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<?xml version="1.0" standalone="yes"?> <Paper uid="C82-1054"> <Title>AN IMPROVED LEFT-CORNER PARSING ALGORITHM</Title> <Section position="3" start_page="333" end_page="333" type="metho"> <SectionTitle> 334 K,M. ROSS </SectionTitle> <Paragraph position="0"> If A is on top of stack alpha, B is on top of stack beta, C is on top of stack gamma, and &quot;Conditions&quot; are satisfied then replace A by D, B by E, and C by F.</Paragraph> </Section> <Section position="4" start_page="333" end_page="333" type="metho"> <SectionTitle> THE NBT ALGORITHM </SectionTitle> <Paragraph position="0"> The NBT algorithm is a nonselective version of the SBT (Selective Bottom to Top) algorithm, also given in G+P. The only difference between the two is that the SBT algorithm employs a reachability matrix to selectively eliminate bad paths before trying them. For more on this, see G+P and Ross (1981). For the purpose of this paper, it is not necessary to say anything more than that the addition of a reachability matrix modifies the algorithm only slightly and serves only to make the algorithm more efficient.</Paragraph> <Paragraph position="1"> A version of NBT modified to employ a third stack and to parse rather than recognize strings follows. This algorithm will be modified further throughout the paper.</Paragraph> <Paragraph position="2"> (1) \[VI,X,Y\] --> \[~,V2 ... Vn,t X,A Y\] if A --> Vl V2 ... Vn is a rule of the phrase structure grammar X is in the set of nonterminals and Y is anything (2) \[X,t,A\] --> \[A X,~,~\] if A is in the set of nonterminals (3) \[B,B,Y\] --> \[~,~,Y\] if B is in the set of nonterminals or terminals To begin, put the terminal string to be parsed followed by END on stack alpha. Put the nonterminal which is to be the root node of the tree to be constructed followed by END on stack beta. Put END on stack gamma. The symbol t is neither a terminal nor a nonterminal. If END is on top of each stack, the string has been recognized. If none of the turing machine instructions apply and END is not on the top of each stack, the path which led to this situation was a bad path and does not yield a valid parse.</Paragraph> <Paragraph position="3"> The rules necessary to give a parse tree can be stated informally (i.e., not in terms of turing machine instructions) as follows: When (I) is applied, attach V1 beneath A.</Paragraph> <Paragraph position="4"> When (3) is applied, attach ~he B on alpha B as the right daughter of the top,symbol on gamma.</Paragraph> <Paragraph position="5"> Note that there is a formal statement of the parsing version of NBT in Griffiths (1965). However, it is somewhat more complicated and obscures what is going on during the parse. Therefore, the informal procedure given above will be used instead.</Paragraph> <Paragraph position="6"> Intuitively, what NBT does is put the symbols on alpha together in a bottom-up manner with the ultimate goal of constructing a tree that has, at its top, whatever nonterminal symbol is on top of beta. So, to parse a sentence of English, NBT would begin with the lexical categories of the words to be parsed as a sentence on alpha and the nonterminal &quot;S&quot; on beta. An application of turing machine instruction (I) reduces this problem to a simpler one. (I) finds some phrase structure rule containing the symbol that is on top of alpha immediately AN ff~PROVED LEFT-CORNER PARSING ALGORITHM 335 after the arrow. So, if the first symbol on alpha was &quot;det&quot;, the phrase structure rule NP --> det AdjP N would qualify. By this application of (I), the problem is reduced to building an AdjP and #inding an N from the symbols on alpha. Once this is done, the trees for the &quot;det&quot;, the &quot;AdjP&quot; and the &quot;N&quot; would be combined into an NP. By application of (2), the NP would be put on alpha. Then a rule with NP immediately following the arrow would be looked for so that (I) could apply again.</Paragraph> <Paragraph position="7"> NBT is a nondeterministic algorithm. The nondeterminism comes from two places. Firstly, rule (I) can apply in more than one way. For this to happen, there would need to be two phrase structure rules with the same nonterminal symbol immediately after the arrow. The following two rules are an example of this.</Paragraph> <Paragraph position="9"> Secondly, rule (3) and rule (I) could apply in the same situation. Intuitively, an application of rule (3) indicates that a tree topped by node B was being searched for and a tree topped by node B has been found~ so use the tree just found as the tree that was sought. Rule (I) could apply as well if a phrase structure rule of the form X --> B Y1 Y2 ... Yn existed. Applying (I) indicates that the B being sought is not the B that was just built. Rather, the B that was just built is an initial subtree of the B being sought.</Paragraph> </Section> <Section position="5" start_page="333" end_page="333" type="metho"> <SectionTitle> RULZS WITH ABBREVIATIONS </SectionTitle> <Paragraph position="0"> ~ important aspect of the modified algorithm being proposed is that it can deal directly with rules which employ abbreviatory conventions which are utilized by linguists. Thus, parentheses (expressing optionalnodes) and curly brackets (expressing the fact that one of the set of nodes in brackets should be chosen) can appear in the rules that the parser accesses when parsing a string.