A Parsing: Fast Exact Viterbi Parse Selection

Transcription

1 A Parsing: Fast Exact Viterbi Parse election Dan Klein and Christopher D Manning Compter cience Department tanford University tanford CA klein Paper ID: P0348 Keywords: Contact Athor: Dan Klein (klein@csstanforded) Under consideration for other conferences (specify) None The athors have written a previos paper (IWPT2001) in which the correspondence between parsing and hypergraph analysis was detailed and sed to constrct and prove correct a Viterbi parser The application of A estimates to redcing parser work is entirely new as are all of the estimates introdced experiments on their performance and data analysis Abstract A PCFG parsing can dramatically redce the time reqired to find the exact Viterbi parse by conservatively estimating otside Viterbi probabilities We discss varios estimates and give efficient algorithms for compting them On Penn treebank sentences or most detailed estimate redces the total nmber of edges processed to less than 3% of that reqired by exhastive parsing and even a simpler estimate which can be pre-compted in nder a minte still redces the work by a factor of 5 The algorithm extends the classic A graph search procedre to a certain hypergraph associated with parsing Unlike best-first and finite-beam methods for achieving this kind of speed-p the A parser is garanteed to retrn the most likely parse not jst an approximation The algorithm is also correct for a wide range of parser control strategies and maintains a worst-case cbic time bond

2 A Parsing: Fast Exact Viterbi Parse election Paper ID: P0348 Abstract A PCFG parsing can dramatically redce the time reqired to find the exact Viterbi parse by conservatively estimating otside Viterbi probabilities We discss varios estimates and give efficient algorithms for compting them On Penn treebank sentences or most detailed estimate redces the total nmber of edges processed to less than 3% of that reqired by exhastive parsing and even a simpler estimate which can be pre-compted in nder a minte still redces the work by a factor of 5 The algorithm extends the classic A graph search procedre to a certain hypergraph associated with parsing Unlike best-first and finite-beam methods for achieving this kind of speedp the A parser is garanteed to retrn the most likely parse not jst an approximation The algorithm is also correct for a wide range of parser control strategies and maintains a worst-case cbic time bond 1 Introdction PCFG parsing algorithms with worst-case cbictime bonds are well-known However when dealing with wide-coverage grammars and long sentences even cbic algorithms can be far too expensive in practice Two primary methods of accelerating parse selection have been proposed Roark (2001) and Ratnaparkhi (1999) se a beam-search strategy in which only the best parses are tracked at any moment Parsing time is linear and can be made arbitrarily fast by redcing This amonts to a greedy strategy and like any greedy strategy the actal Viterbi parse can be prned from the beam becase thogh globally optimal it is not locally optimal at every parse stage Chitrao and Grishman (1990) Caraballo and Charniak (1998) and Charniak et al (1998) describe best-first parsing which is intended for a tablar item-based framework In best-first parsing one bilds a figre-ofmerit (FOM) over parser items and ses the FOM to decide the order in which agenda items shold be processed This approach also dramatically redces the work done dring parsing We discss best-first parsing frther in section 31 Note that it does not garantee that the first parse retrned is the actal Viterbi parse (nor does it preserve the worst-case cbic time bond) oth of these speed-p techniqes are based on greedy models of parser actions The beam search makes greedy selections at each parse step and a best-first FOM greedily compares parse items However if we view parsing as best-path finding in a certain hypergraph then the natral way to avoid processing bad edges is to consider both the distance from the start point (the sentence) to an edge (the Viterbi inside score of that edge) and the distance remaining to the goal (the Viterbi otside score of that edge) We propose an A algorithm (Rssell and Norvig 1995) which relies on combining a known best inside score of an edge with a conservative estimate of the otside score The A formlation provides two benefits First it sbstantially redces the work reqired to parse a sentence withot sacrificing either the optimality of the answer or the worst-case cbic time bonds on the parser In section 3 we describe the estimates that we se along with algorithms to efficiently calclate them and illstrate their effectiveness in parsing sentences from the Penn Treebank econd it allows s to easily prove the correctness of or algorithm over a broad range of control strategies rle encodings and inpt lattices 2 An A Algorithm The parser is an agenda-based parser and operates on edges sch as :[02] which represent a category over a span It maintains two data strctres: a chart or table which records edges for which (best) 1

