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Agi-conf.orgferent amounts of Shannon information, they have one thingin common: Entropy within a chunk is relatively low, en- We argue that the ability to find meaningful chunks in se- tropy at chunk boundaries is relatively high. Two kinds of quential input is a core cognitive ability for artificial generalintelligence, and that the Voting Experts algorithm, which evidence argue that this signature of chunks is general for searches for an information theoretic signature of chunks, the task of chunking sequences and series (see for provides a general implementation of this ability. In sup- a similar idea applied to two-dimensional images). First, port of this claim, we demonstrate that VE successfully finds the Voting Experts (VE) chunking algorithm and its several chunks in a wide variety of domains, solving such diverse variants, all of which detect this signature of chunks, per- tasks as word segmentation and morphology in multiple lan- form very well in many domains. Second, when sequences guages, visually recognizing letters in text, finding episodes are chunked all possible ways and ranked by a “chunkiness in sequences of robot actions, and finding boundaries in the score” that combines within- and between-chunk entropy, instruction of an AI student. We also discuss further desirable the highest-ranked chunks are almost always real chunks ac- attributes of a general chunking algorithm, and show that VE cording to a gold standard. Here, we focus primarily on the former kind of evidence, but also provide some early evi-dence of the latter kind.
To succeed, artificial general intelligence requires domain- independent models and algorithms that describe and imple- What properties should a general-purpose chunking algo- ment the fundamental components of cognition. Chunking rithm have? It must not simply exploit prior knowledge of a is one of the most general and least understood phenomena particular domain, but rather must be able to learn to chunk in human cognition. George Miller described chunking as novel input. It must operate without supervision in novel do- “a process of organizing or grouping the input into familiar mains, and automatically set any parameters it has to appro- units or chunks.” Other than being “what short term mem- priate values. For both humans and artificial agents, work- ory can hold 7 +/- 2 of,” chunks appear to be incommen- ing memory is finite, and decisions must be made online, surate in most other respects. Miller himself was perplexed so the algorithm must be efficient and rely on local informa- because the information content of chunks is so different. A tion rather than global optimization. Finally, learning should telephone number, which may be two or three chunks long, be rapid, meaning that the algorithm should have relatively is very different from a chessboard, which may also con- tain just a few chunks but is vastly more complex. Chunks VE has these properties. Its name refers to the “experts” contain other chunks, further obscuring their information that vote on possible boundary locations. The original ver- content. The psychological literature describes chunking sion of VE had two experts: One votes to place bound- in many experimental situations (mostly having to do with aries after sequences that have low internal entropy, given long-term memory) but it says nothing about the intrinsic, by HI (seq) = −log(p(seq)), the other places votes af- mathematical properties of chunks. The cognitive science ter sequences that have high boundary entropy, given by literature discusses algorithms for forming chunks, each of which provides a kind of explanation of why some chunks set of successors to seq. All sequences are evaluated locally, rather than others are formed, but there are no explanations within a sliding window, so the algorithm is very efficient.
of what these algorithms, and thus the chunks they find, have The statistics required to calculate HI and HB are stored efficiently using an n-gram trie, which is constructed in asingle pass over the corpus. The trie depth is 1 greater than the size of the sliding window. Importantly, all statistics in Miller was close to the mark when he compared bits with the trie are normalized so as to be expressed in standard devi- chunks. Chunks may be identified by an information the- ation units. This allows statistics from sequences of different oretic signature. Although chunks may contain vastly dif- lengths to be compared to one another.
The sliding window is passed over the corpus and each Tanaka-Ishii and Jin developed an algorithm expert votes once per window for the boundary location that called Phoneme to Morpheme (PtM) to implement ideas best matches its criteria. VE creates an array of vote counts, originally developed by Harris in 1955. Harris no- each element of which represents a location and the number ticed that if one proceeds incrementally through a sequence of times an expert voted to segment at that location. The of phonemes and asks speakers of the language to list all the result of voting on the string thisisacat could be repre- letters that could appear next in the sequence (today called sented as t0h0i1s3i1s4a4c1a0t, where the numbers
the successor count), the points where the number increases between letters are the total votes cast to split at the corre- often correspond to morpheme boundaries. Tanaka-Ishii and Jin correctly recognized that this idea was an early version With vote totals in place, VE segments at locations that of boundary entropy, one of the experts in VE. They de- meet two requirements: First, the number of votes must signed their PtM algorithm based on boundary entropy in be locally maximal (this is called the zero crossing rule).
both directions (not merely the forward direction, as in VE), Second, the number of votes must exceed a threshold.
and PtM was able to achieve scores similar to those of VE Thus, VE has three parameters: the window size, the vote on word segmentation in phonetically-encoded English and threshold, and whether to enforce the zero crossing rule.
