context sensitive grammar - dronadrona.csa.iisc.ernet.in/~deepakd/atc-2011/csl.pdf · de nition...
TRANSCRIPT
![Page 1: Context Sensitive Grammar - Dronadrona.csa.iisc.ernet.in/~deepakd/atc-2011/csl.pdf · De nition Context Sensitive Grammar(Type1 Grammar) A context-sensitive grammar (CSG) is an unrestricted](https://reader030.vdocument.in/reader030/viewer/2022020104/5aba2c297f8b9ad13d8e8854/html5/thumbnails/1.jpg)
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
![Page 2: Context Sensitive Grammar - Dronadrona.csa.iisc.ernet.in/~deepakd/atc-2011/csl.pdf · De nition Context Sensitive Grammar(Type1 Grammar) A context-sensitive grammar (CSG) is an unrestricted](https://reader030.vdocument.in/reader030/viewer/2022020104/5aba2c297f8b9ad13d8e8854/html5/thumbnails/2.jpg)
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 2 / 12
![Page 3: Context Sensitive Grammar - Dronadrona.csa.iisc.ernet.in/~deepakd/atc-2011/csl.pdf · De nition Context Sensitive Grammar(Type1 Grammar) A context-sensitive grammar (CSG) is an unrestricted](https://reader030.vdocument.in/reader030/viewer/2022020104/5aba2c297f8b9ad13d8e8854/html5/thumbnails/3.jpg)
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 2 / 12
![Page 4: Context Sensitive Grammar - Dronadrona.csa.iisc.ernet.in/~deepakd/atc-2011/csl.pdf · De nition Context Sensitive Grammar(Type1 Grammar) A context-sensitive grammar (CSG) is an unrestricted](https://reader030.vdocument.in/reader030/viewer/2022020104/5aba2c297f8b9ad13d8e8854/html5/thumbnails/4.jpg)
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 2 / 12
![Page 5: Context Sensitive Grammar - Dronadrona.csa.iisc.ernet.in/~deepakd/atc-2011/csl.pdf · De nition Context Sensitive Grammar(Type1 Grammar) A context-sensitive grammar (CSG) is an unrestricted](https://reader030.vdocument.in/reader030/viewer/2022020104/5aba2c297f8b9ad13d8e8854/html5/thumbnails/5.jpg)
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 2 / 12
![Page 6: Context Sensitive Grammar - Dronadrona.csa.iisc.ernet.in/~deepakd/atc-2011/csl.pdf · De nition Context Sensitive Grammar(Type1 Grammar) A context-sensitive grammar (CSG) is an unrestricted](https://reader030.vdocument.in/reader030/viewer/2022020104/5aba2c297f8b9ad13d8e8854/html5/thumbnails/6.jpg)
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 2 / 12
![Page 7: Context Sensitive Grammar - Dronadrona.csa.iisc.ernet.in/~deepakd/atc-2011/csl.pdf · De nition Context Sensitive Grammar(Type1 Grammar) A context-sensitive grammar (CSG) is an unrestricted](https://reader030.vdocument.in/reader030/viewer/2022020104/5aba2c297f8b9ad13d8e8854/html5/thumbnails/7.jpg)
Definition
Context Sensitive Grammar(Type1 Grammar)
A context-sensitive grammar (CSG) is an unrestricted grammar inwhich every production has the formα→ β with |β| ≥ |α| (where α and β are strings of nonterminals andterminals).
The concept of context-sensitive grammar was introduced by NoamChomsky in the 1950.
In every derivation the length of the string never decreases.
The term ”context-sensitive” comes from a normal form for thesegrammars,where each production is of the form α1Aα2→α1βα2,withβ 6= ε.
They permit replacement of variable A by string β only in the”context ” α1 - α2.
Aparna S Vijayan (CSA) CSG December 2, 2011 3 / 12
![Page 8: Context Sensitive Grammar - Dronadrona.csa.iisc.ernet.in/~deepakd/atc-2011/csl.pdf · De nition Context Sensitive Grammar(Type1 Grammar) A context-sensitive grammar (CSG) is an unrestricted](https://reader030.vdocument.in/reader030/viewer/2022020104/5aba2c297f8b9ad13d8e8854/html5/thumbnails/8.jpg)
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 α1Aα2→α1βα2.
A ∈ N ( A is a single nonterminal)α1, α2, β ∈ (N ∪ Σ)+.