3 :[0n] X words (a) :[02] :[03] DT:[01] NN:[12] VZ:[23] (b) :[02] Figre 1: A edge costs The cost of an edge is a combination of the cost to bild the edge (the Viterbi inside score) and the cost to incorporate it into a root parse (the otside Viterbi score) We will constrct exact vales for the inside scores and pper bonds on the otside scores parses have already been fond and an agenda of newly-formed edges waiting to be processed The core loop as in any agenda-based tablar parser involves removing an edge from the agenda and combining that edge with other edges to potentially create new edges For example :[02] might be removed from the agenda and if :[28] is already in the table the edge :[08] will be formed and added to the agenda if it is not in the chart already Psedocode is shown in figre 10 The way in which this algorithm differs from a classic chart parser is that as in best-first parsing agenda items are processed according to an edge priority In best-first parsing this priority is called a figre-of-merit (FOM) and is based on varios approximations to the (conditional) inside probability which is the fraction of parses of which inclde Edges which are in few parses of are left to wait on the agenda indefinitely For pars-! " ing this figre is heristic; even if we know exactly we do not know whether occrs in any best parse of Nonetheless a good FOM leads qickly to good parses becase of the strong correlation between smmed inside scores and Viterbi inside scores est-first parsing aims to find a good parse qickly bt gives no garantee that the first parse discovered is the Viterbi parse nor does it allow one to verify a Viterbi parse withot exhastive parsing In A parsing we wish to constrct priorities which will speed p parsing bt still garantee that the first parse retrned is indeed a best parse With a categorical CFG rn to exhastion it does not matter in what order one removes edges from the agenda; all edges involved in fll parses of the sentence will be constrcted at some point With PCFG parsing there is a sbtlety involved We want to score edges with best-parse score estimates as we go If dring parsing we find a better way to constrct some edge that has already been entered into the chart we have also fond a better way to constrct all edges which were bilt from estfirst parsers can deal with this by having an pward propagation step which pdates sch edges scores (Caraballo and Charniak 1998) If rn to exhastion all edges Viterbi scores will be correct bt the propagation destroys the cbic time bond of the parser In order to ensre correctness it is sfficient that all edges # which are contained in a best parse of an edge get processed before itself If we have a priority over edges which ensres this property we can avoid pward propagation and be sre that each edge leaves the agenda with its correct Viterbi score If the grammar is in CNF then letting shorter spans have higher priority than longer ones works; this priority essentially gives the CKY algorithm t one needs non-trivial modifications to handle grammars with nary -ary and empty prodctions and cannot incorporate non-bottom-p control If we se the crrent best known Viterbi score of an edge as that edge s priority Klein and Manning (2001a) show that one gets a correct algorithm over arbitrary PCFGs and a wide range of control strategies This is proved sing an extension of Dijkstra s algorithm to a certain kind of hypergraph associated with parsing shown in figre 1(b) We do not present the fll details here bt the correspondence shold be clear enogh: parser edges are nodes in the hypergraph hyperarcs take sets of edges to their reslt edge and paths map to parses Reachability from %$'&)( $ corresponds to parseability and shortest paths to Viterbi parses The graph shown is a parse of the goal :[03] which incldes :[02] This path or parse can be split into an inside portion (solid lines) and otside portion (dashed lines) as indicated in figre 1(a) The algorithm corresponds to a generalization of niform cost search (Rssell and Norvig 1995) with the backward cost (the cost of the solid path) being the inside Viterbi score of an edge Using an analogos generalization of A search we can also 2

