1. Construct a regular expression defining each of the following languages over the alphabet {a,b}: a) All strings of as and bs such that every pair of adjacent as appears before any pair of adjacent bs. (2 pts) b) All strings in which the total number of as is divisible by four, such as aabaaaabbaba. (2 pts) 2. Find finite accepter for the language L = {w: there is a symbol ai Îå not appearing in w} if the alphabetå = {a1, a2, a3}. (2 pts) 3. What is the complement of the language L accepted by the nfa onå = {a, b}: (2 pts) a l

1. Construct a regular expression defining each of the following languages over the alphabet {a,b}: a) All strings of as and bs such that every pair of adjacent as appears before any pair of adjacent bs. (2 pts) b) All strings in which the total number of as is divisible by four, such as aabaaaabbaba. (2 pts) 2. Find finite accepter for the language L = {w: there is a symbol ai Îå not appearing in w} if the alphabetå = {a1, a2, a3}. (2 pts) 3. What is the complement of the language L accepted by the nfa onå = {a, b}: (2 pts) a l l 4. a) Find a npda that accept the language L = { am bn, m£ 2 n }. (4 pts) b) Find a context-free grammar that defines the given language L. (3 pts) 5. Find CFG for the languages: a) L1 = {a x b y a zb w where x, y, z, w 0 and y x and z w and x+z=y+w } overå = {a, b}. (3 pts) b) L2 = { an bmck: çn mç = k, n ³ 0, m³ 0, k ³ 0 } over å={a,b,c}. (3. pts) 6. Find a left linear grammar for the language L((bb+a*b*ab)*). (3 pts.) 7. Transform the grammar G with productions into Chomsky Normal Form S ® ACaS A ® BC B ® bB ½ ? C ® c ½ ? (3 pts) 8. Construct the Turing machine that will accept the language of balanced strings of parentheses on ={ ( , ) , X }. For example, the string w = (((X)XX)(XXX)) belongs to the language and will be accepted by TM, and the strings: ((X)X((XX)) and )X( are not accepted. (5 pts.) 9. Let L = { aibj ck : i¹ j or j¹ k} a) Show that the language L is a context-free language. (3 pts.) b) Show that a complementof the language L is not a context-free language. (1 pts.) 10. Construct npda that accept the following language on {a, b, c}: L = { w: a n+1 b n+m c m , n = 0, m = 0} ( 4 pts.)

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