Aparna S Vijayan (CSA) CSG December 2, 2011 4 / 12
![Page 9: Context Sensitive Grammar - Dronadrona.csa.iisc.ernet.in/~deepakd/atc-2011/csl.pdf · De nition Context Sensitive Grammar(Type1 Grammar) A context-sensitive grammar (CSG) is an unrestricted](https://reader030.vdocument.in/reader030/viewer/2022020104/5aba2c297f8b9ad13d8e8854/html5/thumbnails/9.jpg)
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 α1Aα2→α1βα2.
A ∈ N ( A is a single nonterminal)α1, α2, β ∈ (N ∪ Σ)+.
Aparna S Vijayan (CSA) CSG December 2, 2011 4 / 12
![Page 10: Context Sensitive Grammar - Dronadrona.csa.iisc.ernet.in/~deepakd/atc-2011/csl.pdf · De nition Context Sensitive Grammar(Type1 Grammar) A context-sensitive grammar (CSG) is an unrestricted](https://reader030.vdocument.in/reader030/viewer/2022020104/5aba2c297f8b9ad13d8e8854/html5/thumbnails/10.jpg)
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 α1Aα2→α1βα2.
A ∈ N ( A is a single nonterminal)α1, α2, β ∈ (N ∪ Σ)+.
Aparna S Vijayan (CSA) CSG December 2, 2011 4 / 12
![Page 11: Context Sensitive Grammar - Dronadrona.csa.iisc.ernet.in/~deepakd/atc-2011/csl.pdf · De nition Context Sensitive Grammar(Type1 Grammar) A context-sensitive grammar (CSG) is an unrestricted](https://reader030.vdocument.in/reader030/viewer/2022020104/5aba2c297f8b9ad13d8e8854/html5/thumbnails/11.jpg)
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 α1Aα2→α1βα2.
A ∈ N ( A is a single nonterminal)α1, α2, β ∈ (N ∪ Σ)+.
Aparna S Vijayan (CSA) CSG December 2, 2011 4 / 12
![Page 12: Context Sensitive Grammar - Dronadrona.csa.iisc.ernet.in/~deepakd/atc-2011/csl.pdf · De nition Context Sensitive Grammar(Type1 Grammar) A context-sensitive grammar (CSG) is an unrestricted](https://reader030.vdocument.in/reader030/viewer/2022020104/5aba2c297f8b9ad13d8e8854/html5/thumbnails/12.jpg)
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 α1Aα2→α1βα2.
A ∈ N ( A is a single nonterminal)α1, α2, β ∈ (N ∪ Σ)+.
Aparna S Vijayan (CSA) CSG December 2, 2011 4 / 12
![Page 13: Context Sensitive Grammar - Dronadrona.csa.iisc.ernet.in/~deepakd/atc-2011/csl.pdf · De nition Context Sensitive Grammar(Type1 Grammar) A context-sensitive grammar (CSG) is an unrestricted](https://reader030.vdocument.in/reader030/viewer/2022020104/5aba2c297f8b9ad13d8e8854/html5/thumbnails/13.jpg)
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 α1Aα2→α1βα2.
A ∈ N ( A is a single nonterminal)α1, α2, β ∈ (N ∪ Σ)+.
Aparna S Vijayan (CSA) CSG December 2, 2011 4 / 12
![Page 14: Context Sensitive Grammar - Dronadrona.csa.iisc.ernet.in/~deepakd/atc-2011/csl.pdf · De nition Context Sensitive Grammar(Type1 Grammar) A context-sensitive grammar (CSG) is an unrestricted](https://reader030.vdocument.in/reader030/viewer/2022020104/5aba2c297f8b9ad13d8e8854/html5/thumbnails/14.jpg)
Context Sensitive Language
The language generated by the Context Sensitive Grammar is calledcontext sensitive language.
If G is a Context Sensitive Grammar thenL(G)=
{w∣∣(w ∈ Σ∗) ∧
(S ⇒+
G w)}
.
Eg 1 of a context sensitive grammarG = {{S ,A,B,C , a, b, c} , {a, b, c} ,P,S}where P is the set of rules.
S→ aSBCS → aBCCB → BCaB → abbB → bbbC → bccC →cc
The language generated by this grammar is {anbncn|n ≥ 1} .
Aparna S Vijayan (CSA) CSG December 2, 2011 5 / 12
![Page 15: Context Sensitive Grammar - Dronadrona.csa.iisc.ernet.in/~deepakd/atc-2011/csl.pdf · De nition Context Sensitive Grammar(Type1 Grammar) A context-sensitive grammar (CSG) is an unrestricted](https://reader030.vdocument.in/reader030/viewer/2022020104/5aba2c297f8b9ad13d8e8854/html5/thumbnails/15.jpg)
Context Sensitive Language
The language generated by the Context Sensitive Grammar is calledcontext sensitive language.
