sliek môn máy tự động chương 5 context-free grammar

Chapter 5: Context-Free GrammarNovember 9, 2009 Chapter 5: Context-Free Grammar... Leftmost and Rightmost DerivationsLeftmost: in each step the leftmost variable in the sententialform is

Chapter 5: Context-Free Grammar

November 9, 2009

Chapter 5: Context-Free Grammar

Non-regular Languages

Not all languages are regular

L1 = {anbn: n ≥ 0} is not regular

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

Some properties of programming languages are not regular

Chapter 5: Context-Free Grammar

CFG and Regular Language

A regular language is also a context-free language

Why the name?

Chapter 5: Context-Free Grammar

G = ({S, A, B}, {a, b}, S, {S → AB, A → aaA, A → λ, B →

Bb, B → λ})

Chapter 5: Context-Free Grammar

Leftmost and Rightmost Derivations

Leftmost: in each step the leftmost variable in the sententialform is replaced

Rightmost: in each step the rightmost variable in the

sentential form is replaced

Chapter 5: Context-Free Grammar

Example 5.5

G = ({S, A, B}, {a, b}, S, {S → AB, A → aaA, A → λ, B →

Bb, B → λ})

Chapter 5: Context-Free Grammar

Ordered Tree for a Production

Derivation Tree

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

An ordered tree is aderivation treeiff:

1 The root is labeled S

2 Every leaf has a label from T ∪ {λ}

3 Every interior vertex has a label from V

4 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.

5 A leaf labeled λ has no siblings.

Chapter 5: Context-Free Grammar

Partial Derivation Tree

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

An ordered tree is apartial derivation tree iff:

1 The root is labeled S (not need).

2 Every leaf has a label from V ∪ T ∪ {λ}.

3 Every interior vertex has a label from V

4 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.

5 A leaf labeled λ has no siblings.

Chapter 5: Context-Free Grammar

Yield of a Tree

The string of symbols obtained by reading the leaves of a tree fromleft to right (omitting any λ’s encountered) is called theyieldofthe tree

Chapter 5: Context-Free Grammar

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 t G is a partial derivation tree of G whose root label is S, then the yield of tG is a sentential form of G.

Chapter 5: Context-Free Grammar

Membership and Parsing

Membership algorithm: tells if w ∈ L(G)

Parsing: finding a sequence of productions by which w ∈ L(G)

is derived

Chapter 5: Context-Free Grammar

Exhaustive Search Parsing

Top-down parsing

S → SS | aSb | bSa | λw = aabb

Chapter 5: Context-Free Grammar

Exhaustive Search Parsing Disadvantages

Restricted Grammar for Exhaustive Search Parsing

Suppose G = (V, T, S, P) is a context-free grammar which

L(G) or not

Chapter 5: Context-Free Grammar

Theorem 5.1

For every context-free grammar G, there exists an algorithmthat parses any w ∈ L(G) in a number of steps proportional to

|w |3

Not satisfactory as linear timeparsing algorithm (proportional

to the length of a string)

Chapter 5: Context-Free Grammar

Simple Grammars (S-Grammars)

Parsing in Simple Grammars (S-Grammars)

w = a1a2 an

S ⇒ a1A1A2 Am ⇒a1a2B1B2 BkA1A2 Am ⇒

Chapter 5: Context-Free Grammar

S-Grammars and Contemporary Programming Languages

Many features of Pascal-like programming languages can be

expressed with s-grammars

Chapter 5: Context-Free Grammar

Definition

Chapter 5: Context-Free Grammar

Unambiguous Grammar

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

unambiguous grammar, then L is said to be unambiguous

Chapter 5: Context-Free Grammar

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 iscalledinherently ambiguous

Chapter 5: Context-Free Grammar

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 morecomplicated aspects

Chapter 5: Context-Free Grammar

Example 5.14

S-grammars:

<if-statement> ::= if <expression> <then clause>

<else clause>

<then clause> ::= then <statement>

<else clause> ::= else <statement>

Chapter 5: Context-Free Grammar

Exercises: 2, 3, 4, 6, 7, 9, 15, 17, 22 of Section 5.1 - Linzsbook

Exercises: 1, 2, 5, 6, 8, 10, 11, 12, 13, 15 of Section 5.2 Linzs book

-Exercises: 1, 2, 3 of Section 5.3 - Linzs book

Chapter 5: Context-Free Grammar