S→ abAB | ba, A→ aaa, BaA | bb and SabAaA | abAbb | ba, A-aaa are equivalent.
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- (1) Grammar G is N→D|ND D→0|1|2|3|4|5|6|7|8|9 What is the language L(G) specified by G? Write the leftmost and rightmost derivations of the sentences 0127, 34 and 568.2. [Based on Problem 2.26, page 157 in Sipser's 3rd Edition] Suppose G is a context-free grammar in Chomsky normal form. This exercise will help you prove that for any string w e L(G) of length n 2 1, exactly 2n - 1 steps are required for any derivation of w. To make the proof simple, we define: a variable rule is any rule of the form A BC, where A, B,C are variables, • a terminal rule is any rule of the form A → a, where A is a variable and a is a terminal. Now consider a derivation in G that starts with the start variable and applies apply variable rules s times (may not be consecutive) and applies terminal rules t times, and let r be the string generated by the derivation (z can contain both variables and terminals). (a) Prove that the string r contains t terminals. (b) Prove that the string r contains (1+s – t) variables. (c) Prove that if r E L(G) and |2| = n, then s +t = 2n – 1. %3DChapter 11 Section 7 3(b) S-> abA | abcA A -> aA |A Use left-factoring to reduce k in this LL(k) grammar. Please indicate what is k in the resultant grammar and why.
- Q5 (a) Reduce the grammar to Chomsky Normal Form. SzaAB|caB AnzaCb| bbA B→ cb| Ac C→ id1. (a) Show that the following grammar is ambiguous S → AB | aaB Aa | a (b) Find an unanbiguous grammar that describes the same language.(1) Grammar G is N→ DIND D→0|1|2|3|4|5|6|7|8|9 What is the language L(G) specified by G? Write the leftmost and rightmost derivations of the sentences 0127, 34 and 568.
- Problem 3: Consider the grammar S → aSlaSbS|e. This grammar is ambiguous. (a) Show in particular that the string aab has two parse trees. (b) Generate an unambiguous grammar for the language generated by the given grammar.8) Transform the grammar with productions Chomsky normal form. SabAB A → baB |λ B→BAA|A|A intoProblem 3 A, B are two regular languages over the alphabet Σ. Prove the language A_Contain_B = {x € Σ* | x € A and x contains some string y € B as a substring} is Regular. For example, when A = {aba, baa, bab} and B = {ab}, A_Contain B = {aba, bab} (Hint: You can make use of closure properties proved in lectures)
- 1 1.12 3 4. 5. . ... Theorem: Let G= (V,T,P.S) be a rghtlimear grammar. Then there exists a language L(G) which is accepted by a FA. i. , the language generated from the regular grammar is regular language. 1/1 lilil1)Let G = (V, ∑, R, S) be a grammar with V = {S, H, Gm J}; ∑ = {h, g, j}; and the set of rules: S → H H → h | hJG G → Gg | Sg | HgJ J → jH | HHJ| Jj a)Convert this grammar to a PDA using the method we described. b)Convert G to Chomsky normal form.(20 Points) Consider the following grammars where S is the start symbol and a, b, and c are terminal symbols and A, B, and C are non-terminal symbols.S ---> aA | bB | cCA --> aA | aB | aB --> bB | bC | bC --> cC | cExplain in your own words what the grammar generates. Which of the following sentences are in the language generated by the grammar? Show derivations. If a sentence cannot be generated by the grammar, explain why abcabcab aaaaaabb cccccaaaa bbbbbbbc aaaaaaccc