This ppt is about the context free grammar (CFG), which must be studied to learn about the parsing phase.

1. Akhil Kaushik
Asstt. Prof., CE Deptt.,
TIT Bhiwani
Context Free Grammar

2. Introduction
• Syntax:- the form or structure of the expressions,
statements, and program units.
• Semantics:- the meaning of the expressions,
statements, and program units.
• Syntax and semantics provide a language’s
definition.

3. Introduction
• A language is a set of sentences.
• A sentence is a string of characters over some
alphabets. Ex: a = b + c; or z = (a + b) * c;
• Syntax: <assign> →<id> = <expr> ;
• <id> →a | b | c
• <expr> →<id> + <expr> | <id> * <expr> | <id>
• Semantics : a = b + c;

4. Introduction
• Recognizers – A recognition device reads input
strings of the language and decides whether the
input strings belong to the language.
• Ex: Syntax analysis part of a compiler.
• Generators – A device that generates sentences of
a language.
• One can determine if the syntax of a particular
sentence is correct by comparing it to the structure
of the generator.

5. Introduction

6. The Chomsky Hierarchy
• Noam Chomsky, - American linguist, philosopher,
scientist and social activist gave Chomsky hierarchy
of grammars in 1956.
• Grammar Definition: It is defined by four tuples:
G = {V,T,P,S} where
– V = Non-terminals
– T = Terminals
– P = Production Rule
– S = Start Symbol

7. The Chomsky Hierarchy

8. The Chomsky Hierarchy
• Unrestricted grammar generates Recursively
enumerable languages (Type-0 Languages)
include all formal grammars.
• They generate exactly all languages that can be
recognized by a Turing machine.
• This grammar has rules of the form α → β (where α
contains non-terminals and β contains terminals or
non terminals).

9. The Chomsky Hierarchy
Examples:-
• AB → A
• AB → aB
• S → ε
• a → AB
• ε → a

10. The Chomsky Hierarchy
• Type-1 grammar generate the Context-sensitive
languages.
• These languages can be recognized by a linear
bounded automaton.
• These grammars have rules of the form α → β with
a restriction that length of | α | ≤ | β | .

11. The Chomsky Hierarchy
• Examples:-
• aAb → bbb
• aA → bbb
• aAb → bb

12. The Chomsky Hierarchy
• Type-2 grammar generate the Context-Free
languages.
• These languages are exactly all languages that can
be recognized by a non-deterministic pushdown
automaton.
• Context-free languages are the theoretical basis for
the syntax of most programming languages.

13. The Chomsky Hierarchy
• These are defined by rules of the form A → α where
A is a nonterminal and α is a string of terminals and
non-terminal (there will be no context on the left and
right of non-terminal ).
• Examples: A → BCD
• A →aBC
• a →AbC

14. The Chomsky Hierarchy
• Type-3 Languages generate the Regular languages.
• These languages are exactly all languages that can
be decided by a finite state automaton.
• Regular languages are commonly used to define
search patterns of programming languages.

15. The Chomsky Hierarchy
• It can be classified into two types:-
(1)Right Linear (2)Left Linear.
• If we have repetition of non terminals on right side
[ A → xB|x] then it is known as Right Linear.
• If we have repetition of non terminals on left side
[ A → Bx|x] then it is known as Left Linear.
(A,B є non terminals and x є Σ*)

16. The Chomsky Hierarchy
Examples:-
• S → aS|b or S → aS|c
• S → Sa|b or A → ba
• S → aS|b or S → Sa|c

17. The Chomsky Hierarchy

18. Context- Free Grammars
(CFG)

19. Introduction
• A context-free grammar is a notation for describing
languages.
• It is more powerful than finite automata or RE’s, but
still cannot define all possible languages.
• Useful for nested structures. Ex. parentheses in
programming languages.

20. Introduction
• Basic idea is to use “variables” to stand for sets of
strings (i.e., languages).
• These variables are defined recursively, in terms of
one another.
• Recursive rules (“productions”) involve only
concatenation.
• Alternative rules for a variable allow union.

21. Introduction
• CFGs are also known as:-
– Phrase Structure Grammars
– Backus-Naur Form (BNF)
• It consists of:-
– Rules (equations or productions)
– Terminals (tokens)
– Non-terminals (Language constituents like noun phrase,
verb phrase, sentence, etc.)

22. Formal Definition
• G (CFG) is defined by G = (V, Σ, P, S)
– V = variables a finite set (non-terminals)
– Σ = alphabet or terminals a finite set
– P = productions a finite set
– S = start variable S ϵ V
• Productions’ form, where A ϵ V, α ϵ (V υ S)*:
• A → a

23. Example
• Here is a formal CFG for { 0n1n | n >= 1}.
• Terminals = {0, 1}, Variables = {S}, Start symbol = S.
• Productions = S -> 01, S -> 0S1
• We derive strings in the language of a CFG by
starting with the start symbol, and repeatedly
replacing some variable A by the right side of one of
its productions.

24. RE to CFG

25. Derivation
• A derivation is a sequence of rules applied to a
string that accounts for that string.
– Covers all the elements in the string.
– Covers only the elements in the string.
• Derivations allow us to replace any of the variables
in a string.
• Leads to many different derivations of the same
string.

26. Derivation Tree
A parse tree of a derivation is a tree
in which:-
• Each internal node is labeled with a
non-terminal.
• If a rule A →A1A2…An occurs in the
derivation then is a parent node of
nodes labeled A1, A2, …, An

27. Derivation Tree

28. Leftmost vs Rightmost Derivation
A left-most derivation
of a sentential form is
one in which rules
transforming the left-
most non-terminal are
always applied.
A right-most derivation of
a sentential form is one in
which rules transforming
the right-most non-terminal
are always applied.

29. Leftmost vs Rightmost Derivation

30. Ambiguous Grammar
• A grammar G is ambiguous if there is a word
w ϵ L(G) having are least two different parse trees.

31. Ambiguous Grammar

32. Ambiguous Grammar
• Given an ambiguous grammar, would like an
equivalent unambiguous grammar.
– Allows you to know more about structure of a given
derivation.
– Can lead to more efficient parsing algorithms.
– In programming languages, want to impose a canonical
structure on derivations. E.g., for 1+2´3.
• Strategy: Force an ordering on all derivations.

33. Ambiguous Grammar

34. Ambiguous Grammar

35. Ambiguous Grammar

36. Ambiguous Grammar

37. Ambiguous Grammar

38. Ambiguous Grammar

39. Ambiguous Grammar

40. Ambiguous Grammar

41. Ambiguous Grammar

42. Ambiguous Grammar

43. Ambiguous Grammar

44. Ambiguous Grammar

45. Disambiguation

46. Disambiguation

47. Disambiguation

48. Disambiguation
By forcing the leftmost variable (or alternatively, the
rightmost variable) to be replaced, we avoid these
“distinctions without a difference.”

49. Akhil Kaushik
akhilkaushik05@gmail.com
9416910303
CONTACT ME AT:
Akhil Kaushik
akhilkaushik05@gmail.com
9416910303
THANK YOU !!!