4 & Estimate X XL XLR TRUE mmary (16) (16VZ) (16VZ ) (entire context) est Tree PP IN DT JJ NN VD VZ IN VZ PP DT N N N N VZ CC VZ DT JJ NN VZ VZ PRP VZ DT NN PRP VZ core (a) (b) (c) (d) Figre 2: est otside parses given richer smmaries of edge context (a X) Knowing only the edge state () and the left and right otside spans (b XL) also knowing the left tag (c XLR) knowing the left and right tags and (d TRUE) knowing the entire otside context DT NN se estimates & of the forward cost + (the cost of the dashed path) to focs exploration on regions of the graph which appear to have good total cost A is correct as long as & is satisfies two conditions First it mst be admissible meaning that it mst not overestimate the actal cost to the goal econd it mst be monotonic meaning that as one explores a path to the goal the combined cost - does not decrease For parsing this means we can add any admissible monotonic estimate & of + to or crrent estimate of We will still have a correct algorithm and even rogh heristics can dramatically ct down the nmber of edges processed and therefore the total work involved in parsing We present several estimates describe how to pre-compte them efficiently and show the edge savings when parsing Penn treebank WJ sentences We also sketch proofs of correctness for A parsing 3 A Estimates for Parsing When parsing each edge spans some part / of the sentence Its yield is the terminals in the sentence it spans Its context is the rest of the sentence We called the spans / and / of the context the left and right span respectively or the split otside span as a pair Their sm is the combined otside span For concreteness all scores will be log-probabilities and lower cost is higher logprobability To avoid confsion : or better will mean higher probability One way to constrct an admissible estimate is to smmarize the context in some way and to find the score of the best parse of any context that fits that smmary If we give no information in the smmary the estimate will be 0 This is the trival estimate NULL and corresponds to simply sing inside scores alone as priorities On the other hand the ideal estimate for & is + the tre otside Viterbi score of that exact context which we call TRUE It wold be impractical to precompte TRUE in practice bt we can still find its vale for any given context by exhastive parsing It is worth noting that or ideal estimate is not like the ideal FOM rather it is <;>=@ AC<;9AD where ;E=F A is a best parse of the goal G among those which contain and ;9A is a best parse of over the yield of We sed varios smmaries some illstrated in figre 2 H consists of only the combined otside span while is the split otside span X also incldes s label XL and XR add the tags adjacent to on the left and right respectively H XLR incldes both the left and right tags bt ses the combined otside span This was done solely to redce memory sage; it is no qicker to calclate with combined otside span than split otside span bt more compact to store As the smmaries become richer the estimates become sharper As an example consider an in the context VZ PRP VZ DT NN shown in figre 2 1 The smmary X tells s only that there is an with 6 words to the left and 1 to the right and gives an estimate of IKJLJLMN This score is backed by the concrete parse shown in figre 2(a) Now this is a best parse of a context compatible with what little we specified bt very optimistic It assmes very common tags in very common patterns XL adds that the tag to the left is VZ and the hope 1 Or examples and or experiments sed delexicalized sentences from the Penn treebank 3

5 P P P F XL TRUE F XMLR X O NULL XR O XLR Figre 3: A estimates form a lattice Lines indicate sbsmption Q indicates estimates which are the explicit join of lower estimates that the is part of a sentence-initial PP mst be abandoned; the best score drops to IRJFN!M backed by the parse in figre 2(b) pecifying the right tag to be drops the score frther to IKJ@T)MJ given by figre 2(c) The actal best parse is figre 2(d) with a score of IRJFU!MJ All of the smmaries described above se somewhat local information combined with spans F is a less local estimate It tracks for each state what tags mst occr to the left (and in what seqence) in order for that state to be reached For example a state labeled V IN W cannot