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Pushdown Accepters & Context-Free Grammars

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Pushdown Accepters & Context-Free Grammars

Sipser, Theorem 2.12

Denning, Chapter 8

- Theorem 2.12 (Sipser) A language L is context-free if and only if (iff) there exists a pushdown accepter M that recognizes L

- Given a PDA, M, that recognizes language L, then there exists a CFG, G, that generates language L.
- Proof: Denning, Section 8.4, using traverse sets

- Given a CFL, L, generated by grammar G, we can build a PDA, M, which recognizes language L.
- Proof: Denning, Section 8.3

- Since L is context free, there is a context free grammar, G = (N,T,P,S) that generates L.
- Construct the PDA, M=(Q,T,U,P’,q0,{q3}), as follows:
- T = same input alphabet
- U = N T {S} = stack alphabet
- Q = {q0,q1,q2,q3} {qx | x U}

- q0: , push($), goto q1q1: , push(S), goto q2q2: , pop($), goto q3
- For each production A->wq2: , pop(A), goto qAqA: , push(wR), goto q2
- For each terminal symbol aq2: a, NOP, goto qaqa: , pop(a), goto q2

- Grammar G has productions:SA; A01; A0A1

push 1

push A

push 0

pop A

push A

push 0

pop S

push S

push $

pop $

pop 0

in 1

pop 1

in 0

- SA; A0A2; A0B2; B12; B1B2