# Pushdown Accepters & Context-Free Grammars - PowerPoint PPT Presentation

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Pushdown Accepters & Context-Free Grammars. Sipser, Theorem 2.12 Denning, Chapter 8. Fundamental theorem of CFLs and PDAs. Theorem 2.12 (Sipser) A language L is context-free if and only if (iff) there exists a pushdown accepter M that recognizes L. CFL if PDA; PDA -> CFL.

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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

### Fundamental theorem of CFLs and PDAs

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

### CFL if PDA; PDA -> CFL

• 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

### CFL only if PDA; CFL -> PDA

• Given a CFL, L, generated by grammar G, we can build a PDA, M, which recognizes language L.

• Proof: Denning, Section 8.3

### Proof: CFL -> PDA

• 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}

### Program of M

• 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

### Example: L = {0k1k | k  0}

• Grammar G has productions:SA; A01; A0A1

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

### Ex 2: L={0j1k2j+k| j,k > 0}

• SA; A0A2; A0B2; B12; B1B2