Context-Free Grammars - Automata and Complexity Theory - Lecture Slides, Slides of Theory of Automata

Download Context-Free Grammars - Automata and Complexity Theory - Lecture Slides and more Slides Theory of Automata in PDF only on Docsity!

1

Context-Free Grammars

Formalism Derivations

Backus-Naur Form

Left- and Rightmost Derivations

2

Informal Comments

A

context-free grammar

is a notation

for describing languages.

It is more powerful than finite automata or RE’s, but still cannot defineall possible languages.

Useful for nested structures, e.g., parentheses in programming languages.

4

Example: CFG for { 0

n

n

| n > 1}

Productions:

S -> 01S -> 0S

Basis: 01 is in the language.

Induction: if w is in the language, then so is 0w1.

5

CFG Formalism

Terminals

= symbols of the alphabet

of the language being defined.

Variables

nonterminals

= a finite

set of other symbols, each of whichrepresents a language.

Start symbol

= the variable whose

language is the one being defined.

7

Example: Formal CFG

Here is a formal CFG for { 0

n

n

| n > 1}.

Terminals = {0, 1}.

Variables = {S}.

Start symbol = S.

Productions =

S -> 01S -> 0S

8

Derivations – Intuition

We

derive

strings in the language of a

CFG by starting with the start symbol,and repeatedly replacing some variableA by the right side of one of itsproductions.



That is, the “productions for A” are thosethat have A on the left side of the ->.

10

Iterated Derivation

=>* means “zero or more derivation steps.”

Basis:

for any string

Induction: if

and

, then

11

Example: Iterated Derivation

S -> 01; S -> 0S1.

S => 0S1 => 00S11 => 000111.

So S =>* S; S =>* 0S1; S =>* 00S11; S =>* 000111.

13

Language of a Grammar

If G is a CFG, then L(G), the

language

of G, is {w | S =>* w}.



Note: w must be a terminal string, S is thestart symbol.

Example: G has productions S ->

ε

and

S -> 0S1.

L(G) = {

n

n

| n > 0}.

Note:

ε

is a legitimate

right side.

14

Context-Free Languages

A language that is defined by some CFG is called a

context-free language.

There are CFL’s that are not regular languages, such as the example justgiven.

But not all languages are CFL’s.

Intuitively: CFL’s can count two things, not three.

16

BNF Notation – (2)

Symbol ::= is often used for ->.

Symbol | is used for “or.”



A shorthand for a list of productions withthe same left side.

Example: S -> 0S1 | 01 is shorthand for S -> 0S1 and S -> 01.

17

BNF Notation – Kleene Closure

Symbol … is used for “one or more.”

Example: ::= 0|1|2|3|4|5|6|7|8|

::= …



Note: that’s not exactly the * of RE’s.

Translation: Replace

… with a new

variable A and productions A -> A

19

BNF Notation: Optional Elements

Surround one or more symbols by […] to make them optional.

Example: ::=

if

then

[;

else

]

Translation: replace [

] by a new

variable A with productions A ->

ε

20

Example: Optional Elements

Grammar for if-then-else can be replaced by:

S -> iCtSAA -> ;eS |

ε