1. Earley’s algorithm Earley’s algorithm employs the dynamic programming technique to address the weaknesses of general top-down parsing. Dynamic programming involves storing of results so they don’t ever need to be recomputed. Dynamic programming reduces exponential time requirement to polynomial time requirement: O(N 3 ), where N is length of input in words.

2. Data structure Earley’s algorithm uses a data structure called a chart to store information about the progress of the parse. A chart contains an entry for each position in the input A position occurs before the first word, between words, and after the last word.  word1  word2  …  wordN  A position is represented by a number; positions in the input are numbered from 0 (at the left) to N (at the right).

3. Chart details A chart entry consists of a sequence of states. A state represents – a subtree corresponding to a single grammar rule – information about how much of a rule has been processed – information about the span of the subtree w.r.t. the input A state is represented by an annotated grammar rule – a dot (  ) is used to show how much of the rule has been processed – a pair of positions, [x,y], indicates the span of the subtree w.r.t. the input; x is the position of the left edge of the subtree, and y is the position of the dot.

4. Three operators on a chart Predictor – applies when NonTerminal to right of  in a state is not a POS category (i.e. is not a pre-terminal) – adds states to current chart entry Scanner – applies when NonTerminal to right of  in a state is a POS category (i.e. is a pre-terminal) – adds states to next chart entry Completer – applies when there is no NonTerminal (and hence no Terminal) to right of  in a state (i.e.  is at end) – adds states to current chart entry

5. Predictor Suppose state to which Predictor applies is: X    NT  [x,y] Predictor adds, to the current chart entry, a new state for each possible expansion of NT For each expansion EX of NT, state added is NT   EX [y,y]

6. Scanner Suppose rule to which Scanner applies is: X    POS  [x,y] Scanner adds, to the next chart entry, a new state if the word in the next position can be a member of the category POS. The new state added is POS  word  [y,y+1]

7. Completer Suppose rule to which Completer applies is: X    [x,y] Completer adds, to the current chart entry, a new state for each possible reduction using the (now completed) state For each state (from any earlier chart entry) of the form Y    X  [w,x] a new state of the following form is added Y   X   [w,y]

8. Completer (modification) In order to recover parse tree information from the chart once parsing is complete, we need to modify the completer slightly. Each state in the chart must be given a unique identifier (  N for state N) Each time the completer adds a state, it also adds the unique identifier of the state completed to the list of previous states for that new state (which is a copy of an already existing state, waiting for the category which the current state just completed).

9. Initial state of chart for “book that flight” chart[0]chart[1]chart[2]chart[3]  0 :    S

10. chart[0] – initial state Idstatespanprevious states  0 :    S[0,0][] This is a dummy start state.

11. chart[0] – after  0 :    S (Predictor) Idstatespanprevious states  0 :    S[0,0][]  1 :S   NP VP[0,0][]  2 :S   Aux NP VP[0,0][]  3 :S   VP[0,0][]

12. chart[0] – after  1 : S   NP VP (Predictor) Idstatespanprevious states  0 :    S[0,0][]  1 :S   NP VP[0,0][]  2 :S   Aux NP VP[0,0][]  3 :S   VP[0,0][]  4 :NP   Det Nominal[0,0][]  5 :NP   ProperNoun[0,0][]

13. chart[1] – after  2 : S   Aux NP VP (Scanner) Since the input does not start with an auxiliary verb, the scanner does not add any state to chart[1], which therefore remains empty.

14. chart[0] – after  3 : S   VP (Predictor) Idstatespanprevious states  0 :    S[0,0][]  1 :S   NP VP[0,0][]  2 :S   Aux NP VP[0,0][]  3 :S   VP[0,0][]  4 :NP   Det Nominal[0,0][]  5 :NP   ProperNoun[0,0][]  6 :VP   Verb[0,0][]  7 :VP   Verb NP[0,0][]

15. chart[1] – after  4 : NP   Det Nominal (Scanner) Since the input does not start with an determiner, the scanner does not add any state to chart[1], which therefore remains empty.

