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Chapter 13: Parsing with Context-Free Grammars

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  1. Chapter 13: Parsing with Context-Free Grammars Heshaam Faili University of Tehran

  2. Context-Free Grammars • Context-Free Grammars are of the form: • A , where  is a string of terminals and/or non-terminals • Note that the regular grammars are a proper subset of the context-free grammars. • This means that every regular grammar is context-free, but there are context-free grammars that aren’t regular • CFGs only specify what trees look like, not how they should be computationally derived  We need to parse the sentences

  3. Parsing Intro Input: a string Output: a (single) parse tree A useful step in the process of obtaining meaning We can view the problem as searching through all possible parses (tree structures) to find the right one Strategies Top-Down (goal-directed) vs. Bottom-Up (data-directed) Breadth-First vs. Depth-First Adding Bottom-Up to Top-Down: Left-Corner Parsing Example Book that flight!

  4. Grammar and Desired Tree

  5. Top-Down Parsing • Expand rules, starting with S and working down to leaves • Replace the left-most non-terminal with each of its possible expansions. • While we guarantee that any parse in progress will be S-rooted, it will expand non-terminals that can’t lead to the existing input • e.g., 5 of 6 trees in third ply = level of the search space • None of the trees take the properties of the lexical items into account until the last stage

  6. Expansion techniques • Breadth-First Expansion All the nodes at each level are expanded once before going to the next (lower) level. • This is memory intensive when many grammar rules are involved • Depth-First • Expand a particular node at a level, only considering an alternate node at that level if the parser fails as a result of the earlier expansion • i.e., expand the tree all the way down until you can’t expand any more

  7. Top-Down Depth-First Parsing • There are still some choices that have to be made: • 1. Which leaf node should be expanded first? • Left-to-right strategy moves through the leaf nodes in a left-to-right fashion • 2. Which rule should be applied first? • There are multiple NP rules; which should be used first? • Can just use the textual order of rules from the grammar • There may be reasons to take rules in a particular order (e.g., probabilities)

  8. Top-Down breath-First Parsing • Search states are kept in an agenda • Search states consist of partial trees and a pointer to the next input word in the sentence • Based on what we’ve seen before, apply the next item on the agenda to the current tree • Add new items to (the front of) the agenda, based on the rules in the grammar which can expand at the (leftmost) node • We maintain the depth-first strategy by adding new hypotheses (rules) to the front of the agenda • If we added them to the back, we would have a breadth-first strategy

  9. S S S NP VP Aux NP VP Det Nom Does S Does NP VP S Aux NP VP Does FAIL Det Nom this Top-down (depth-first) parsing Does this flight include a meal ?

  10. S S S Aux NP VP VP NP VP S S S S S S NP VP NP VP Aux NP VP Aux NP VP VP VP V Det Nom PropN Det Nom PropN V NP Top-down (breadth-first) parsing S

  11. Bottom-Up Parsing • Bottom-Up Parsing is input-driven  start from the words and move up to form a tree • Here we match one or more nodes on the upper fringe of the parse tree against the right-hand side of a CFG rule, building the left-hand side as a parent node of those nodes. • We can also have breadth-first and depth-first approaches • The example on the next slide (p. 362, Fig. 10.4) moves in a breadth-first fashion • While any parse in progress will be tied to the input, many may not lead to an S! • e.g., left-most trees in plies 1-4 of next Figure

  12. Bottom-up parsing

  13. Comparing Top-Down and Bottom-Up Parsing • Top-Down: • While we guarantee that any parse in progress will be S-rooted, it will expand non-terminals that can’t lead to the existing input, e.g., first 4 trees in third ply. • Bottom-Up: • While any parse in progress will be tied to the input, many may not lead to an S, e.g., left-most trees in plies 1-4 of Figure. • So, both pure top-down and pure bottom up approaches are highly inefficient.

