B O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O B I O O B O O O O O B I O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O