Chinese. PtM can be viewed as detecting an information- For further details of the VE algorithm see Cohen et al.
theoretic signature similar to that of VE, but relying only on boundary entropy and detecting change-points in the abso- unsupervised version to the algorithm, which sets its own lute boundary entropy, rather than local maxima in the stan- parameters, is described briefly later in the paper.
Also within the morphology domain, Johnson and Mar- tin’s HubMorph algorithm constructs a trie from a Some of the best unsupervised sequence-segmentation re- set of words, and then converts it into a DFA by the pro- sults in the literature come from the family of algorithms cess of minimization. Within this DFA, HubMorph searches derived from VE. At an abstract level, each member of the for stretched hubs, which are sequences of states in the DFA family introduces an additional expert that refines or gener- that have a low branching factor internally, and high branch- alizes the boundary information produced by the two origi- ing factor at the edges (shown in Figure This is a nearly nal VE experts to improve segmentation quality. Extensions identical chunk signature to that of VE, only with succes- to VE include Markov Experts Hierarchical Vot- sor/predecessor count approximating boundary entropy. The ing Experts - 3 Experts (HVE-3E) and Bootstrap generality of this idea was not lost on Johnson and Martin, either: Speaking with respect to the morphology problem, The first extension to VE introduced a “Markov Expert,” Johnson and Martin close by saying “We believe that hub- which treats the segmentation produced by the original ex- automata will be the basis of a general solution for Indo- perts as a data corpus and analyzes suffix/prefix distributions European languages as well as for Inuktitut.” within it. Boundary insertion is then modeled as a Markovprocess based on these gathered statistics. HVE-3E is sim-pler: The third expert votes whenever it recognizes an entirechunk found by VE on the first iteration.
The new expert in BVE is called the knowledge expert.
The knowledge expert has access to a trie (called the knowl-edge trie) that contains boundaries previously found by thealgorithm, and votes to place boundaries at points in thesequence that are likely to be boundaries given this in- Figure 1: The DFA signature of a hub (top) and stretched formation. In an unsupervised setting, BVE generates its hub in the HubMorph algorithm. Figure from Johnson and own supervision by applying the highest possible confidence threshold to the output of VE, thus choosing a small, high-precision set of boundaries. After this first segmentation,BVE repeatedly re-segments the corpus, each time con-structing the knowledge trie from the output of the previous iteration, and relaxing the confidence threshold. In this way,BVE starts from a small, high-precision set of boundaries To demonstrate the domain-independent chunking ability and grows it into a larger set with higher recall.
of VE, we now survey a variety of domains to which VEhas been successfully. Some of these results appear in the literature, others are new and help to explain previous re- While Cohen and Adams were the first to formulate sults. Unless otherwise noted, segmentation quality is mea- the information-theoretic signature of chunks that drives VE, sured by the boundary F-measure: F = (2 × Precision × similar ideas abound. In particular, simpler versions of the Recall)/(Precision+Recall), where precision is the percent- chunk signature have existed within the morphology domain age of the induced boundaries that are correct, and recall is the percentage of the correct boundaries that were induced.
not been examined previously. Morph segmentation is a VE and its variants have been tested most extensively in lin- harder task to evaluate than word segmentation, because guistic domains. Language arguably contains many levels intra-word morph boundaries are typically not indicated of chunks, with the most natural being the word. The word when writing or speaking. We constructed a gold standard segmentation task also benefits from being easily explained, corpus of Latin text segmented into morphs with the mor- well-studied, and having a large amount of gold-standard data available. Indeed, any text can be turned into a cor-pus for evaluating word segmentation algorithms simply by have been reported in nearly every VE-related paper, and so is the most general comparison that can be drawn. This cor-pus is the first 50,000 characters of George Orwell’s 1984.
Table shows the aggregated results for VE and its deriva- Table 2: Morph-finding results by algorithm. All Points is a baseline that places a boundary at every possible location.