occr nless there is actally an IN somewhere to the left and possibly the IN mst be at least one terminal away (if there are no empties in the grammar) The estimate based on F retrns IYX if the reqired tags do not exist and 0 otherwise Unlike the other estimates it either rles an edge ot entirely or says nothing abot it A standard chart parser with top-down control will render F redndant by never constrcting an edge nless the reqisite tags have already been fond and added to the chart It is only sefl for these experiments becase we se a tre bottom-p parsing scheme to separate ot the A effects from gains gotten by selective parser control Rnning the other estimates along with top-down control is qantitatively and qalitatively similar to combining them with F F can be compted very efficiently at parse-time for each left span and edge label the rest are precompted and reqire constant access time to retrieve ection 32 discsses the precomtation time vs benefit trade-offs involved These estimates form a join lattice illstrated in figre 3; adding context information sbdivides the Original Rles D-Trie Rles N-Trie Rles Z DT JJ NN 03 Z DT X [ DT\ 04 Z DT X JJ NN 03 Z DT NN NN 01 X [ DT\ Z JJ NN 075 X JJ NN Z JJ NN 10 X [ DT\ Z NN NN 025 X NN NN Z NN NN 10 Edges locked Figre 4: Two trie encodings of rles NULL pan X XR 1XLR XL XMLR D-Tries N-Tries Figre 5: Fraction of edges saved by sing varios estimate methods for two rle encodings D-TRIE is a deterministic right-branching trie encoding with weights pshed left N-TRIE is a non-deterministic left-branching trie with weights on rle entry as in (Charniak et al 1998) partitions and can only sharpen the individal otside estimates In this sense for example ] X The lattice top is TRUE and the bottom is NULL In addition the maximm of a set of admissible estimates is still an admissible estimate We can se this to combine or basic estimates into composite estimates: XMLR = ^ (XL XR) will be vaild and a better estimate than either XL or XR individally imilarly is ^ (XMLR H XLR) 31 Parsing Performance After (Charniak et al 1998) we parsed nseen sentences of length from the Penn Treebank sing the grammar indced from the remainder of the treebank We tried all estimates described above Rles were encoded as both deterministic (D) and non-deterministic (N) tries shown in figre 4 ch an encoding binarizes the grammar and compacts it N-tries are as in (Charniak et al 1998) where V DT JJ NN becomes V DT X JJ NN and X JJ NN V JJ NN D-tries as in (Klein and Manning 2001b) trn V DT JJ NN into V DT X _ DT ` and X _ DT ` V JJ NN Figre 5 shows the overall savings for several estimates for each type The N-tries were sperior for the coarser estimates while D-tries were sperior for the finer estimates In addition D-tries have a simpler (and more effective) version of F: X _ DT ` clearly mst have a DT somewhere to the left while X JJ NN might be compatible with any context depending on the rest 4

6 entences Parsed 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% Edges Processed F XF XL XR X Figre 6: Nmber of sentences parsed as more edges are expanded entences are Penn treebank sentences of length parsed with the treebank grammar A typical nmber of edges in an exhastive parse is Even relatively simple Aa estimates allow sbstantial savings of the grammar Additionally with N-tries only the top-level rle has probability less than 1 while for D-tries one can back-weight probability as in (Mohri 1997) also shown in figre 4 enabling sbparts of rare rles to be penalized even before they are completed For all ftre reslts we discss only the D-trie nmbers Figre 8 lists the overall savings for each estimate with and withot the non-local F Comparing the A estimates we see that the NULL estimate is not very effective alone it only blocks 13% of the edges It is still better than exhastive parsing: with it one stops parsing when the best parse is fond while in exhastive parsing one contines ntil no edges remain ince in all cases we stop when the &b" combined score :dc NULL can still exploit For even the simplest non-trivial otside estimate already 40% of the edges have been blocked and the best estimate F blocks over 97% of the edges a speed-p of over 35 times withot sacrificing optimality or algorithmic complexity For comparison to previos FOM work figre 6 shows for an edge cont and an estimate the proportion of sentences for which a first parse was fond sing at most that many edges To sitate or reslts the FOMs sed by (Caraballo and Charniak 1998) reqire 10K edges to