Suppose the PDA P = (Q= {go, q1}, Σ={0,1}, r={Zo, X}, &, qo, z=Zo, F={q1} ) has the following transition function: Automata theory, Assignment, UMS, Malaysia

1. Suppose the PDA P = (Q= {go, q1}, Σ={0,1}, r={Zo, X}, &, qo, z=Zo, F={q1} ) has the following transition function:
a. (go, 0, Zo) = {(go, XZo)}.
b. (go, 0, X)= {(go, XX)}.
c. (go, 1, X)= {(go, X)}.
d. (qo, λ, X)= {(1, 2)}.
e. &(q1, λ, X)= {(91, λ)}. f. &(q1, 1, X)= {(91, XX)}. g. (q1, 1, Zo) = {(1, 2)}.

Starting from the initial ID (go, w, Zo), show ALL the reachable ID’s and list the transition functions that are used when the input w is:
a. 01. [Hint: there are two ways]
b. 0011. [Hint: there are three ways] c. 010.
(6 marks) (19 marks) (5 marks)

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5. Let C be a context-free language and R be a regular language. Let P be the PDA that recognizes C, and D be the DFA that recognizes R. If Q is the set of states of P and Q’ is the set of states of D, we can construct a PDA P’ that recognizes CR with the set of states Q x Q’. P’ will do what P does and also keep track of the states of D. It will accept a string wiff it stops a a state qe FpxFD, where Fp is the set of accept states of P and FD is the set of accept states of D. Since CR is recognized by P’, it is context free.

Let A= {w we (a, b, c)* and w contains equal numbers of a’s, b’s, and c’s}. Use the fact that CR is context free to show that A is not a context free language.

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