From the table above (Table it is clear that VE in its standard form has some difficulty finding the correct morphs. Still, its performance is comparable to PtM on this task, as expected due to the similarity in the two al- gorithms. PtM’s advantage probably is due to its bidirec- tionality: VE only actually examines the boundary entropy at the right (forward) boundary. VE was modified with theaddition of an expert that places its votes before sequencesthat have high boundary entropy in the backward direction.
Table 1: Results for VE and VE variants for word segmen- This bidirectional version of VE, referred to as BidiVE, is a more faithful implementation of the idea that chunks aresequences with low internal entropy and high boundary en- Similar results can be obtained for different underlying tropy. BidiVE performed better than VE at finding morphs languages, as well as different writing systems. Hewlett and Cohen showed similar scores for VE in Latin (F=0.772) and For reference, when the task is to find word boundaries, German (F=0.794) texts, and also presented VE results for the F-score for VE is approximately 0.77 on this same cor- word segmentation in orthographic Chinese (“Chinese char- pus. The reason for this is somewhat subtle: Because VE acters”). VE achieved an F-score of 0.865 on a 100,000 only looks at entropy in the forward direction, it will only word section of the Chinese Gigaword Corpus.
consider the entropy after a morph, not before it. Consider The higher score for Chinese than for the other languages a word like senat.us: The entropy of the next character has a simple explanation: Chinese characters correspond following senat is actually fairly low, despite the fact that roughly to syllable-sized units, while the letters in the Latin it is a complete morph. This is because the set of unique alphabet correspond to individual phonemes. By grouping endings that can appear with a given stem like senat is letters/phonemes into small chunks, the number of correct actually fairly small, usually less than ten. Furthermore, in boundary locations remains constant, but the number of po- any particular text a word will only appear in certain syntac- tential boundary locations is reduced. The means that even a tic relationships, meaning the set of endings it actually takes baseline like All Locations, which places a boundary at ev- will be smaller still. However, the entropy of the character ery possible location, will perform better when segmenting preceding us is very high, because us appears with a large a sequence of syllables than a sequence of letters.
number of stems. This fact goes unnoticed by VE.
VE has also been tested on phonetically-encoded English, in two areas: First, transcripts of of child-directed speech dence relevant to an important debate within the child lan- from the CHILDES database Second, on a phone- guage learning literature: How do children learn to seg- mic encoding of 1984 produced with the CMU pronounc- ment the speech stream into words? Famously, Saffran et ing dictionary. On the CHILDES data, VE was able to find al. showed that 8-month-old infants were able to word boundaries as well or better (F=0.860) than several distinguish correctly and incorrectly segmented words, even other algorithms, even though the other algorithms require when those words were nonsense words heard only as part their inputs to be sequences of utterances from which in- of a continuous speech stream. This result challenges mod- formation about utterance beginnings and endings can be els of word segmentation, such as Brent’s MBDP-1 gathered VE achieved an F-score of 0.807 on the which cannot operate without some boundary information.
Saffran et al. proposed that children might segment continu- While the word segmentation ability of VE ous sequences at points of low transitional probability (TP), has been studied extensively, its ability to find morphs has the simplest method which would successfully segment their robot wandered around a large playpen for 20-30 minutes However, TP alone performs very poorly on natural lan- looking for interesting objects, which it would orbit for a guage, a fact which has not escaped opponents of the view few minutes before moving on. At one level of abstraction, that word segmentation is driven by distributional properties the robot engaged in four types of behaviors: wandering, rather than innate knowledge about language. Linguistic na- avoiding, orbiting and approaching. Each behavior was im- tivists such as Gambell and Yang argue that this plemented by sequences of actions initiated by controllers failure of TP to scale up to natural language suggests that such as move-forward and center-camera-on-object. The the statistical segmentation ability that children possess is challenge for Voting Experts was to find the boundaries of limited and likely orthogonal to a more powerful segmenta- the four behaviors given only information about which con- tion ability driven by innate linguistic knowledge. Gambell and Yang demonstrate that an algorithm based on linguis- This experiment told us that the encoding of a sequence tic constraints (specifically, constraints on the pattern of syl- matters: When the coding produced shorter behaviors (aver- lable stress in a word) significantly outperforms TP when age length of 7.95 time steps), VE’s performance was com- segmenting a corpus of phonetically-encoded child-directed parable to that in earlier experiments (F=0.778), but when speech. In fact, VE can further outperform Gambell and the coding produced longer behaviors, performance is very Yang’s method (F=0.953 vs. F=0.946) even though VE has much worse (F=0.183). This is because very long episodes no prior knowledge of linguistic constraints, suggesting that are unique, so most locations in very long episodes have zero adding innate knowledge may not be as useful as simply in- boundary entropy and frequency equal to one. And when the creasing the power of the chunking method.