16. chart[1] – after  5 : NP   ProperNoun (Scanner) Since the input does not start with an proper noun, the scanner does not add any state to chart[1], which therefore remains empty.

17. chart[1] – after  6 : VP   Verb (Scanner) Idstatespanprevious states  8 :Verb  book  [0,1][]

18. chart[1] – after  7 : VP   Verb NP (Scanner) Idstatespanprevious states  8 :Verb  book  [0,1][] The state to be added is already in chart[1], so no change.

19. After finishing processing of chart[0] chart[0]  0:    S[0,0][]  1:S   NP VP[0,0][]  2:S   Aux NP VP[0,0][]  3:S   VP[0,0][]  4:NP   Det Nominal[0,0][]  5:NP   ProperNoun[0,0][]  6:VP   Verb[0,0][]  7:VP   Verb NP[0,0][] chart[1]  8:Verb  book  [0,1][]

20. chart[1] – after  8 : Verb  book  (Completer) Idstatespanprevious states  8 :Verb  book  [0,1][]  9 :VP  Verb  [0,1][  8 ]  10 :VP  Verb  NP[0,1][  8 ] The completer moves the dot in those states already in a chart state with annotation [0,0] More generally, for a completed state with annotation [j,k], the completer moves the dot in those states already in a chart state with annotation [i,j].

21. chart[1] – after  9 : VP  Verb  (Completer) Idstatespanprevious states  8 :Verb  book  [0,1][]  9 :VP  Verb  [0,1][  8 ]  10 :VP  Verb  NP[0,1][  8 ]  11 :S  VP  [0,1][  9 ]

22. chart[1] – after  10 : VP  Verb  NP (Predictor) Idstatespanprevious states  8 :Verb  book  [0,1][]  9 :VP  Verb  [0,1][  8 ]  10 :VP  Verb  NP[0,1][  8 ]  11 :S  VP  [0,1][  9 ]  12 :NP   Det Nominal[1,1][]  13 :NP   ProperNoun[1,1][]

23. chart[1] – after  11 : S  VP  (Completer) Idstatespanprevious states  8 :Verb  book  [0,1][]  9 :VP  Verb  [0,1][  8 ]  10 :VP  Verb  NP[0,1][  8 ]  11 :S  VP  [0,1][  9 ]  12 :NP   Det Nominal[1,1][]  13 :NP   ProperNoun[1,1][] The book does not process this rule. I’m not sure why. However, if it were processed it would clearly not indicate a successful parse since it does not span entire input.

24. chart[2] – after  12 : NP   Det Nominal (Scanner) Idstatespanprevious states  14 :Det  that  [1,2][]

25. chart[2] – after  13 : NP   ProperNoun (Scanner) Idstatespanprevious states  14 :Det  that  [1,2][] Since the input does not start with an proper noun, the scanner does not add any state to chart[2], which therefore remains the same.

26. After finishing processing of chart[1] chart[1]  8 :Verb  book  [0,1][]  9 :VP  Verb  [0,1][  8 ]  10 :VP  Verb  NP[0,1][  8 ]  11 :S  VP  [0,1][  9 ]  12 :NP   Det Nominal[1,1][]  13 :NP   ProperNoun[1,1][] chart[2]  14 :Det  that  [1,2][]

27. chart[2] – after  14 : Det  that  (Completer) Idstatespanprevious states  14 :Det  that  [1,2][]  15 :NP  Det  Nominal[1,2][  14 ]

28. chart[2] – after  15 : NP  Det  Nominal (Predictor) Idstatespanprevious states  14 :Det  that  [1,2][]  15 :NP  Det  Nominal[1,2][  14 ]  16 :Nominal   Noun[2,2][]  17 :Nominal   Noun Nominal[2,2][]

29. chart[3] – after  16 : Nominal   Noun (Scanner) Idstatespanprevious states  18 :Noun  flight  [2,3][]

30. chart[3] – after  17 : Nominal   Noun Nominal ( Scanner ) Idstatespanprevious states  18 :Noun  flight  [2,3][] The state to be added is already in chart[3], so no change.

