1. Formal Language & Automata Theory
Shyamanta M Hazarika
Computer Sc. & Engineering
Tezpur University

2. Introduction
Finite Automata accept all regular languages and only regular languages
Even very simple languages are non regular ( = {a,b}):
- {anbn : n = 0, 1, 2, …}
- {w : w is palindrome word}
We are going to define a new class of languages, called context-free languages that contain all regular languages and many more (including the 2 above)

3. Context-Free Grammar A context-free grammar is a kind of program
Languages that are generated by context-free grammars are called context-free languages
Context-free grammars are more expressive than finite automata: if a language L is accepted by a finite automata then L can be generated by a context-free grammar

4. Context-Free Grammar (CFG)
Definition. A context-free grammar (CFG) is a 4-tuple (V, , R, S), where:
 is an alphabet (characters  are called terminals)
V is a set (elements in NT are called variables)
R is a subset of NT  (  NT)*
S, the start variable, is one of the variables in NT
V   = 
If (,)  R, we write   
   is called a rule

5. Derivations
Definition. u yields v in one-step, written u  v, if: for some u,v in (V  )* the following 3 conditions hold:
u = xz
v = xz
   in R
Definition. u derives v, written u * v, if:
There is a chain of one-step yields of the form:
u  u1  u2  …  v

6. Example
= {a,b}
V = {S}
R = { S aSb,
S  e }

7. Context-Free Languages
Definition. Given a context-free grammar
G = (V, , R, S), the language generated or
derived from G is the set:
L(G) = {w  *: }
S * w
Definition. A language L is context-free if there is a context-free grammar G = (, NT, R, S), such that L is generated from G

8. Example
= {a,b}
NT = {S}
R = { S aS,
S  Sb,
S  e}

9. Example
= {a,b}
NT = {S}
R = { S aSa,
S  bSb,
S  e}

10. Parse Tree A parse tree of a derivation u  u1  u2  …  v
is a tree in which:
Each internal node is labeled with a variable
If a rule A  A1A2…An occurs in the derivation then A is a parent node of nodes labeled A1, A2, …, An S a e b

11. Leftmost, Rightmost Derivations
Definition. A leftmost derivation of a sentential form is one in which rules transforming the left-most nonterminal are always applied
Definition. A rightmost derivation of a sentential form is one in which rules transforming the right-most nonterminal are always applied

12. Ambiguous Grammar
Definition. A grammar G is ambiguous if there is a word w  L(G) having are least two different leftmost derivations
S  A
S  B
S  AB
A  aA
B  bB
A  e
B  e
Notice that the word a has at least two left-most derivations
Some ambiguous grammars G can be disambiguated:
find an unambiguous grammar G’ such that
L(G) = L(G’)
Some languages cannot be disambiguated