window size is very much smaller than the episode length, Algorithms like VE and PtM provide a counter-argument then there will be a strong bias to cut the sequence inappro- to the nativist position, by fully explaining the results that Saffran et al. observed, and also performing very well at seg-menting natural language. When represented symbolically as a sequence of phonemes, VE perfectly segments the sim- The goal of the DARPA’s Bootstrapped Learning (BL) ple artificial language generated by Saffran et al. project is to develop an “electronic student” that can be in- while also performing well in the segmentation of child- structed by human teachers, in a natural manner, to perform directed speech. Miller et al. reinforce this case complex tasks. Currently, interaction with the electronic stu- by replicating the experimental setup of Saffran et al., but dent is not very different from high-level programming. Our feeding the speech input to VE instead of a child. The audio goal is to replace many of the formal cues or “signposts” that signal had to be discretized before VE could segment it, but enable the electronic student to follow the teacher, making VE was able to achieve an accuracy of 0.824.
the interaction between them more natural. VE can largelyreplace one of these cues: the need to inform the student whenever the teacher’s instruction method changes.
Miller and Stoytchev applied VE in a hierarchical In BL, teachers communicate with the student in a lan- fashion to perform a visual task similar to optical charac- guage called Interlingua language (IL). Some IL messages ter recognition (OCR). The input was an image containing serve only to notify the student that a “Lesson Epoch” (LE) words written in a particular font. VE was to first segment this image into short sequences corresponding to letters, and Several curricula have been developed for BL. VE finds then chunk the short sequences into longer sequences cor- LE boundaries with high accuracy in all of them – and can responding to words. The image was represented as a se- be trained on one and tested on another to good effect. To quence of columns of pixels, where each pixel was either illustrate, we will present results for the Unmanned Aerial black or white. Each of these pixel columns can be repre- Vehicle (UAV) domain. To study the detection of LE bound- sented by a symbol denoting the particular pattern of black aries, a training corpus was generated from version 2.4.01 and white pixels within it, thus creating a sequence of sym- of the UAV curriculum by removing all of the messages that bols to serve as input to VE. Depending on the font used, VE indicate boundaries between LEs. This training corpus con- scored between F=0.751 and F=0.972 on segmenting this tains a total of 742 LEs. A separate corpus consisting of 194 LEs served as a test corpus. As the teacher should never have After finding letters, VE had to chunk these letters to- to provide LE boundaries, the problem is treated as unsuper- gether into words, which is essentially the same as the well- vised and both the training and test corpora are stripped of studied word segmentation problem except with some noise added to the identification of each character. VE was still Each individual message in the corpus is a recursive struc- able to perform the task, with scores ranging from F=0.551 ture of IL objects that together express a variety of relations to F=0.754 for the three fonts. With perfect letter identifica- about the concepts being taught and the state of teaching.
LEs are defined more by the structure of the message se-quence than the full content of each message. Thus, we rep- resent each message as a single symbol, formed by concate- Cohen et al. tested VE on data generated by a nating the IL type of the two highest composite IL objects mobile robot, a Pioneer 2 equipped with sonar and a pan- (generally equivalent to the message’s type and subtype).