If G is a Context Sensitive Grammar thenL(G)=
{w∣∣(w ∈ Σ∗) ∧
(S ⇒+
G w)}
.
Eg 1 of a context sensitive grammarG = {{S ,A,B,C , a, b, c} , {a, b, c} ,P,S}where P is the set of rules.
S→ aSBCS → aBCCB → BCaB → abbB → bbbC → bccC →cc
The language generated by this grammar is {anbncn|n ≥ 1} .
Aparna S Vijayan (CSA) CSG December 2, 2011 5 / 12
![Page 16: Context Sensitive Grammar - Dronadrona.csa.iisc.ernet.in/~deepakd/atc-2011/csl.pdf · De nition Context Sensitive Grammar(Type1 Grammar) A context-sensitive grammar (CSG) is an unrestricted](https://reader030.vdocument.in/reader030/viewer/2022020104/5aba2c297f8b9ad13d8e8854/html5/thumbnails/16.jpg)
Context Sensitive Language
The language generated by the Context Sensitive Grammar is calledcontext sensitive language.
If G is a Context Sensitive Grammar thenL(G)=
{w∣∣(w ∈ Σ∗) ∧
(S ⇒+
G w)}
.
Eg 1 of a context sensitive grammarG = {{S ,A,B,C , a, b, c} , {a, b, c} ,P,S}where P is the set of rules.
S→ aSBCS → aBCCB → BCaB → abbB → bbbC → bccC →cc
The language generated by this grammar is {anbncn|n ≥ 1} .Aparna S Vijayan (CSA) CSG December 2, 2011 5 / 12
![Page 17: Context Sensitive Grammar - Dronadrona.csa.iisc.ernet.in/~deepakd/atc-2011/csl.pdf · De nition Context Sensitive Grammar(Type1 Grammar) A context-sensitive grammar (CSG) is an unrestricted](https://reader030.vdocument.in/reader030/viewer/2022020104/5aba2c297f8b9ad13d8e8854/html5/thumbnails/17.jpg)
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 6 / 12
![Page 18: Context Sensitive Grammar - Dronadrona.csa.iisc.ernet.in/~deepakd/atc-2011/csl.pdf · De nition Context Sensitive Grammar(Type1 Grammar) A context-sensitive grammar (CSG) is an unrestricted](https://reader030.vdocument.in/reader030/viewer/2022020104/5aba2c297f8b9ad13d8e8854/html5/thumbnails/18.jpg)
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 6 / 12
![Page 19: Context Sensitive Grammar - Dronadrona.csa.iisc.ernet.in/~deepakd/atc-2011/csl.pdf · De nition Context Sensitive Grammar(Type1 Grammar) A context-sensitive grammar (CSG) is an unrestricted](https://reader030.vdocument.in/reader030/viewer/2022020104/5aba2c297f8b9ad13d8e8854/html5/thumbnails/19.jpg)
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 6 / 12
![Page 20: Context Sensitive Grammar - Dronadrona.csa.iisc.ernet.in/~deepakd/atc-2011/csl.pdf · De nition Context Sensitive Grammar(Type1 Grammar) A context-sensitive grammar (CSG) is an unrestricted](https://reader030.vdocument.in/reader030/viewer/2022020104/5aba2c297f8b9ad13d8e8854/html5/thumbnails/20.jpg)
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 6 / 12
![Page 21: Context Sensitive Grammar - Dronadrona.csa.iisc.ernet.in/~deepakd/atc-2011/csl.pdf · De nition Context Sensitive Grammar(Type1 Grammar) A context-sensitive grammar (CSG) is an unrestricted](https://reader030.vdocument.in/reader030/viewer/2022020104/5aba2c297f8b9ad13d8e8854/html5/thumbnails/21.jpg)
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 6 / 12
![Page 22: Context Sensitive Grammar - Dronadrona.csa.iisc.ernet.in/~deepakd/atc-2011/csl.pdf · De nition Context Sensitive Grammar(Type1 Grammar) A context-sensitive grammar (CSG) is an unrestricted](https://reader030.vdocument.in/reader030/viewer/2022020104/5aba2c297f8b9ad13d8e8854/html5/thumbnails/22.jpg)
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 6 / 12
![Page 23: Context Sensitive Grammar - Dronadrona.csa.iisc.ernet.in/~deepakd/atc-2011/csl.pdf · De nition Context Sensitive Grammar(Type1 Grammar) A context-sensitive grammar (CSG) is an unrestricted](https://reader030.vdocument.in/reader030/viewer/2022020104/5aba2c297f8b9ad13d8e8854/html5/thumbnails/23.jpg)