</Paragraph> <Paragraph position="1"> Assume ~hat ieft and right parentheses are put o~ stack beta as separate elements.</Paragraph> <Paragraph position="2"> Also assume chat left and right curly brackets are put on stack beta as separate items. Given these assumptions~ to modify NBT to handle rules with parenthesized elements, the following turing machine instructions must be added.</Paragraph> <Paragraph position="3"> (4) IX, ( Cl C2 ,.. Cn ),Y\] --> IX,C\] C2 ... C,~,Y\] (5) \[X,( C! C2 ... Cn ),Y\] --~ \[X,~,V\] For all i, Ci = ( Cj Cj+l ... Cp ) or C1 Cl+i ... Cm ~ or X if X is i.i the sec of terminals.</Paragraph> <Paragraph position="4"> The first rule will apply when the parenthesized node is present. The second ru\]e will apply when the node is not present. The Ci variable handle cases of nested parentheses or curly brackets. Informally, a Ci is a variable that stands for a nonterminal, a terminal, a left parenthesis followed by some number of expressions which are Ci's followed by a right parenthesis, or a left curly bracket followed by some number of expressions which are Ci's followed by a right curly bracket. The following rules are necessary to directly parse with rules containing curly brackeLs.</Paragraph> <Paragraph position="6"> (8) Ix,: }ov\] .... \[x,o,Y\] (~ \[x~ < Cl },Y=\] --> \[x,c~,Y\] 336 K.M. ROSS (I0) \[X,: CI,Y\] --> \[X,:,Y=\] where : is a special symbol which is neither a terminal or a nonterminal symbol, C1 is a Ci type variable as defined earlier. Once these modifications are incorporated, the resulting algorithm will be more efficient than if the NBT algorithm were used with abbreviated rules completely expanded into many distinct rules. To see why this is so, consider a situation in which there was a rule of the form X --> ~I A2 ... An (Z). If this was replaced by two rules, X --> A1 A2 ... An Z and X --> A1 A2 ... An, the parse would have to be split immediately upon encounteging X. However, if the alternative solution being proposed were used, rather than parsing for AI, A2 ..... to An twice, they would only be parsed for once. The parse path would not split until it came time to decide whether we wanted to look for Z or not. In general, every rule which has, following the arrow, some number of obligatory elements followed by a parenthesized element will result in a savings. Thus, any such rule can be parsed with more efficiently than the two rules ~t would be turned into if parentheses were eliminated. Note that the additional cost here is quite small. For each parenthesized element, (4) and (5) will each apply once. In the alternative solution, many rules might apply unnecessarily to the parse for the nodes which came before the parenthesized node.</Paragraph> <Paragraph position="7"> There is a class of grammars for which the solution proposed here will require a bit more work than the solution where parentheses are simply eliminated from the grammar. These are grammars that only have rules inwhich parenthesized items come first and have no rules in which obligatory items precede optional ones. In a grammar with both kinds of rules, the savings made far outweigh t~e amount of extra work needed. Since the classes of grammars used in parsing systems generally have both kinds of rules, my solution will result in a savings for these. Note that a similar efficiency argument can be made for the curly bracket case.</Paragraph> <Paragraph position="8"> The above rules will handle all occurrences of parentheses and curly brackets except for those in which the item immediately following the arrow in a phrase structure rule is in parentheses or curly brackets. The algorithm could be modified to handle these cases directly, however, this will not increase efficiency. Items in curly brackets or parentheses that immediately follow the arrow in a phrase structure rule must be expanded immediately upon encountering them. There is no savings in postponing this expansion until run time. Putting off the choice of how to expand such a phrase structure rule will not allow paths to be mergedtbgether. Therefore, the best way to handle these is to expand all such rules into rules that do not have this property.</Paragraph> <Paragraph position="9"> Linguistically, the above is an interesting result. Linguists have claimed that use of parentheses simplifies the grammar. Since simpler grammars are preferred to more complex ones, a solution which collapses two rules to one by parentheses is preferable to .a solution that has two distinct rules. In parsing, we see that in many instanc'~s, the ~se of one rule with parentheses rather than two rules without results in the parser operating more efficiently. It is able to merge parse paths together which would have been distinct had several context-free rules not been collapsed together as one, using the abbreviatory conventions. Thus, a notationaldevice which was originally proposed to simplify phrase structure rules actually results in a more efficient parse in many cases. Therefore, at least for some cases, we have additional evidence for the use of parentheses in phrase structure rules.