parse 96% of these sentences while F reqires only 6K edges On the other hand the tned more complex FOM in (Charniak et al 1998) is able to parse all of these sentences sing arond 2K edges while F reqires 7K edges In short or estimates do not redce the total edge cont qite so far as the best FOMs can bt are in the same range Crcially however or first parses are always optimal while the FOM parses need not be and are not 2 Also or parser never needs to propagate score changes pwards and so may be expected to do less work overall per edge all else being eqal In is interesting that XL is so mch more effective than XR; this is primarily becase of the way that the rles have been encoded If we factor the rles in the other direction we get the opposite effect Also when combined with F the difference in their performance drops from 103% to 08%; F is a left-filter and is partially redndant when added to XL bt orthogonal to XR 32 Estimate Comptation The amont of work reqired to calclate A estimates depends on how easy it is to efficiently take the max over all parses compatible with each context smmary The benefit provided by an estimate will depend on how well the restrictions in that smmary nail down the important featres of the fll context We do not claim that the estimates above are the best possible at the latter one cold easily imagine featres of the context sch as nmber of verbs to the right or nmber of nsal tags in the context which wold partition the contexts more effectively than adjacent tags To calclate these A estimates efficiently we sed a memoization approach rather than a dynamic programming approach This reslted in code which was simpler to reason abot and more importantly allowed s to exploit sparseness when present There are two kinds of sparseness The first is reqest spareseness: not all context signatres will be asked abot when parsing For example with right-factored trie encodings 76% of (state left tag) combinations are simply impossible The second is hierarchical sparsity For example 86% of the estimate vales in the XL tables were within a log-score of ef7mj of the vales of the X entries which sbsmed them This cold have been exploited sing a back-off method over the lattice where the most specific entry which was present was sed However the increase in space reqired 2 In fact the bias from the FOM commonly raises the bracket accracy slightly over the Viterbi parses bt that difference nevertheless demonstrates that the first parses are not always the Viterbi ones 5

7 otsidex(state lspan rspan) if (lspan+rspan == 0) if state is the top then 0 else gih score = gih % cold have a left sibling for sibsize in [0lspan-1] for (xj y state) in grammar cost = insidex(ysibsize)+ otsidex(xlspan-sibsizerspan) score = max(scorecost) % cold have a right sibling for sibsize in [0rspan-1] for (xj state y) in grammar cost = insidex(ysibsize)+ otsidex(xlspanrspan-sibsize) score = max(scorecost) retrn score; insidex(state span) if (span == 0) if state is a terminal then 0 else gih score = gih % choose a split point for split in [1span-1] for (statej x y) in grammar cost = insidex(xsplit)+insidex(yspan-split) retrn score; Figre 7: Psedocode for the X estimate in the case where the grammar is in CNF Other estimates and grammars are similar by the hashtables negated the tility of this back-off for or particlar estimate family For space reasons example code is only provided in figre 7 for the X estimate in the case where the grammar is in CNF bt the other cases are similar Tables which mapped argments to retrned reslts were sed to memoize each procedre In or experiments we forced these tables to be filled in a precomptation step bt depending on the sitation it might be advantageos to allow them to fill as needed with early parses proceeding slowly while the tables poplate The optimal forward estimate the actal distance to the closest goal wold mean that we never expand edges other than those in a best parse bt its comptation wold reqire parsing the rest of the sentence On the other hand no precomptation is needed for NULL In between is a tradeoff of space/time reqirements for precomptation and the online savings dring the parsing of new sentences Figre 8 shows the average savings verss the precomptation time 3 Where on this crve one chooses to be depends on many factors; 9 hors may 3 All times are for a Java implementation rnning on a 2G 700MHz Intel machine Estimate avings w/ Filter torage Precomp K 1 min X M 1 min XR M 30 min k XLR G 480 min XL M 30 min XMLR G 60 min G 540 min Figre 