Context Sensitive Grammar
Aparna S Vijayan
Department of Computer Science and Automation
December 2, 2011
Aparna S Vijayan (CSA)
CSG
December 2, 2011
1 / 12
Contents
Aparna S Vijayan (CSA)
CSG
December 2, 2011
2 / 12
Contents
1
Definition of CSG
Aparna S Vijayan (CSA)
CSG
December 2, 2011
2 / 12
Contents
1
Definition of CSG
2
Context Sensitive Language(CSL)
Aparna S Vijayan (CSA)
CSG
December 2, 2011
2 / 12
Contents
1
Definition of CSG
2
Context Sensitive Language(CSL)
3
Example of CSL
Aparna S Vijayan (CSA)
CSG
December 2, 2011
2 / 12
Contents
1
Definition of CSG
2
Context Sensitive Language(CSL)
3
Example of CSL
4
Chomsky Hierachy
Aparna S Vijayan (CSA)
CSG
December 2, 2011
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Definition
Context Sensitive Grammar(Type1 Grammar)
A context-sensitive grammar (CSG) is an unrestricted grammar in
which every production has the form
α → β with |β| ≥ |α| (where α and β are strings of nonterminals and
terminals).
The concept of context-sensitive grammar was introduced by Noam
Chomsky in the 1950.
In every derivation the length of the string never decreases.
The term ”context-sensitive” comes from a normal form for these
grammars,where each production is of the form α1 Aα2 →α1 βα2 ,with
β 6= .
They permit replacement of variable A by string β only in the
”context ” α1 - α2 .
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Formal Definition
A Context Sensitive Grammar is a 4-tuple , G = (N,Σ, P, S) where:
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Formal Definition
A Context Sensitive Grammar is a 4-tuple , G = (N,Σ, P, S) where:
N=Set of non terminal symbols.
Aparna S Vijayan (CSA)
CSG
December 2, 2011
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Formal Definition
A Context Sensitive Grammar is a 4-tuple , G = (N,Σ, P, S) where:
N=Set of non terminal symbols.
Σ=Set of terminal symbols.
Aparna S Vijayan (CSA)
CSG
December 2, 2011
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Formal Definition
A Context Sensitive Grammar is a 4-tuple , G = (N,Σ, P, S) where:
N=Set of non terminal symbols.
Σ=Set of terminal symbols.
S=Start symbol of the production.
Aparna S Vijayan (CSA)
CSG
December 2, 2011
4 / 12
Formal Definition
A Context Sensitive Grammar is a 4-tuple , G = (N,Σ, P, S) where:
N=Set of non terminal symbols.
Σ=Set of terminal symbols.
S=Start symbol of the production.
P=Finite set of productions.
Aparna S Vijayan (CSA)
CSG
December 2, 2011
4 / 12
Formal Definition
A Context Sensitive Grammar is a 4-tuple , G = (N,Σ, P, S) where:
N=Set of non terminal symbols.
Σ=Set of terminal symbols.
S=Start symbol of the production.
P=Finite set of productions.
All rules in P are of the form α1 Aα2 →α1 βα2 .
A ∈ N ( A is a single nonterminal)
α1 , α2 , β ∈ (N ∪ Σ)+ .
Aparna S Vijayan (CSA)
CSG
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Context Sensitive Language
The language generated by the Context Sensitive Grammar is called
context sensitive language.
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Context Sensitive Language
The language generated by the Context Sensitive Grammar is called
context sensitive language.
If G is a Context
Sensitive Grammar
then
+
∗
L(G)= w (w ∈ Σ ) ∧ S ⇒G w .
Aparna S Vijayan (CSA)
CSG
December 2, 2011
5 / 12
Context Sensitive Language
The language generated by the Context Sensitive Grammar is called
context sensitive language.
If G is a Context
Sensitive Grammar
then
+
∗
L(G)= w (w ∈ Σ ) ∧ S ⇒G w .
Eg 1 of a context sensitive grammar
G = {{S, A, B, C , a, b, c} , {a, b, c} , P, S}where P is the set of rules.
S→ aSBC
S → aBC
CB → BC
aB → ab
bB → bb
bC → bc
cC →cc
The language generated by this grammar is {an b n c n |n ≥ 1} .
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Context Sensitive Language
The derivation for the string aabbcc is
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Context Sensitive Language
The derivation for the string aabbcc is
S ⇒ aSBC
Aparna S Vijayan (CSA)
CSG
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Context Sensitive Language
The derivation for the string aabbcc is
S ⇒ aSBC
⇒ aaBCBC
Aparna S Vijayan (CSA)
CSG
December 2, 2011
6 / 12
Context Sensitive Language
The derivation for the string aabbcc is
S ⇒ aSBC
⇒ aaBCBC
⇒ aabCBC
Aparna S Vijayan (CSA)
CSG
December 2, 2011
6 / 12
Context Sensitive Language
The derivation for the string aabbcc is
S ⇒ aSBC
⇒ aaBCBC
⇒ aabCBC
⇒ aabBCC
Aparna S Vijayan (CSA)
CSG
December 2, 2011
6 / 12
Context Sensitive Language
The derivation for the string aabbcc is
S ⇒ aSBC
⇒ aaBCBC
⇒ aabCBC
⇒ aabBCC
⇒ aabbCC
Aparna S Vijayan (CSA)
CSG
December 2, 2011
6 / 12
Context Sensitive Language
The derivation for the string aabbcc is
S ⇒ aSBC
⇒ aaBCBC
⇒ aabCBC
⇒ aabBCC
⇒ aabbCC
⇒ aabbcC
Aparna S Vijayan (CSA)
CSG
December 2, 2011
6 / 12
Context Sensitive Language
The derivation for the string aabbcc is
S ⇒ aSBC
⇒ aaBCBC
⇒ aabCBC
⇒ aabBCC
⇒ aabbCC
⇒ aabbcC
⇒ aabbcc
Aparna S Vijayan (CSA)
CSG
December 2, 2011
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Example2 CSG L = (#a = #b = #c)
G2 =({S, A, B, C , a, b, c} , {a, b, c} , P, S)
S→ ABC
S →ABCS
AB→ BA
AC → CA
BC → CB
BA → AB
CA →AC
CB → BC
A→a
B →b
C →c
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Hierarchy of Formal Languages
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Relation between Formal Languages
The CFL’s not containing are properly contained in the context
sensitive languages
Not all Context Sensitive Languages are Context Free.
Every Context Sensitive Language is recusive.
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Chomsky Hierachy
Described by Chomsky in 1956.
Four classes of language Type 3,Type2,Type1,Type 0 from most
restrictive to most general(Unestricted).
Each level of hierarchy can be characterized by a class of grammar.
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CSG
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Chomsky Hierarchy
Type
3
2
1
0
Language
Regular
CFL
CSL
RE
productions
A → aB, A → a
A→α
α → β,|β| ≥ |α|
α→β
Device
Finite Automaton
Pushdown Automaton
Linear Bounded Automaton
Turing Machine
A,B→ Nonterminals
α,β→ string of terminals and nonterminals
a→ terminal symbol
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References
Introduction to Automata Theory Languages and Computation by ”JOHN E
HOPCROFT”,”JEFFERY D.ULLMAN”.
Introduction to Languages and the Theory of Computation ”JOHN MARTIN”
http://en.wikipedia.org/wiki/Context-sensitive grammar
http://adammikeal.org/courses/compute/Chomsky
Aparna S Vijayan (CSA)
CSG
December 2, 2011
12 / 12