ONLY NEED AN ANSWER 10.C

THE ENTIRE QUESTION IS POSTED FOR CONTEXT

Let be finite alphabet. A code is a mapping f : {0, 1} + . For example, if = {a, b, c}, a code f might be f(a) = 101, f(b) = 01100, and f(c) = 10. (To simplify things, we assume for any a 6= b, we have f(a) 6= f(b). ) For a string w1w2 wm , we dene f(w) = f(w1)f(w2) f(wm). In the above code, f(abcba) = 101 01100 10 01100 101. = 101011001001100101.

10 (100 PTS.) Prove Regular Let be finite alphabet. A code is a mapping f: + {0,1}+. For example, if = {a,b,c}, a code f might be f(a) = 101, f(0) = 01100, and f(c) = 10. (To simplify things, we assume for any a #b, we have f(a) + f(6). ) For a string w1w2... Wm E E*, we define f(w) = f(wi) f(w2) ... f(wm). In the above code, f(abcba) = 101. 01100 10 01100 101. = 101011001001100101. 10.A. (10 PTS.) Let L be the language of the following DFA M. What language does L represent ? a,b,c a.b.c 10.B. (20 PTS.) Working on the DFA M from 10.A. construct an NFA for the language f(L). Here, f(L) = {f(w) | We L} is the code language where f is code from the above example. 10.C. (70 PTS.) Let L C I* be a arbitrary regular language. Prove that the coded language f(L) = {f(w) | We L} is regular. Specifically, given a DFA M = (0, 2, 8, , A) for L, describe how to build an NFA N for f(L). Then, prove the correctness of your construction, i.e., the language of the constructed NFA is indeed the desired language f(L). Your construction and proof should be for any arbitrary code f and not just the code in the example above. (30 points for a correct construction, and 40 points for a correct proof of correctness.)