8: The trade-off between online savings and precomptation time be too mch to spend compting bt an hor for XMLR gives nearly the same performance and the one minte reqired for X is comparable to the I/O time taken to simply read the Penn treebank in or system 33 Estimate Effectiveness A disadvantage of admissible estimates is that necessarily they are overly optimistic as to the contents of the otside context The larger the otside context the farther the gap between the tre cost and the estimate Figre 9 shows average otside estimates for Viterbi edges as span size increases For small otside spans all estimates are fairly good approximations of TRUE As the span increases the approximations fall behind eyond the smallest otside spans all of the crves are approximately linear bt the actal vale s slope is roghly twice that of the estimates The gap between or empirical methods and the tre cost grows fairly steadily bt the differences between the empirical methods themselves stay relatively constant This reflects the natre of these estimates: they have differing local information in their smmaries bt all are eqally ignorant abot the more distant context elements The varios local environments can be more or less costly to integrate into a parse bt within a few words the local restrictions have been incorporated one way or another and the estimates are all free to be eqally optimistic abot the remainder of the parse The cost to package p the local restrictions creates their contant differences and the shared ignorance abot the wider context cases their same-slope linear drop-off 4 Correctness and Completeness Thogh not in terms of A search Klein and Manning (2001a) prove the correctness of the nll estimate for a range of grammars encodings control 6

8 # # & Average A Estimate Relative Average A Estimate Otside pan (a) Otside pan (b) X XR TRUE X XR XL TRUE Figre 9: (a) The average estimate by otside span length for varios methods (b) The average difference from the X method; for large otside spans the estimates differ by relatively constant amonts strategies and scan policies All of those proofs go throgh with any monotonic admissible score fnction added to the edge priorities For space reasons we omit general proofs and assme bottom-p control and at-most-binary grammar rles First we need to know that or estimates are admissible and monotonic Their admissibility shold be clear from their constrction: the max over all otside parses of all contexts compatible with a given smmary incldes the actal best otside parse of the actal context For monotonicity we need that if an edge # is contained in some best ;9A parse of some edge then &9 &9 ml This is also tre for all of or estimates since the max over all otside parses in- inclded all parses with &b volved in calclating # the particlar strctre otside # ;9A in For the following we say an edge is finished when it is entered into the chart Claim: (Correctness) When the algorithm above terminates every finished edge is correctly scored Proof: First we need to know that the inside estimates always nderestimate the tre best score: on " Leaf edges are correctly scored at initialization For all other edges let traversal( ) be the traversal which either discovered or last pdated its score by relaxation Then it is easy to see that we can take any edge at any time and se the recorded traversal vales to (recrsively) extract a concrete parse of with score That parse and therefore can be scored no better than a best parse of The rest of the proof is by contradiction Let be "qp r the first edge which is finished with sl " y or first fact it mst be that that ; it is scored too low t consider a best parse of ; The leaves of have all been finished ;ot becase s discovery means that some parse of has had all of its non-root nodes finished and all parses of share the same leaves Therefore there mst be at least one node ; in (possibly ) which has the property that both of its children #8H and #Cv have been finished bt itself has not been finished t whenever the latter of #8H and #Cv was finished was discovered (if it had not been earlier) and relaxed with the traversal wv #8Hf#Cv ince #8H and #Cv were finished before they were correctly scored at their finishing y first lemma they still are o it mst ; be that is correctly scored since its score in a best parse mst be its tre best score If is nfinished bt correctly scored when is removed from the agenda it mst be that &9 &9 :x We know that r &b at that time so r & &b y the monotonicity assmption on we also