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 6 / 12
![Page 24: Context Sensitive Grammar - Dronadrona.csa.iisc.ernet.in/~deepakd/atc-2011/csl.pdf · De nition Context Sensitive Grammar(Type1 Grammar) A context-sensitive grammar (CSG) is an unrestricted](https://reader030.vdocument.in/reader030/viewer/2022020104/5aba2c297f8b9ad13d8e8854/html5/thumbnails/24.jpg)
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 6 / 12
![Page 25: Context Sensitive Grammar - Dronadrona.csa.iisc.ernet.in/~deepakd/atc-2011/csl.pdf · De nition Context Sensitive Grammar(Type1 Grammar) A context-sensitive grammar (CSG) is an unrestricted](https://reader030.vdocument.in/reader030/viewer/2022020104/5aba2c297f8b9ad13d8e8854/html5/thumbnails/25.jpg)
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 →cAparna S Vijayan (CSA) CSG December 2, 2011 7 / 12
![Page 26: Context Sensitive Grammar - Dronadrona.csa.iisc.ernet.in/~deepakd/atc-2011/csl.pdf · De nition Context Sensitive Grammar(Type1 Grammar) A context-sensitive grammar (CSG) is an unrestricted](https://reader030.vdocument.in/reader030/viewer/2022020104/5aba2c297f8b9ad13d8e8854/html5/thumbnails/26.jpg)
Hierarchy of Formal Languages
Aparna S Vijayan (CSA) CSG December 2, 2011 8 / 12
![Page 27: Context Sensitive Grammar - Dronadrona.csa.iisc.ernet.in/~deepakd/atc-2011/csl.pdf · De nition Context Sensitive Grammar(Type1 Grammar) A context-sensitive grammar (CSG) is an unrestricted](https://reader030.vdocument.in/reader030/viewer/2022020104/5aba2c297f8b9ad13d8e8854/html5/thumbnails/27.jpg)
Relation between Formal Languages
The CFL’s not containing ε are properly contained in the contextsensitive languages
Not all Context Sensitive Languages are Context Free.
Every Context Sensitive Language is recusive.
Aparna S Vijayan (CSA) CSG December 2, 2011 9 / 12
![Page 28: Context Sensitive Grammar - Dronadrona.csa.iisc.ernet.in/~deepakd/atc-2011/csl.pdf · De nition Context Sensitive Grammar(Type1 Grammar) A context-sensitive grammar (CSG) is an unrestricted](https://reader030.vdocument.in/reader030/viewer/2022020104/5aba2c297f8b9ad13d8e8854/html5/thumbnails/28.jpg)
Chomsky Hierachy
Described by Chomsky in 1956.
Four classes of language Type 3,Type2,Type1,Type 0 from mostrestrictive to most general(Unestricted).
Each level of hierarchy can be characterized by a class of grammar.
Aparna S Vijayan (CSA) CSG December 2, 2011 10 / 12
![Page 29: Context Sensitive Grammar - Dronadrona.csa.iisc.ernet.in/~deepakd/atc-2011/csl.pdf · De nition Context Sensitive Grammar(Type1 Grammar) A context-sensitive grammar (CSG) is an unrestricted](https://reader030.vdocument.in/reader030/viewer/2022020104/5aba2c297f8b9ad13d8e8854/html5/thumbnails/29.jpg)
Chomsky Hierarchy
Type Language productions Device3 Regular A → aB, A → a Finite Automaton
2 CFL A → α Pushdown Automaton
1 CSL α→ β,|β| ≥ |α| Linear Bounded Automaton
0 RE α→ β Turing Machine
A,B→ Nonterminals
α,β→ string of terminals and nonterminals
a→ terminal symbol
Aparna S Vijayan (CSA) CSG December 2, 2011 11 / 12
![Page 30: Context Sensitive Grammar - Dronadrona.csa.iisc.ernet.in/~deepakd/atc-2011/csl.pdf · De nition Context Sensitive Grammar(Type1 Grammar) A context-sensitive grammar (CSG) is an unrestricted](https://reader030.vdocument.in/reader030/viewer/2022020104/5aba2c297f8b9ad13d8e8854/html5/thumbnails/30.jpg)
References
Introduction to Automata Theory Languages and Computation by ”JOHN EHOPCROFT”,”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