  14. Left-Corner Parsing • Motivation: • Both pure top-down and bottom-up approaches are inefficient • The correct top-down parse has to be consistent with the left-most word of the input • Left-corner parsing: a way of using bottom-up constraints as part of a top-down strategy. • Left-corner rule: expand a node with a grammar rule only if the current input can serve as the left corner from this rule. • Left-corner from a rule:first word along the left edge of a derivation from the rule • Put the left-corners into a table, which can then guide parsing

  15. S NP VP S VP S Aux NP VP NP Det Nominal | ProperNoun Nominal  Noun Nominal | Noun VP Verb | Verb NP Noun  book | flight | meal | money Verb  book | include | prefer Aux  does ProperNoun  Houston | TWA Left Corners S => NP …=> Det, ProperNoun VP => Verb Aux … => Aux NP => Det, ProperNoun VP => Verb Nominal => Noun Left-Corner Example

  16. Other problems: Left-Recursion • Left-corner parsers still guided by top-down parsing • Consider rules like: S  S and S NP  NP PP • A top-down left-to-right depth-first parser could apply a rule to expand a node (e.g., S), and then apply that same rule again, and again, ad infinitum. • Left Recursion: A grammar is left-recursive if a non-terminal leads to a derivation that includes itself as its leftmost immediate or non-immediate child (i.e., along its leftmost branch). • PROBLEM: Top-Down parsers may not terminate on a left-recursive grammar

  17. Other problems: Repeated Parsing of Subtrees • When parser backtracks to an alternative expansion of a non-terminal, it loses all parses of sub-constituents that it built. • There is a good chance that it will rebuild the parses of some of those constituents again. • This can occur repeatedly. • a flight from Indianopolis to Houston on TWA • NP  Det Nom • Will build an NP for a flight, before failing when the parser realizes the input PPs aren’t covered • NP  NP PP • Will again build an NP for a flight, before failing when the parser realizes the two remaining PPs in the input aren’t covered

  18. Duplicated effort caused by backtracking in top-down parsing

  19. Other problems: Ambiguity • Repeated parsing of sub-trees is even more of a problem for ambiguous sentences • 2 kinds of ambiguities: attachment, coordination • PP attachment: • NP or VP: I shot an elephant in my pajamas. • NP bracketing: the meal [on flight 286] [from SF] [to Denver] • Coordination: • [old men] and women vs. old [men and women] • Parsers have to disambiguate between lots of valid parses or return all parses • Using statistical, semantical and pragmatic knowledge as the source of disambiguation • Local ambiguity: even if the sentence isn’t ambiguous it can be inefficient because of local ambiguity: e.g: parsing “Book” in sentence “Book that flight”

  20. Ambiguity (PP-attachment)

  21. VP  VP PP NP  NP PP

  22. Addressing the problems: Chart Parsing • More or less a standard method for carrying out parsing; keeps tables of constituents that have been parsed earlier, so it doesn’t reduplicate the work. • Each possible sub-tree is represented exactly once. • This makes it a form of dynamic programming (which we saw with minimum edit distance and the Viterbi algorithm) • Combines bottom-up and top-down parsing

  23. CYK Parsing • The DP method by using CNF grammar • ABC • Am • Any CFG can be converted to CNF: should be care of 3 types of rules • Rule that mix terminal and non-terminal on the right-hand side • Adding dummy non-terminal to handle terminals • Rules that have a single non-terminal on the right-hand side (unit productions ) • AB : unit productions (can be rewrited by A for any B) • Rule where the right-hand side length more that two

  24. CYK Parsing, cont,d • Like other DP methods, a simple (n+1)*(n+1) matrix used to encode the structure of the sentence (n: sentence length) • Indexed is the gap between words • [0 Book 1 that 2 flight 3 ] • [i,j] : is a set of non-terminals that represent all the constituents that span positions i through j of the input

  25. CYK Parsing, cont,d • Since our grammar is in CNF, the non-terminal entries in the table have exactly two daughters in the parse. • for each constituent represented by an entry [i, j] in the table there must be a position in the input, k, where it can be split into two parts such that i < k < j. • Given such a k, the first constituent [i,k] must lie to the left of entry [i, j] somewhere along row i, and the second entry [k, j] must lie beneath it, along column j

  26. CYK Algorithm

  27. CYK example(CNF Grammar)

  28. CYK example(Book the flight through Houston)

  29. CYK in practice • Does not have major problem theoretically • The resulted parse tree are not consistent to syntacticians…(because of CNF formal) • Syntax to Semantic approach complicated … • Post-processing needed to return-back the result to more acceptable form

