Earleys algorithm

1
Earleys algorithm

• Earleys algorithm employs the dynamic
  programming technique to address the weaknesses
  of general top-down parsing.
• Dynamic programming involves storing of results
  so they dont ever need to be recomputed.
• Dynamic programming reduces exponential time
  requirement to polynomial time requirement
  O(N3), where N is length of input in words.

2
Data structure

• Earleys 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 NonTermial (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
• X ? ? word ? ? x,y1

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
10
chart0 initial state

• Id state span previous states
• ?0 ? ? ? S 0,0
• This is a dummy start state.

11
chart0 after ?0 ? ? ? S (Predictor)

• Id state span previous states
• ?0 ? ? ? S 0,0
• ?1 S ? ? NP VP 0,0
• ?2 S ? ? Aux NP VP 0,0
• ?3 S ? ? VP 0,0

12
chart0 after ?1 S ? ? NP VP (Predictor)

• Id state span previous 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
chart1 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
  chart1, which therefore remains empty.

14
chart0 after ?3 S ? ? VP (Predictor)

• Id state span previous 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
chart1 after ?4 NP ? ? Det Nominal (Scanner)

• Since the input does not start with an
  determiner, the scanner does not add any state to
  chart1, which therefore remains empty.

16
chart1 after ?5 NP ? ? ProperNoun (Scanner)

• Since the input does not start with an proper
  noun, the scanner does not add any state to
  chart1, which therefore remains empty.

17
chart1 after ?6 VP ? ? Verb (Scanner)

• Id state span previous states
• ?8 Verb ? book ? 0,1

18
chart1 after ?7 VP ? ? Verb NP (Scanner)

• Id state span previous states
• ?8 Verb ? book ? 0,1
• The state to be added is already in chart1, so
  no change.

19
After finishing processing of chart0

• chart0
• ?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

• chart1
• ?8 Verb ? book ? 0,1

20
chart1 after ?8 Verb ? book ? (Completer)

• Id state span previous 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
chart1 after ?9 VP ? Verb ? (Completer)

• Id state span previous 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
chart1 after ?10 VP ? Verb ? NP (Predictor)

• Id state span previous 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
chart1 after ?11 S ? VP ? (Completer)

• Id state span previous 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. Im not
  sure why.
• However, if it were processed it would clearly
  not indicate a successful parse since it does not
  span entire input.

24
chart2 after ?12 NP ? ? Det Nominal (Scanner)

• Id state span previous states
• ?14 Det ? that ? 1,2

25
chart2 after ?13 NP ? ? ProperNoun (Scanner)

• Id state span previous states
• ?14 Det ? that ? 1,2
• Since the input does not start with an proper
  noun, the scanner does not add any state to
  chart2, which therefore remains the same.

26
After finishing processing of chart1
chart1 ?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
chart2 ?14 Det ? that ? 1,2
27
chart2 after ?14 Det ? that ? (Completer)

• Id state span previous states
• ?14 Det ? that ? 1,2
• ?15 NP ? Det ? Nominal 1,2 ?14

28
chart2 after ?15 NP ? Det ? Nominal
(Predictor)

• Id state span previous 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
chart3 after ?16 Nominal ? ? Noun (Scanner)

• Id state span previous states
• ?18 Noun ? flight ? 2,3

30
chart3 after ?17 Nominal ? ? Noun Nominal
(Scanner)

• Id state span previous states
• ?18 Noun ? flight ? 2,3
• The state to be added is already in chart3, so
  no change.

31
After finishing processing of chart2
chart2 ?14 Det ? that ? 1,2 ?15 NP ?
Det ? Nominal 1,2 ?14 ?16 Nominal ? ?
Noun 2,2 ?17 Nominal ? ? Noun
Nominal 2,2
chart3 ?18 Noun ? flight ? 2,3
32
chart3 after ?18 Noun ? flight ? (Completer)

• Id state span previous states
• ?18 Noun ? flight ? 2,3
• ?19 Nominal ? Noun ? 2,3 ?18
• ?20 Nominal ? Noun ? Nominal 2,3 ?18

33
chart3 after ?19 Nominal ? Noun ? (Completer)

• Id state span previous 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
chart3 after ?20 Nominal ? ? Noun Nominal
(Predictor)

• Id state span previous 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
chart3 after ?21 NP ? Det Nominal ?
(Completer)

• Id state span previous 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
chart3 after ?22 Nominal ? ? Noun (Scanner)

• Id state span previous 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
chart3 after ?23 Nominal ? ? Noun Nominal
(Scanner)

• Id state span previous 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
chart3 after ?24 VP ? Verb NP ? (Completer)

• Id state span previous 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
Were done!

• All states in chart3 have been processed, no
  new states have been added to chart4, and a
  state with LHS S spanning all the input is in
  chart3
• ?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
NP
Verb
Det
Nominal
Noun
book
that
flight