Context Sensitive Grammar Habeeb and Alvin ATC Seminar 30 NOV 2018 - PowerPoint PPT Presentation

Context Sensitive Grammar Habeeb and Alvin ATC Seminar 30 NOV 2018 Habeeb and Alvin CSG 30 NOV 2018 1 / 15

Overview Introduction 1 Formal definition 2 Context Sensitive Language 3 Examples Closure properties 4 Relation Between Recursive and CSL 5 Habeeb and Alvin CSG 30 NOV 2018 2 / 15

Definition A context sensitive grammar (CSG) is a grammar where all productions are of the form α A β → αγβ where γ � = ǫ During derivation non-terminal A will be replaced by γ only when it is present in context of α and β . This definition shows clearly one aspect of this type of grammar; it is noncontracting , in the sense that the length of successive sentential forms can never decrease. Habeeb and Alvin CSG 30 NOV 2018 3 / 15

Formal definition A context sensitive grammar G = ( N , Σ , P , S ) , where N is a set of nonterminal symbols Σ is a set of terminal symbols S is the start symbol, and P is a set of production rules, of the form α A β → αγβ where A in N, α, β ∈ ( N ∪ Σ) ∗ and γ ∈ ( N ∪ Σ) + The production S → ǫ is also allowed if S is the start symbol and it does not appear on the right side of any production. Habeeb and Alvin CSG 30 NOV 2018 4 / 15

Context Sensitive Language A language L is said to be context-sensitive if there exists a context-sensitive grammar G, such that L = L ( G ). If G is a Context Sensitive Grammar then, S ⇒ + � � L ( G ) = { w | ( w ∈ Σ ∗ ) ∧ } G w Habeeb and Alvin CSG 30 NOV 2018 5 / 15

Context Sensitive Language : Example Example. The following grammar(G) is context-sensitive S → aTb | ab aT → aaTb | ac L ( G ) = { ab } ∪ { a n cb n | n > 0 } Habeeb and Alvin CSG 30 NOV 2018 6 / 15

Context Sensitive Language L ( G ) = { ab } ∪ { a n cb n | n > 0 } This language is also a context-free. For example, Context free grammar(G1) for this. S → aTb | ab T → aTb | c Any context-free language is context sensitive Not all context-sensitive languages are context-free. Habeeb and Alvin CSG 30 NOV 2018 7 / 15

Context Sensitive Language : Example Example L = { a n b n c n | n > 0 } Habeeb and Alvin CSG 30 NOV 2018 8 / 15

Context Sensitive Language : Example Example L = { a n b n c n | n > 0 } Context sensitive grammar(G) 1 . S → aBC 2 . S → a SBC 3 . aB → ab 4 . bB → bb 5 . bc → bc 6 . cC → cc 7 . CB → CZ 8 . CZ → WZ 9 . WZ → WC 10 . WC → BC Habeeb and Alvin CSG 30 NOV 2018 9 / 15

Closure properties Context Sensitive Languages are closed under Union Intersection Complement Concatenation Kleene closure Habeeb and Alvin CSG 30 NOV 2018 10 / 15

Closure properties : Union Let G 1 = ( N 1 , T 1 , P 1 , S 1 ) and G 2 = ( N 2 , T 2 , P 2 , S 2 ), s.t L ( G 1 ) = L 1 and L ( G 2 ) = L 2 . Construct G = ( S ∪ N 1 ∪ N 2 , T 1 ∪ T 2 , { S → S 1 , S → S 2 } ∪ P 1 ∪ P 2 , S ) s.t N 1 ∩ N 2 = ∅ and S / ∈ { N 1 ∪ N 2 } . G also CSG and any derivation has the form S ⇒ S i ⇒ ∗ G i w ∈ L ( G i ) for some i ∈ { 1 , 2 } We can derive only words and all words of L ( G 1 ) ∪ L ( G 2 ) = L 1 ∪ L 2 Therefore L 1 ∪ L 2 = L ( G ) ∈ L ( CS ) Habeeb and Alvin CSG 30 NOV 2018 11 / 15

Closure properties : Concatenation Let G 1 = ( N 1 , T 1 , P 1 , S 1 ) and G 2 = ( N 2 , T 2 , P 2 , S 2 ), s.t L ( G 1 ) = L 1 and L ( G 2 ) = L 2 . Construct G = ( S ∪ N 1 ∪ N 2 , T , { S → S 1 S 2 } ∪ P 1 ∪ P 2 , S ) s.t N 1 ∩ N 2 = ∅ and S / ∈ { N 1 ∪ N 2 } Any derivation in G has the form S ⇒ S 1 S 2 ⇒ ∗ G 1 w 1 S 2 ⇒ ∗ G 2 w 1 w 2 S ⇒ w i is a derivation in G i . i.e. the derivation only uses rules of P i . The derivations in G 1 and G 2 cannot be influenced by the contexts of the other part. So G is a context sensitive grammar, L ( G ) is a CSL. Habeeb and Alvin CSG 30 NOV 2018 12 / 15

Relation Between Recursive and CSL Theorem Every context-sensitive language L is recursive. For CSL L, CSG G, Derivation of w S ⇒ x 1 ⇒ x 2 ⇒ x 3 · · · · ⇒ w has bound on no of steps.(Bound on possible derivations). We know that | x i | ≤ | x i +1 | (G is non contracting). We can check whether w is in L(G) as follows Construct a transition graph whose vertices are the strings of length ≤ | w | Paths correspond to derivation in grammars. Add edge from x to y if x ⇒ y w ∈ L ( G ) iff there is a path from S to w Habeeb and Alvin CSG 30 NOV 2018 13 / 15

References An Introduction to Formal Languages and Automata by Peter Linz https: / / en. wikipedia. org/wiki/Context-sensitive grammar https://gyires.inf.unideb.hu/GyBITT/14/index.html old seminars. Habeeb and Alvin CSG 30 NOV 2018 14 / 15

The End Habeeb and Alvin CSG 30 NOV 2018 15 / 15