1. Given the following regular grammar: S→aSlalld, I→K1al, K→aK1d, J→ all bJ | ∧ one can use the algorithm given in slide 31 of the notes "Regular Languages and Finite Automata- IV" to construct an NFA to recognize the language of that grammar. Fill out blanks in the following figure so that the resulting NFA would fulfill such a task, i.e., would recognize the language of the given grammar. ㅁㅁ 80000 S K

THEORY of Automata. grammar G

1. Prove that the language L(G) is not regular where G is the following context-free grammar: G = ({S,A,B,C}, {a,b}, {S->bB|C, A>b|C, B->Sb, C->aAa }, S). Note: You must first determine L(G).

2. Using Pumping Lemma (slides 30-35 of the notes 'Regular Languages & Finite Automata-IV') one can show the language L= {a"b" | n€N } is not regular (We need this property in the notes 'Context-free Languages and Pushdown Automata I). This is done by way of contradiction. We assume L is regular. Since L is infinite, Pumping Lemma applies. We then consider the string s = ambm where m is the number of states in the DFA that recognizes L. Since the length of s is bigger than m, by Pumping Lemma, there exists strings x,y and z such that s = xyz, y + A, |xy| < 2m and xykz eL for all k e N. If |xy| < 2m then the first repeated state on the acceptance path cannot be a final state. Is this statement true, why or why not?

Given a regular grammar G and a list of strings, identify the strings that are in L(G).