31. After finishing processing of chart[2] chart[2]  14 :Det  that  [1,2][]  15 :NP  Det  Nominal[1,2][  14 ]  16 :Nominal   Noun[2,2][]  17 :Nominal   Noun Nominal[2,2][] chart[3]  18 :Noun  flight  [2,3][]

32. chart[3] – after  18 : Noun  flight  (Completer) Idstatespanprevious states  18 :Noun  flight  [2,3][]  19 :Nominal  Noun  [2,3][  18 ]  20 :Nominal  Noun  Nominal[2,3][  18 ]

33. chart[3] – after  19 : Nominal  Noun  (Completer) Idstatespanprevious states  18 :Noun  flight  [2,3][]  19 :Nominal  Noun  [2,3][  18 ]  20 :Nominal  Noun  Nominal[2,3][  18 ]  21 :NP  Det Nominal  [1,3][  14,  19 ]

34. chart[3] – after  20 : Nominal   Noun Nominal ( Predictor ) Idstatespanprevious states  18 :Noun  flight  [2,3][]  19 :Nominal  Noun  [2,3][  18 ]  20 :Nominal  Noun  Nominal[2,3][  18 ]  21 :NP  Det Nominal  [1,3][  14,  19 ]  22 :Nominal   Noun[3,3][]  23 :Nominal   Noun Nominal[3,3][]

35. chart[3] – after  21 : NP  Det Nominal  (Completer) Idstatespanprevious states  18 :Noun  flight  [2,3][]  19 :Nominal  Noun  [2,3][  18 ]  20 :Nominal  Noun  Nominal[2,3][  18 ]  21 :NP  Det Nominal  [1,3][  14,  19 ]  22 :Nominal   Noun[3,3][]  23 :Nominal   Noun Nominal[3,3][]  24 :VP  Verb NP  [0,3][  8,  21 ]

36. chart[3] – after  22 : Nominal   Noun (Scanner) Idstatespanprevious states  18 :Noun  flight  [2,3][]  19 :Nominal  Noun  [2,3][  18 ]  20 :Nominal  Noun  Nominal[2,3][  18 ]  21 :NP  Det Nominal  [1,3][  14,  19 ]  22 :Nominal   Noun[3,3][]  23 :Nominal   Noun Nominal[3,3][]  24 :VP  Verb NP  [0,3][  8,  21 ] Since there is no more input, no new states are added.

37. chart[3] – after  23 : Nominal   Noun Nominal ( Scanner ) Idstatespanprevious states  18 :Noun  flight  [2,3][]  19 :Nominal  Noun  [2,3][  18 ]  20 :Nominal  Noun  Nominal[2,3][  18 ]  21 :NP  Det Nominal  [1,3][  14,  19 ]  22 :Nominal   Noun[3,3][]  23 :Nominal   Noun Nominal[3,3][]  24 :VP  Verb NP  [0,3][  8,  21 ] Since there is no more input, no new states are added.

38. chart[3] – after  24 : VP  Verb NP  (Completer) Idstatespanprevious states  18 :Noun  flight  [2,3][]  19 :Nominal  Noun  [2,3][  18 ]  20 :Nominal  Noun  Nominal[2,3][  18 ]  21 :NP  Det Nominal  [1,3][  14,  19 ]  22 :Nominal   Noun[3,3][]  23 :Nominal   Noun Nominal[3,3][]  24 :VP  Verb NP  [0,3][  8,  21 ]  25 :S  VP  [0,3][  24 ]

39. We’re done! All states in chart[3] have been processed, no new states have been added to chart[4], and a state with LHS S spanning all the input is in chart[3]:  25 :S  VP  [0,3][  24 ]

40. Recovering the tree The basic idea is to trace back through the “previous state” links:  25 :S  VP  [0,3][  24 ]  24 :VP  Verb NP  [0,3][  8,  21 ]  21 :NP  Det Nominal  [1,3][  14,  19 ]  19 :Nominal  Noun  [2,3][  18 ]  18 :Noun  flight  [2,3][]  14 :Det  that  [1,2][]  8 :Verb  book  [0,1][] S VP NPVerb DetNominal bookthat Noun flight