Let L, and L2 be the languages defined as follows: L1 ={ xe{0,1}* | x has an odd number of Os}. L2 ={ 0"1m0" | n, m 20} Which of the following context-free grammars generates the language L1 U L2? (Here all uppercase letters are variables) O E- 1E| OF F 1F| OE je A- OAO | B B - 1B|E OS-EJA E → OF | 1E F- OE | 1F|E A- OAO | 1A |E OS- S1 | S2 S1→ 00S | 15 | 01A| 10A | B A- 10S | 01S B -1|8 S2 → OS,0| A2 A2 → 1| S2|E S- EJA E - 1E | OF F 1F| OE je A- OAO | B B - 1B|E

Let Σ be a non-empty alphabet and LCE* be a context-free language. Which of the following languages is/are necessarily context-free? Note: For some string w = Σ*, w is the string reversal. € 1. reverse_1(L) = {xy : ³x, y ≤ Σ*, x € L and yR € L} R 2. reverse_2(L) = {x : 3x € Σ*, x € L and x² EL If a language is necessarily context-free, prove your claim. Any construction you make does not require a proof of correctness. If a language is not necessarily context-free, give a counterexample language L which is context- free but for which reverse_i(L) (i 1, 2) is not context-free. To do so, you must show that your language L is context-free by giving either a PDA which accepts it or a CFG which generates it. You do not need to prove the correctness of your construction but you should explain how it works. In addition, you must give a Pumping lemma proof which shows that reverse; (L) is not context-free. =

Assumptions ...   Alphabet = {a, b}   • L1 = { a"b" : n ≥ 0 ^ n mod 3 = 0}   L2 = {anbm : n ≥ 0 ^ m ≥ 0 ^ n = m −1}   To do in this exercise ...   ⚫ Find a context-free grammar G such that L(G) = L1 L2, the concatenation of languages L1 and L2.   ⚫ Save your context-free grammar as a JFLAP.jff file, and submit that .jff file to Canvas as your solution to th is exercise.

9. Show that the language +kin L = {a*b"+k : n > 0, k > 1} U {a"+*b" : n 2 0, k > 3} is not regular.

a) Construct a push-down automaton that accepts the language L = {a't°c*[i = j or i=k for i> 0}.

6.. The following is a grammar for JH-Lisp 2.1, modified version of the language you encountered in homework 4, with slightly different syntax, and expanded set of functions. It uses underscores rather than spaces, square brackets rather than parentheses, and words as function names instead of symbolic operators. S → [0.E_E] E → [0.E.E]|N N → 0|1|2|3|4|5|6|7|8|9|NN 0 → add|sub|mul|div|mod|shl|shr Create a PDA that recognizes JH-Lisp 2.1.

rt- Suppose that the language LO has three elements and L1 has two. Knowing only that information, for each of these, what is the least number of elements possible and what is the greatest number possible? (a) LO u L1 (b) LO n L1 (c) LO_L1 (d) L21 (e) L1R (f) LO+ n L1.

Q1. Design Context free grammars for the following: 1. {w € {0, 1}*| w contains at least three 1s } 2. { w € {0, 1}* | w =w* and |w| is even } Where R indicates reverse. 3. { w e {0, 1}*| the length of w is odd and the middle symbol is 0 } 4. { a'b'c* | i, j, k 20, and i = j or i > k} 5. { a'bic* |i, j, k20 and i +j=k } 6. The language A of strings of properly balanced left and right brackets: every left bracket can be paired with a unique subsequent right bracket, and every right bracket can be paired with a unique preceding left bracket. Moreover, the string between any such pair has the same property. For example, [] [[ D]E A. 7. { a"b" | n21} 8. L= {w|w E (0,1)* and |w| = 2} 9. L= {w|w E (0,1)* and |w|2 2} 10. L= {w|w E (0,1)* and |w| 1} 17.{ a"bm*c" | n,m 21}

Let the following languages Li be given over the alphabet E = {O, I, 2}. . L1-u-2 | υε Σ') . L- 11 . υ | υ ε Σ} V • L3 = {2"10" | n € N} • L4 = {0"12"+m | n, m E N} Subtask (a) Answer the following question and give a reason for your answer. Which of these languages Li has a Nerode index greater than or equal to three? Subtask (b) Reason that the two languages LI and L2 have a finite Nerode index. Subtask (c) In order to justify that the language L3 has an infinite Nerode index, one must specify an infinite number of character strings uj from E* to which different sets of suffixes SL3 (uj) are assigned in pairs. For this purpose it is advisable to choose the character string uj = 2-1 for each je IN. Answer the following questions. (i) State how many and which character strings from E*the one String contains the suffix set ŞL3 (uj) assigned to uj. Kii) Explain why it follows from j1 #ja that the suffix sets assigned to the two strings ujl and uj2 are different, ie. H. that SL3 ujil ESL3…

d.