Problem 7(a). Trace out an accepting computation of P on the input x = aabbabaababa (please try to use my "mountain diagram format). For each step show the input consumed (if any), the resulting state, and the resulting stack. Problem 7(b). Using the textbook construction, convert the PDA P to a grammar G, with start symbol A14. Try to omit all rules from G that are not necessary to generate L(P). Problem 7(c). Draw a parse tree for aabbabaababa, corresponding to the computation in the first part. Problem 8 Problem 8(a). Let L = {a'bickd': i=0 or j=k=1}. Consider the context-free pumping game for L. State a winning strategy for C. Problem 8(b). See "Ogden's Lemma" on wikipedia. State a "game version of it. (This should resemble our context-free puming game, but now one of the players gets to "mark some positions in s. Ogden's Lemma should say that C wins your game, if L is context-free.) Problem 8(c). Describe a winning strategy for N in your game. (Remark: this shows the converse of Pumping Claim 2 is false.) Problem 9 Consider this modified version of the regular pumping game: 1. R' chooses an integer p > 0. 2. N chooses a string s E L such that s p . 3. R' chooses strings x,y,z such that s = xyz, [x2]

0. 4. N chooses an integer i >0 such that cy'z & L. (The difference is that R is renamed R', and we require |xz

0. 2. N chooses a string s E L such that s p . 3. R' chooses strings x,y,z such that s = xyz, [x2]

0. 4. N chooses an integer i >0 such that cy'z & L. (The difference is that R is renamed R', and we require |xz