Pushdown Accepters & Context-Free Grammars

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

Pushdown Accepters & Context-Free Grammars