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.
- (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.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.
- 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.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 lilil(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
- 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)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.