context sensitive grammar - indian statistical institute ansuman/flat2018/csl-lba.pdfآ  de nition...

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

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

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

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

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

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

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 α1Aα2→α1βα2,with β 6= �. 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

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

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

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

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

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

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

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 {anbncn|n ≥ 1} .

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 {anbncn|n ≥ 1} .

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 {anbncn|n ≥ 1} . Aparna S Vijayan (CSA) CSG December 2, 2011 5 / 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 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 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 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 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 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 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 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 6 / 12

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 Aparna S Vijayan (CSA) CSG December 2, 2011 7 / 12

Hierarchy of Formal Languages

Aparna S Vijayan (CSA) CSG December 2, 2011 8 / 12

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.

Aparna S Vijayan (CSA) CSG December 2, 2011 9 / 12

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.

Aparna S Vijayan (CSA) CSG December 2, 2011 10 / 12

Chomsky Hierarchy

Type Language productions Device 3 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