pushdown automata (pda)

Proof L(G) ⊆ L(M)Consider partial derivationS ∗ ⇒ a 1 ...a n AA 2 ...A m ⇒ a 1 ...a n bB 1 ...B k A 2 ...A mBy construction of M, if the stack content is AA 2 ...A mafter reading a 1 ...a n then it is B 1 ...B k A 2 ...A m afterreading a 1 ...a n b.That is, the stack content matches the variable string inthe sentential form, and the input position matches theterminal prefix of the sentential form. This can beformally proved by induction on the number of steps.Thus, if S ∗ ⇒ w, then eventually M will empty the stackand end up in q f using w as input.Pushdown Automata – p. 12