tilt-zoom camera running a subsumption architecture. The The sequence of structured messages is thus translated into Though the success of VE in a given domain provides in-direct evidence that the chunk signature successfully iden- tifies chunks in that domain, we can evaluate the validity of the chunk signature much more directly. To evaluate the ability of the chunk signature to select the true segmentation from among all possible segmentations of a given sequence,we developed a “chunkiness” score that can be assigned to Table 3: BVE Results on UAV Domain trained on different each possible segmentation, thus ranking all possible seg- subsets of the training corpus. “Size” is percentage of the mentations by the quality of the chunks they contain. The chunkiness score rewards frequent sequences that have highentropy at both boundaries (Equation just as in VE. Thescore for a complete segmentation is simply the average ofthe chunkiness of each segment. If the chunk signature is a sequence of symbols, and it is this symbol sequence that correct, the true segmentation should have a very high score, and so will appear close to the top of this ranking. Unfor- BVE is allowed to process the training corpus repeatedly tunately, due to the exponential increase in the number of to gather statistics and segment it, but the segmentation of segmentations (a sequence of length n has 2n−1 segmenta- the test corpus must be done in one pass, to model more tions), this methodology can only be reasonably applied to closely the constraints of a real teacher-student interaction.
short sequences. However, it can be applied to many such If allowed to operate on the full UAV corpus, BVE finds LE short sequences to better gain a better estimate of the de- boundaries handily, achieving an F-score of 0.907. How- gree to which optimizing chunkiness optimizes segmenta- ever, this domain is non-trivial: VE achieves an F-score of 0.753, only slightly lower than its score for word segmenta-tion in English text. As a baseline comparison, segmenting the corpus at every location results in an F-score of 0.315, which indicates that LE boundaries are roughly as frequent For each 5-word sequence (usually between 18 and 27 as word boundaries in English, and thus that high perfor- characters long) in the Bloom73 corpus from CHILDES, we mance is not guaranteed simply by the frequency of bound- generated all possible segmentations and ranked them all by chunkiness. On average, the true segmentation was in the Results from segmenting a test corpus (not drawn from 98.7th percentile. All probabilities needed for computing the training corpus) consisting of 194 lesson epochs are the chunkiness score were estimated from a training corpus, shown in Table “Training Size” refers to the percentage the Brown73 corpus (also from CHILDES). Preliminarily, it of the training corpus processed by BVE before segmenting appears that syntax is the primary reason that the true seg- the test corpus. From these results, it is evident that BVE mentation is not higher in the ranking: When the word-order can perform very well on a new corpus when the training in the training corpus is scrambled, the true segmentation is corpus is sufficiently large. However, with a small training in the 99.6th percentile. Still, based on these early results we corpus BVE does not encounter certain boundary situations, can say that, in at least one domain, optimizing chunkiness and thus fails to recognize them during the test, resulting in very nearly optimizes segmentation quality.
Automatic Setting of ParametersVE has tunable parameters, and Hewlett and Cohen showed that these parameters can greatly affect perfor-mance. However, they also demonstrated how these pa- So far, we have discussed in detail one kind of evidence for rameters can be tuned without supervision. Minimum De- the general applicability of VE, namely that VE success- scription Length (MDL) provides an unsupervised way to fully performs unsupervised segmentation in a wide variety set these parameters indirectly by selecting among the seg- of domains. In order for VE to be successful in a given do- mentations each combination of parameters generates. The main, chunks must exist in that domain that adhere to the Description Length for a given hypothesis and data set refers VE’s signature of chunks, and VE must correctly identify to the number of bits needed to represent both the hypoth- these chunks. Thus, the success of VE in each of these esis and the data given that hypothesis. The Minimum De- domains is evidence for the presence of chunks that ad- scription Length, then, simply refers to the principle of se- here to the signature in each domain. Also, VE’s chunk lecting the hypothesis that minimizes description length. In signature is similar to (or a direct generalization of) sev- this context, the data is a corpus (sequence of symbols), and eral other independently-developed signatures, such as PtM, the hypotheses are proposed segmentations of that corpus, HubMorph, and the work of Kadir and Brady The each corresponding to a different combination of parameter independent formulation of similar signatures by researchers settings. Thus, we choose the vector of parameter settings working in different domains suggests that a common prin- that generates the hypothesized segmentation which has the ciple is at work across those domains.
Strictly speaking, VE can only operate over sequences of [Bre99] Michael R Brent. An Efficient, Probabilistically Sound Algorithm for Segmentation and Word Discovery.
Miller et al.’s applications of VE to the visual and auditory Machine Learning, pages 71–105, 1999.
domains, many sequences of multivariate or continuous- [CA01] P Cohen and N Adams. An algorithm for segment- valued data can be transformed into a symbolic representa- ing categorical time series into meaningful episodes. Lec- tion for VE. Also, the SAX algorithm provides ture notes in computer science, 2001.
a general way to convert a stream of continuous data into a [CAH07] Paul Cohen, Niall Adams, and Brent Heeringa.
