Pushdown Accepters & Context-Free Grammars

Sipser, Theorem 2.12Denning, 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 q3For each production A->wq2: , pop(A), goto qAqA: , push(wR), goto q2For each terminal symbol aq2: a, NOP, goto qaqa: , pop(a), goto q2

Example: L = {0k1k | k 0}

Grammar G has productions:SA; A01; A0A1

push $ push Spop S

push A

pop A

push 1

push 0push A

push 0

pop $

in 0

pop 0 in 1pop 1

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

SA; A0A2; A0B2; B12; B1B2