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Graph of a Grammar

- Represents leftmost derivations of a CFG.
- A path from node S to a node w is a leftmost derivation.

L7Parse

Properties of Graph of a Grammar

- Every node has a finite number of children.
- Simple breadth-first enumeration feasible.
- The number of leaves is infinite if the language is infinite.
- Typical case.
- There can be infinite long paths (derivations).
- Loops in depth-first traversals.

L7Parse

Parser

A program that determines if a string

by constructing a derivation. Equivalently,

it searches the graph of G.

- Top-down parsers
- Constructs the derivation tree from root to leaves.
- Leftmost derivation.
- Bottom-up parsers
- Constructs the derivation tree from leaves to root.
- Rightmost derivation in reverse.

L7Parse

Search the graph of a grammar breadth-first

Uses: Queue

(+) Always terminates with shortest derivation

(-) Inefficient in general.

Search the graph of a grammar depth-first

Uses: Stack

(-) Can get into infinite loops

(e.g., left recursion)

(+) Efficient in general.

Top-down parsers: Breadth-first vs Depth-firstL7Parse

Determining when

- Number of terminals in sentential form

>length of w

- Prefix of sentential form preceding the leftmost non-terminal not a prefix of w.
- No rules applicable to sentential form.

L7Parse

Parsing Examples

L7Parse

Breadth-first top-down parser

Queue-up left

sentential forms

level by level

Parse

successful

S

A

T

A+T

b

(A)

A+T+T

T+T

(A)+T

T+T+T

(T)

(A+T)

…

A+T+T+T

…

(T)+T

…

…

(b)

((A))

(b)+T

…

(b)+b

L7Parse

Depth-first top-down parser

Use stack to

pursue entire

path from left

Parse

fails

S

A

T

A+T

b

(A)

A+T+T

T+T

Backtrack

On failure

(T)

(A+T)

T+T+T

A+T+T+T

…

…

…

(b)

((A))

L7Parse

Summary

- In BFTD version, all left derivations investigated in parallel.
- In DFTD version, one specific derivation is pursued to completion.
- Done, if succeeds.
- Otherwise, backtrack and investigate another path.

(Incomplete strategy)

(Used by Prolog interpreter)

L7Parse

T+b

A+b

A+T

Bottom-up parsing(b)+b

Parse

successful

(T)+b

(b)+T

(A)+b

(T)+T

(T)+T

Not

allowed

(b)+A

…

(S)+b

(A)+T

…

A

S

L7Parse

Practical Parsers

- Language/Grammar designed to enable deterministic (directed and backtrack-free) searches.
- Uses lookahead tokens and/or exploits the context in the sentential form constructed so far.

“Look before you leap.” vs “Procrastination principle.”

- Top-down parsers : LL(k) languages
- E.g., Pascal, Ada, etc.
- Better error diagnosis and recovery.
- Bottom-up parsers : LALR(1), LR(k) languages
- E.g., C/C++, Java, etc.
- Handles left recursion in the grammar.
- Backtracking parsers
- E.g., Prolog interpreter.

L7Parse

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