</Paragraph> </Section> <Section position="6" start_page="333" end_page="333" type="metho"> <SectionTitle> AN IMPROVED LEFT-CORNER PARSING ALGORITHM 337 DEPTH OR BREADTH FIRST? </SectionTitle> <Paragraph position="0"> There has of yet been no discussion of the order in which the algorithm proceeds.</Paragraph> <Paragraph position="1"> The statement of the algorithm is completely neutral in this respect. However, an implementation must impose some control structure. When a parse is started, there is one 3-tuple containing the information on stacks alpha, beta, and gamma. In general, there are many different rules of the parsing algorithm that can be applied after this point. In order to assure that all possible paths are pursued to completion, it is necessary to proceed in a principled way.</Paragraph> <Paragraph position="2"> One strategy is to push one state as far as it will go. That is, apply one of the rules that are applicable, get a new state, and then apply one of the applicable rules to that new state. This can continue until either no rules apply or a parse is found. If no rules apply, it was a bad parse path. If a parse is found, it is one of possibly many parses for the sentence. In either case, the algorithm must continue on and pursue all other alternative paths. An easy way to do this and assure that all alternatives are pursued is to backtrack to the last choice point, pick another applicable rule, and continue in the manner described earlier.</Paragraph> <Paragraph position="3"> By doing this until the parser has backed up through all possible choice points, all parses of the sentence will be found. A parser that works in this manner is a depth-first backtracking parser. This is probably the easiest control structure to use for a left-corner parser.</Paragraph> <Paragraph position="4"> Alternative control structures are possible. For instance, rather than pursuing one path as far as possible, one could go down a parse path to some desired distance, save that state for later, and come back up to the top and start some other parse path. The original parse path could be pursued later from the point at which it was stopped. The problem with such an approach is keeping track of all the options.</Paragraph> <Paragraph position="5"> In the algorithm being proposed here, the decision of whether the parse proceeds in a depth-first or breadth-first manner is governed by a parameter which is adjustable. Thus, the parser can proceed to a setable depth down each parse path before going off and pursuing others. This mechanism works by saving the state of the parser when it reaches the desired depth down a particular parse path.</Paragraph> <Paragraph position="6"> Once all paths are pursued to this depth, the parser is called again with each of the states that were saved.</Paragraph> <Paragraph position="7"> To enable the parser to function as described-above, the control structure for a depth-first parser described earlier is used. To introduce the ability to proceed in a breadth-first manner, the parser is only given a subset of the input string.</Paragraph> <Paragraph position="8"> Then, the item MORE is inserted after the last item that is given to the parser.</Paragraph> <Paragraph position="9"> If no other instructions apply and MORE is on top of stack beta, the parser must begin to backtrack as described earlier. Additionally, the state must be saved.</Paragraph> <Paragraph position="10"> Once all backtracking is completed, more input is put on beta and parsing begins again with each of the saved states.</Paragraph> <Paragraph position="11"> By changing the amount of input that is given, the degree to which the parser proceeds either depth or breadth first can be controlled. If one word is given at a time, the parser is compleZely breadth-first. If the entire sentence is given, it is completely depth-first. Any other amount results in some combination of the two.</Paragraph> <Paragraph position="12"> . This mechanism enables the algorithm to easily incorporate a well-formed substring table. All that needs to be done is compare the set of saved states and merge the ones that have subgoals in common. By setting the parameter to different values, the degree to which the well-formed substring table is used can be controlled.</Paragraph> <Paragraph position="13"> This is particularly important in light of Slocum's results which indicate that the overhead involved in maintaining such a table can exceed the savings that it</Paragraph> </Section> <Section position="7" start_page="333" end_page="333" type="metho"> <SectionTitle> 338 K.M. ROSS </SectionTitle> <Paragraph position="0"> gives. By having the degree to which the table :s used be adjustable, the proper setting can be determined, based on the grammar and the sorts of queries that are asked most often.</Paragraph> <Paragraph position="1"> Additionally, the algorithm can be used to process the sentence word by word as it is typed in. When used as the parser in a natural language interface, this can increase the speed of a parse since work can proceed as the user is typing and composing his input.</Paragraph> </Section> class="xml-element"></Paper>