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Regular Grammars

Regular Grammars. Right-linear Grammar. A grammar G=(V,T,S,P) is right-linear if all productions are of the form A -> xB, or A -> x where A,B  V, and x  T*. Language of RL Grammar. Claim: For all right-linear grammars G, L(G) is regular Proof by construction. A RL Grammar for a RL.

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Regular Grammars

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  1. Regular Grammars

  2. Right-linear Grammar • A grammar G=(V,T,S,P) is right-linear if all productions are of the form • A -> xB, or • A -> x where A,B  V, and x  T*

  3. Language of RL Grammar • Claim: For all right-linear grammars G, L(G) is regular • Proof by construction

  4. A RL Grammar for a RL • For any regular language L over alphabet ∑, there is a right-linear grammar G=(V,∑,S,P) such that L = L(G) • Proof by construction

  5. Questions (1/3) • Is the language of the grammar G1 = ({S,A},{a,b},S,{S -> aS | A, A -> bA}) a regular language? • If it is, give an FA with the same language.

  6. Questions (2/3) • Is the language of the grammar G2 = ({S,B}, {a,b}, S, {S -> aB | λ, B -> Sb}) a regular language? • If so, give an FA with the same language as G2.

  7. Questions (3/3) • Give a regular grammar with the same language as the following DFA. c q1 a,b b a,b,c a q0 q3 c b q2 c a

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