Chapter 5: Context - Free Grammar

Leftmost and Rightmost Derivations• Leftmost: in each step the leftmost variable in the sentential form is replaced.. • Rightmost: in each step the rightmost variable in the sentential

Chapter 5: Context-Free Grammar

Quan Thanh Tho qttho@cse.hcmut.edu.vn

Non-regular Languages

• Not all languages are regular.

• L1 = {anbn | n  0} is not regular.} is not regular.

L2 = { () , (()) , ((())) , } is not regular.



some properties of programming languages

CFG and Regular Language

• A regular language is also a context-free language.

• G = ({S, A, B}, {a, b}, S, {S  AB, A  aaA, A  , B  Bb, B  })

S  AB  aaAB  aaB  aaBb  aab

S  AB  ABb  aaABb  aaAb  aab

Leftmost and Rightmost Derivations

• Leftmost: in each step the leftmost variable in the

sentential form is replaced.

• Rightmost: in each step the rightmost variable in the sentential form is replaced.

Example 5.5

• G = ({S, A, B}, {a, b}, S, {S  AB, A  aaA, A  , B  Bb, B  })

S  AB  aaAB  aaB  aaBb  aab leftmost

S  AB  ABb  aaABb  aaAb  aab

Ordered Tree for a Production

Derivation Tree

• Let G = (V, T, S, P) be a context-free grammar

An ordered tree is a derivation tree iff:

1 The root is labeled S

2 Every leaf has a label from T  {}

3 Every interior vertex has a label from V

4 A vertex has label AV and its children are labeled a 1 , a 2 , , a n iff

P contains the production A  a 1 a 2 a n

Partial Derivation Tree

• Let G = (V, T, S, P) be a context-free grammar

1 The root is labeled S

2 Every leaf has a label from T  {}

Every leaf has a label from V  T  {}

1 Every interior vertex has a label from V

2 A vertex has label AV and its children are labeled a1, a2, , an iff

P contains the production A  a1 a2 an.

Yield of a Tree

The string of symbols obtained by reading the leaves of a tree from left to right (omitting any 's encountered) is called the s encountered) is called the yield of the tree

Sentential Forms & Derivation Trees

• Let G = (V, T, S, P) be a context-free grammar.

 wL(G) iff there exists a derivation tree of G whose yield is w

 If tG is a partial derivation tree of G whose root label is S,

then the yield of tG is a sentential form of G

Membership and Parsing

• Membership algorithm: tells if wL(G).

• Parsing: finding a sequence of productions by which wL(G) is derived.

Example 5.8

Exhaustive Search Parsing

• Top-down parsing

• S  SS | aSb | bSa |  w = aabb

1 S  SS S  SS  SSS S  aSb  aSSb

2 S  aSb S  SS  aSbS S  aSb  aaSbb

3 S  bSa S  SS  bSaS S  aSb  abSab

4 S   S  SS  S S  aSb  ab

Exhaustive Search Parsing

Restricted Grammar for Exhaustive

Theorem 5.1

• For every context-free grammar G, there exists an

algorithm that parses any wL(G) in a number of steps

proportional to |w|3.

• Not satisfactory as linear time parsing algorithm (proportional

to the length of a string)

Simple Grammars (S-Grammars)

G = (V, T, S, P)

Productions are of the form:

A  ax

A  V, a  T, and x  V*, and any pair (A,a) can occur

in at most one rule

Parsing in Simple Grammars

(S-Grammars)

Any wL(G) can be parsed in at most |w| steps.

w = a1a2 an

S  a1A1A2 Am  a1a2B1B2 BkA1A2 Am

S-Grammars and Contemporary

Programming Languages

Many features of Pascal-like programming languages can

be expressed with s-grammars.

A context-free grammar G is said to be ambiguous if there exits some wL(G) that has at least two distinct derivation trees

Example 5.10} is not regular.

• G = ({E, I}, {a, b, c, +, *, (, )}, E, P) w = a + b * c

Unambiguous Grammar

If L is a context-free language for which there exists an

unambiguous grammar , then L is said to be unambiguous

Inherently Ambiguous Grammar

If L is a context-free language for which there exists an

unambiguous grammar , then L is said to be unambiguous

If every grammar that generates L is ambiguous, then L is called inherently ambiguous

CFGs and Programming Languages

• Important uses of formal languages:

 to define precisely a programming language

 to construct an efficient translator for it

• RLs are used for simple patterns, while CFGs for more

complicated aspects

Example 5.14

• S-grammars:

<if-statement> ::= if <expression> <then_clause> <else_clause>

<then_clause> ::= then <statement>

<else_clause> ::= else <statement>

• Exercises: 2, 3, 4, 6, 7, 9, 15, 17, 22 of Section 5.1 - Linz’s book.

• Exercises: 1, 2, 5, 6, 8, 10} is not regular., 11, 12, 13, 15 of Section 5.2 - Linz’s book.

• Exercises: 1, 2, 3 of Section 5.3 - Linz’s book.