PARSING WITH CONTEXT-FREE GRAMMARS cc437

PARSING Parsing is the process of recognizing and assigning STRUCTURE Parsing a string with a CFG: Finding a derivation of the string consistent with the grammar The derivation gives us a PARSE TREE

EXAMPLE (CFR LAST WEEK)

PARSING AS SEARCH Just as in the case of non-deterministic regular expressions, the main problem with parsing is the existence of CHOICE POINTS There is a need for a SEARCH STRATEGY determining the order in which alternatives are considered

TOP-DOWN AND BOTTOM-UP SEARCH STRATEGIES The search has to be guided by the INPUT and the GRAMMAR TOP-DOWN search: the parse tree has to be rooted in the start symbol S EXPECTATION-DRIVEN parsing BOTTOM-UP search: the parse tree must be an analysis of the input DATA-DRIVEN parsing

AN EXAMPLE OF TOP-DOWN SEARCH (IN PARALLEL)

AN EXAMPLE OF BOTTOM-UP SEARCH

NON-PARALLEL SEARCH If it’s not possible to examine all alternatives in parallel, it’s necessary to make further decisions: Which node in the current search space to expand first (breadth-first or depth-first) Which of the applicable grammar rules to expand first Which leaf node in a parse tree to expand next (e.g., leftmost)

TOP-DOWN, DEPTH-FIRST, LEFT-TO-RIGHT

TOP-DOWN, DEPTH-FIRST, LEFT-TO-RIGHT (II)

TOP-DOWN, DEPTH-FIRST, LEFT-TO-RIGHT (III)

TOP-DOWN, DEPTH-FIRST, LEFT-TO-RIGHT (IV)

A T-D, D-F, L-R PARSER (Compare with ND-recognize)

TOP-DOWN vs BOTTOM-UP TOP-DOWN: BOTTOM-UP: Only search among grammatical answers BUT: suggests hypotheses that may not be consistent with data Problem: left-recursion BOTTOM-UP: Only forms hypotheses consistent with data BUT: may suggest hypotheses that make no sense globally

LEFT-RECURSION A LEFT-RECURSIVE grammar may cause a T-D, D-F, L-R parser to never return Examples of left-recursive rules: NP  NP PP S  S and S But also: NP  Det Nom Det  NP’s

THE PROBLEM WITH LEFT-RECURSION

LEFT-RECURSION: POOR SOLUTIONS Rewrite the grammar to a weakly equivalent one Problem: may not get correct parse tree Limit the depth during search Problem: limit is arbitrary

LEFT-CORNER PARSING A hybrid of top-down and bottom-up parsing Strategy: don’t consider any expansion unless the current input can serve as the LEFT-CORNER of that expansion

FURTHER PROBLEMS IN PARSING Ambiguity Church and Patel (1982): the number of attachment ambiguities grows like the Catalan numbers C(2) = 2, C(3) = 5, C(4) = 14, C(5) = 132, C(6) = 469, C(7) = 1430, C(8) = 4867 Avoiding reparsing

COMMON STRUCTURAL AMBIGUITIES COORDINATION ambiguity OLD (MEN AND WOMEN) vs (OLD MEN) AND WOMEN ATTACHMENT ambiguity: Gerundive VP attachment ambiguity I saw the Eiffel Tower flying to Paris PP attachment ambiguity I shot an elephant in my pajamas

PP ATTACHMENT AMBIGUITY

AMBIGUITY: SOLUTIONS Use a PROBABILISTIC GRAMMAR (not covered in this module) Use semantics

AVOID RECOMPUTING INVARIANTS Consider parsing with a top-down parser the NP: A flight from Indianapolis to Houston on TWA With the grammar rules: NP  Det Nominal NP  NP PP NP  ProperNoun

INVARIANTS AND TOP-DOWN PARSING

THE EARLEY ALGORITHM

DYNAMIC PROGRAMMING A standard T-D parser would reanalyze A FLIGHT 4 times, always in the same way A DYNAMIC PROGRAMMING algorithm uses a table (the CHART) to avoid repeating work The Earley algorithm also Does not suffer from the left-recursion problem Solves an exponential problem in O(n3)

THE CHART The Earley algorithm uses a table (the CHART) of size N+1, where N is the length of the input Table entries sit in the `gaps’ between words Each entry in the chart is a list of Completed constituents In-progress constituents Predicted constituents All three types of objects are represented in the same way as STATES

THE CHART: GRAPHICAL REPRESENTATION

STATES A state encodes two types of information: DOTTED RULES How much of a certain rule has been encountered in the input Which positions are covered A  , [X,Y] DOTTED RULES VP  V NP  NP  Det  Nominal S   VP

EXAMPLES

SUCCESS The parser has succeeded if entry N+1 of the chart contains the state S   , [0,N]

THE ALGORITHM The algorithm loops through the input without backtracking, at each step performing three operations: PREDICTOR: add predictions to the chart COMPLETER: Move the dot to the right when looked-for constituent is found SCANNER: read in the next input word

THE ALGORITHM: CENTRAL LOOP

EARLEY ALGORITHM: THE THREE OPERATORS

EXAMPLE, AGAIN

EXAMPLE: BOOK THAT FLIGHT

EXAMPLE: BOOK THAT FLIGHT (II)

EXAMPLE: BOOK THAT FLIGHT (III)

EXAMPLE: BOOK THAT FLIGHT (IV)

READINGS Jurafsky and Martin, chapter 10.1-10.4