have &b &b" zl :y ince &9 &9 we know :m bt that is a contradiction Claim (Completeness): When the algorithm above terminates the goal has been finished if it is parsable Moreover all edges whose A score z is less than the score of a best parse of the goal have been finished and no edges whose A scores are more than the best parse score have been finished Proof: If the goal ; edge G is parsable then there is some concrete tree rooted at G All of the leaves are discovered dring initialization At termination the agenda is empty If G is not finished at that time it mst be that there is some lowest node ; in which is nfinished bt whose children are finished t the second child finished wold have triggered s discovery and mst have left the agenda for the agenda to be empty at termination a contradiction Moreover edges enter the agenda only once 7

9 G and there are finitely many edges so the algorithm will terminate For the rest of the claim agenda priorities at extraction are of the form & ince is ad- " &b" missable the priority of G at extraction mst have been Therefore it sffices to show that the priorities of the seqence of items extracted from the agenda is non-increasing & t this follows easily from the monotonicity of and argments similar to those above & Claim (Time and pace): If lookp is constant time the algorithm can be made to rn in amortized 4 ~} 4 time { and in space { v where is the nmber } of non-terminals (states) in the grammar and is maximm nmber of rles with any fixed right symbol on the RH We do not prove this; it is identical to the proof in Klein and Manning (2001a) These bonds will not be tre of a best-first parser (becase of the pward percolation) and if one wants to ensre a best parse is in a beam parsers reslts one mst have an exponentially wide beam (thogh of corse that wold defeat the point of the beam parser) 5 Conclsions We have demonstrated that A estimates can be sed to accelerate parsing work by orders of magnitde withot sacrificing either the garantee that the first retrned parse is the Viterbi parse or the worst-case asymptotic complexity of the parser On averagesize Penn treebank sentences the parser is capable of finding the Viterbi parse in a very few seconds by processing less than 3% of the edges which wold be constrcted by an exhastive parser To do this we sketched a view of parsing as hypergraph analysis which natrally leads to this A formlation rather than to a model which prnes at the parseraction level (sch as finite-beam or best-first parsing) and proved the correctness of or algorithm parse(sentence goal estimate) initialize(sentence goal) while the agenda is non-empty = nextedge(agenda estimate) finishedge( ) nextedge(agenda estimate) priority of is score( )+estimate( ) retrn the in agenda with best priority initialize(sentence goal) create a new chart and new agenda for each word w:[startend] in the sentence add w:[startend] to the agenda exploretraversal(traversal ) = reslt( ) if notyetdiscovered( ) add to the agenda relaxedge( ) relaxedge(edge Traversal ) newcore = combinecores( ) if (newcore is better than score( )) score( ) = score( ) besttraversal( ) = finishedge(edge ) add to the chart for all adjacent edges ƒ in the chart for all labels let = combine( ƒ ) if is valid exploretraversal( ) Figre 10: Psedocode for the Aa parser Natral Langage Workshop Hidden Valley PA pages Morgan Kafmann Dan Klein and Christopher D Manning 2001a Parsing and hypergraphs In Proceedings of the 7th International Workshop on Parsing Technologies (IWPT-2001) Dan Klein and Christopher D Manning 2001b Parsing with treebank grammars: Empirical bonds theoretical models and the strctre of the Penn treebank In ACL 39 pages Mehryar Mohri 1997 Finite-state transdcers in langage and speech processing Comptational Lingistics 23(4) Adwait Ratnaparkhi 1999 Learning to parse natral langage with maximm entropy models Machine Learning 34: rian Roark 2001 Probabilistic top-down parsing and langage modeling Comptational Lingistics 27: tart J Rssell and Peter Norvig 1995 Artificial Intelligence: A Modern Approach Prentice Hall Englewood Cliffs NJ References haron A Caraballo and Egene Charniak 1998 New figres of merit for best-first probabilistic chart parsing Comptational Lingistics 24: Egene Charniak haron Goldwater and Mark Johnson 1998 Edge-based best-first chart parsing In Proceedings of the ixth Workshop on Very Large Corpora pages Morgan Kafmann Mahesh V Chitrao and Ralph Grishman 1990 tatistical parsing of messages In Proceedings of the DARPA peech and 8