  30. Earley Parsing • Uses DP to implement top-down search • Single left-to-right pass and filling a table named Chart(N+1 entry) • 3 kind of information in each entry: • A Subtree corresponding to a single grammar rule • Information about progress made in completing this subtree • Position of the subtree respect to the input

  31. Earley Parsing Representation • The parser uses a representation for parse state based on dotted rules. S  NP  VP • Dotted rules distinguish what has been seen so far from what has not been seen (i.e., the remainder). • The constituents seen so far are to the left of the dot in the rule, the remainder is to the right. • Parse information is stored in a chart, represented as a graph. • The nodes represent word positions. • The labels represent the portion (using the dot notation) of the grammar rule that spans that word position.  In other words, at each position, there is a set of labels (each of which is a dotted rule, also called a state), indicating the partial parse tree produced until then.

  32. Given a trivial grammar NP  D N D  a N  dog Here’s the chart for the complete parse of “a dog” D  a [0,1] (scan) N  dog [1,2] (scan) NP  D N [0,0] (predict) NP  D  N [0,1] (complete) NP  D N  [0,2] (complete) Example: Chart for A Dog

  33. More Early Parsing Terminology • A state is complete if it has a dot at the right-hand side of its rule. Otherwise, it is incomplete. • At each position, there is a list (actually, a queue) of states. • The parser moves through the N+1 sets of states in the chart left-to-right, processing the states in each set in order. • States will be stored in a FIFO (first-in first-out) queue at each start position • The processing applies one of three operators, each of which takes a state and produces new states added to the chart. • Scanner, Predictor, Completer • There is no backtracking.

  34. Earley Parsing Algorithm • In the top level loop, for each position, for each state, it calls the predictor, or else the scanner, or else the completer. • The algorithm never backtracks and never removes states, so we don’t redo any work • The goal is to have S  α• as the last chart entry, i.e. the dot has moved over the entire input to derive an S

  35. The Earley Algorithm

  36. Prediction Procedure PREDICTOR((AB, [i, j])) For each (B) in grammar do Enqueue((B , [j, j]), chart[j]) End • Predicting is the task of saying what kinds of input we expect to see • Add a rule to the chart saying that we have not seen , but when we do, it will form a B • The rule covers no input, so it goes from j to j • Such rules provide the top-down aspect of the algorithm

  37. Scanning Procedure SCANNER ((AB, [i, j])) If B is a part-of-speech for word[j] then Enqueue((B  word[j], [j, j+1]), chart[j+1]) Scanning reads in lexical items • We add a dotted rule indicating that a word has been seen between j and j+1 • This is then added to the following (j+1) chart • Such a completed dotted rule can be used to complete other dotted rules • These rules also show how the Earley parser has a bottom-up component

  38. Completion Procedure COMPLETER((B, [j, k])) For each (AB, [i, j]) in chart[j] do Enqueue((A B, [i, k]), chart[k]) End • Completion combines two rules in order to move the dot, i.e., indicate that something has been seen • A rule covering B has been seen, so any rule A which refers to B in its RHS moves the dot • Instead of spanning from i to j, A now spans from i to k, which is where B ended • Once the dot is moved, the rule will not be created again

  39. Example (Book that flight)

  40. Example(Book that flight)

  41. Example(Book that flight), cont

  42. Example(Book that flight), cont

  43. Example(Book that flight), cont

  44. Earley parsing • The Earley algorithm is efficient, running in polynomial time. • Technically, however, it is a recognizer, not a parser • To make it a parser, each state needs to be augmented with a pointer to the states that its rule covers • For example, a VP would point to the state where its V was completed and the state where its NP was completed

  45. Chart Parser • In both the CKY and Earley algorithms, the order in which events occur (adding entries to the table, reading words, making predictions, etc.) is statically determined by the procedures that make up these algorithms. • Unfortunately, dynamically determining the order in which events occur based on the current information is often necessary • Chart Parsing facilitates just such dynamic determination of the order in which chart entries are processed. • Using Agenda

  46. Chart Parser • fundamental rule: generalized the ideas in CYK and Earley: • if the chart contains two edges A → α • B β , [i, j] and B → γ •, [ j,k] then we should add the new edge A →α B • β [i,k] to the chart • Prediction can be top-down of botton-up