Voting Experts: An Unsupervised Algorithm for Segment-ing Sequences.
[CM05] Jimming Cheng and Michael Mitzenmacher. The While the ability of VE to operate in a fully unsupervised Markov Expert for Finding Episodes in Time Series. In setting is certainly a strength, the fact that VE contains no Proceedings of the Data Compression Conference (DCC natural mechanism for incorporating supervision may be seen as a limitation: If some likely examples of ground truthboundaries are available, the algorithm ought to be able to [GY05] Timothy Gambell and Charles Yang. Word Seg- take advantage of this information. While VE itself cannot mentation: Quick but not Dirty. 2005.
benefit from true boundary knowledge, one of its extensions, BVE, does so handily. BVE’s knowledge trie can store pre- viously discovered boundaries (whether provided to or in- [HC09] Daniel Hewlett and Paul Cohen. Bootstrap Vot- ferred by the algorithm), and the knowledge expert votes for ing Experts. In Proceedings of the Twenty-first Interna- boundary locations that match this prior knowledge. The tional Joint Conference on Artificial Intelligence (IJCAI- Markov Experts version is able to benefit from supervision in a similar way, and, if entire correct chunks are known, [JM03] Howard Johnson and Joel Martin. Unsupervised learning of morphology for English and Inuktitut. Proceed-ings of the 2003 North American Chapter of the Associ- ation for Computational Linguistics on Human LanguageTechnology (NAACL-HLT 03), pages 43–45, 2003.
VE does not represent explicitly a “lexicon” of chunks thatit has discovered. VE produces chunks when applied to a [KB01] Timor Kadir and Michael Brady. Saliency, Scale sequence, but its internal data structures do not represent the and Image Description. International Journal of Computer chunks it has discovered explicitly. By contrast, BVE stores boundary information in the knowledge trie and refines it [LKWL07] Jessica Lin, Eamonn Keogh, Li Wei, and Ste- over time. Simply by storing the beginnings and endings fano Lonardi. Experiencing SAX: a novel symbolic rep- of segments, the knowledge trie comes to store sequences resentation of time series. Data Mining and Knowledge like #cat#, where # represents a word boundary. The set Discovery, 15:107–144, April 2007.
of such bounded sequences constitutes a simple, but accu- [MS85] Brian McWhinney and Cynthia E. Snow. The child rate, emergent lexicon. After segmenting a corpus of child- language data exchange system (CHILDES). Journal of directed speech, the ten most frequent words of this lexicon are you, the, that, what, is, it, this, what’s, to, and look. Of [MS08] Matthew Miller and Alexander Stoytchev. Hierar- the 100 most frequent words, 93 are correct. The 7 errors chical Voting Experts: An Unsupervised Algorithm for Hi- include splitting off morphemes such as ing, and merging erarchical Sequence Segmentation. In Proceedings of the frequently co-occurring word pairs such as do you.
7th IEEE International Conference on Development andLearning (ICDL 2008), pages 186–191, 2008.
[MWS09] Matthew Miller, Peter Wong, and Alexander Stoytchev. Unsupervised Segmentation of Audio Speech Chunking is one of the domain-independent cognitive abili- Using the Voting Experts Algorithm. Proceedings of the ties that is required for general intelligence, and VE provides 2nd Conference on Artificial General Intelligence (AGI a powerful and general implementation of this ability. We have demonstrated that VE and related algorithms performwell at finding chunks in a wide variety of domains, and pro- [SAN96] Jenny R Saffran, Richard N Aslin, and Elissa L vided preliminary evidence that chunks found by maximiz- Newport. Statistical Learning by 8-Month-Old Infants. Sci- ing chunkiness are almost always real chunks. This suggests that the information theoretic chunk signature that drives VE [TIJ06] Kumiko Tanaka-Ishii and Zhihui Jin.
is not specific to any one domain or small set of domains.
Phoneme to Morpheme: Another Verification Using a Cor- We have discussed how extensions to VE enable it to operate pus. In Proceedings of the 21st International Conference over nearly any sequential domain, incorporate supervision on Computer Processing of Oriental Languages (ICCPOL when present, and tune its own parameters to fit the domain.
2006), volume 4285, pages 234–244, 2006.
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