Topdown parsing

1
Top-down parsing

• Janina Kopp
• Miriam Käshammer
• CL2 Parsing Course
• 21 November 2007

2
Overview

• The basic top-down approach
• Pushdown automata
• Breadth-first top-down parsing
• Depth-first top-down parsing

3
The basic approach (1)

• Start with the start symbol on the prediction
  stack.
• Always process the topmost symbol of the stack
• If it is a terminal, match it with the current
  input symbol or reject the parse.
• If it is a non-terminal, replace it with one of
  its right-hand sides.
• ? Left-most derivation

4
The basic approach (2)

• Example
• S ? aB bA
• A ? a aS bAA
• B ? b bS aBB
• Input aabb

Derivation S ? aB ? aaBB ? aabB ? aabb
5
The basic approach (3)
Taken from Dick Grune and Ceriel Jacobs. Parsing
Techniques. Ellis Horwood Limited, 1990. Page 121.
6
Pushdown automata

• Construction of a PDA for a CFG
• Presentation on PDAs (5 November 2007)
• JFLAP Convert to PDA (LL)
• adopts Slide 13 (top down)

7
PDA Shortcomings

• Non-determinism
• ? try all possibilities
• No record of used rules
• ? analysis stack
• For every prediction step, push the non-terminal
  being replaced (with index).
• For every match, push the matched terminal.
• ? instantaneous description

8
Breadth-first top-down parser (1)

• Intuition
• Consider all possible predictions of a given
  grammar, but follow a prediction only until it
  contradicts the input string.

9
Breadth-first top-down parser (2)

• Technique
• Maintain a list of all possible predictions.
• non-terminal in front
• remove this prediction and
• add as many predictions as there are right hand
  sides for this non-terminal
• each with the non-terminal replaced by one of
  these right hand sides.
• -gt repeat this until you have only predictions
  that start with a terminal

10
Breadth-first top-down parser (3)

• Technique (continued)
• Take the next symbol of your input sentence and
  compare it to the terminals of your predictions
• Discard all predictions that do not match your
  current input symbol
• Continue until you reach the end of your input
  sentence.
• the end is marked by appending to the input
  string
• a rule S ? S is added
• Illustration of the principle

11
Breadth-first Example (1)

• Example Grammar
• S ? DC AB
• A ? a aA
• B ? bc bBc
• D ? ab aDb
• C ? c cC
• termination rule
• S ? S

Input aabc
12
Breadth-first Example (2)
1
2
4
3
13
Breadth-first Example (2)
5
4
6
7
14
Breadth-first Example (2)
8
7
9
10
15
Breadth-first Example (2)
11
10
12
16
Depth-first (backtracking) parser (1)

• Intuition
• When faced with a number of possibilities, choose
  one and leave the others for later.
• If this choice turns out to be a failure (or even
  a success, but one needs all solutions), roll
  back the actions until the point of decision and
  continue with the other possibilities.

17
Depth-first (backtracking) parser (2)

• Technique
• Same as the breadth-first parser until the
  topmost symbol on the prediction stack has more
  than one right-hand side
• Chooses the first right-hand side A1
• Continues until
• the prediction stack is empty and there is no
  input left (? a parse is found)
• a match fails (? backtracking)

18
Depth-first (backtracking) parser (3)

• Backtracking
• Removes any terminals from the top of the
  analysis stack and from the matched input.
• Pushes them back onto the prediction stack and
  the rest of the input.

?
19
Depth-first (backtracking) parser (4)

• Backtracking
• Two possibilities
• Empty analysis stack parsing stops
• Non-terminal on top of the analysis stack Pops
  it and pushes it back onto the prediction stack
  replacing its right-hand side

?
20
Depth-first (backtracking) parser (5)

• Two possibilities
• No other right-hand side of this non-terminal
• ? backtracks further
• Another right-hand side of this non-terminal left
  
• ? starts parsing again Replaces the
  non-terminal with the next right-hand side

21
Depth-first (backtracking) parser (6)
Taken from Dick Grune and Ceriel Jacobs. Parsing
Techniques. Ellis Horwood Limited, 1990.
22
Depth-first (backtracking) parser (7)
23
Depth-first (backtracking) parser (8)
24
Depth-first (backtracking) parser (9)
25
Depth-first (backtracking) parser (10)
26
The problem of left-recursion

• Assume we have a grammar
• S ? Sb a

• ? we do not get to a state where a match with the
  input can be attempted
• problem non-terminal derives an infinite
  sequence of sentential forms, all starting with
  non-terminals

• left-recursion
• immediate (A ? A a)
• indirect (A ? B a, B ? A ß)

27
Breadth-first vs. Depth-first