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A - B 1B 2... B n where the B; are nonterminals and n ~ 2 CHAPTER 2. CONTEXT-FREE GRAMMARS 21 To get G 2 from Gh proceed as follows. Let GI = (Nh Th Ph SI). Define G2 = (N 2, T2, P 2, S2), where T2 = Th S2 = Sh and N 2 contains all symbols in NI plus a new nonterminal for each terminal in TI = T2. If a is in TI = T2, then will denote the new nonterminal corresponding to a. P 2 is the set of all productions of the following forms: 4. 5. 6. 7. X2 ... X~ provided A - XIX2 ... X n is in PI and Xi if X; is a terminal and X j = X; if X; is a nonterminal.

5. Prove that the following are not cfl's: a. {anb2nc3n In ~ 0} b. {aa Ia E {a, b}*} c. {a; I i a prime} d. {a; I i = n 2, for some positive integer n} e. {anbncbnan In~ 1} 6. *- L is not a cfl. r. * ~ L and 7. 14 by proving the claim that L (G') = L (G2)ยท 8. 16. 9. 12 to find a nonerasing cfg equivalent to the following cfg: G = (N, T, P, S) where N = {S, X}, T= {a, b} P = {S- XS, S-A, X- aXb, X- A} 10. What is L (G)? Gas in 9. CHAPTER 2. CONTEXT-FREE GRAMMARS 29 11. 14 to find a cfg in Chomsky normal form that is equivalent to G.

Without loss of generality, we will assume that States(M1) and States(M2) are disjoint. If they are not disjoint, we can always make them so by replacing each state of M 1 by a state which is not 44 CHAPTER 3. FINITE-STATE ACCEPTORS in States (M2). cceptor Mp by setting Symbols(Mp) = Symbols(M1) = States(Mp) = States(M1) U States(M2), Symbols(M2), Start(Mp) = Start(M1), Accepting-states(Mp) = Acceptingstates(M2), and Instructions(Mp) = 8", where 8" is defined as follows. For any symbol a and any state q 1 of Mt.

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