1) L1 = the set of all strings that starts with 0 and ends with 1 2) L2 the set of all strings that starts with 1 and have even length 3) L3 the set of all strings that have length at least 3 and its third symbol is 0 %D 4) L1UL3
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- Construct a graphical representation of DFA that accepts L={w∈(a+b)*: na(w) is even, nb(w)>1}no need to include explanations, just the graphical DFA.Please note that “a”s don't need to be consecutive.Consider the language over the alphabet Σ = {0, 1} containing strings which include nonempty binaries(strings with 0’s and 1’s) that start with 1 and end with 0 and have at least three digits. For example, 110and 1100 are such binaries, while 10 and 010 are not.1. Write a regular expression for this language 2. Draw a DFA diagram for this language, with no more than six states. 3. Is the language regular? Why or why not?Given two languages: L1 = { x ∈{a,b}* | x starts with b} L2 = { x ∈ {a,b}* | x ends with a} Construct a FA for each of L1,L2. Be sure that the Q sets for the two FA are disjoint where the names of the states are not the same. Then using the construction technique, generate a FA that accepts L1*L2 where * is intersection. Draw the FAs
- 1) Given the alphabet as {0, 1}, write a DFA for the following three regular languages.(Give the complete description of the DFA with transition function as a table and also as a transition graph) (i) L = {w | w is a string of even length} (ii) L = {w | |w| mod 3 = 0} (iii) L = {w | w contains the string 001 as a substring} 2)Describe the extended transition function on a string recursively using transitions one symbol at a time for the three automata of the above problem. (i) A string (of length >=4) that belongs to the language (II) A string (of length >=4) that does not belong to the languageThe quotient of two languages L_1/L_2 = { x | ∃y ∈ L_2 (xy ∈ L_1) } That is, the set of strings from which a string in L_1 could be created by appending a string in L_2. So, {aaa, aab, aba, abb} / {a, bb} = {aa, ab, a} Are regular languages closed with respect to quotient? What if L_1 is not regular but L_2 is? Is L_1/L_2 guaranteed to be regular? What if L_2 is not regular but L_1 is? Is L_1/L_2 guaranteed to be regular? Exhibit a nonregular language L1 and regular language L2 for which L1/L2 is nonregular.Describe a DFA that accepts each of the following languages. Describe briefly what each state in your DFA means. Do not attempt to draw your DFA, since the number of states could be huge; instead, give a mathematically precise description of the states Q, the start state s, the accepting states A, and the transition function δ All strings x ∈ {0, 1}* such that x contains 0^9 as a substring, or |x| is divisible by 8, or #0(x) − #1(x) is divisible by 7. #0(x) denotes the number of 0's in x, and #1(x) denotes the number of 1's in x.
- Consider the ambiguous grammar S → ABS | ABA → aA | λB → bB | λ What is the shortest string in L(S) other than λ that can be used to show the grammar is ambiguous? Write the string without any spaces or quotation marks. Now show the grammar ambiguous by giving two derivations for the string (each derivation should have a particular important property). Give the two shortest derivations you can find, writing the shorter one in the first box. Your derivations should begin with S, end with your chosen string, and only make one substitution per step. Do not include any spaces, quotes, or lambdas in your answer. Use the ">" character to represent the right arrow. For example, S>aSb>ab might be a legal derivation for a string in another language.Construct DFAs for the following languages. Give both the state diagrams and the state-transition tables. The set of binary strings which, when converted into an integer number, are divisible by 6. The string must not have leading zeros (for example, 110, 0, and 11110 will be accepted, 0110, 1111, and 101 will be rejected). Empty string (ϵ) will be rejected. (Σ = {0,1} The set of binary strings whose length is odd and has 1 in every even position. (The strings 0 and 1 will also be accepted by the DFA). Σ={0,1}. For example: 010, 01111, 11010, 1111010, etc.Consider the languages L = {a'bm qlxm|1, m 2 0} and Lo = {a2lym}|1, m 2 0}. (Recall from the previous exercise that we would state that L is non-context-free). Which is the simplest model of computation one may use on the language that is the intersection of L and L? Show your reasoning.
- Let Cbe the language over {a, b,c, d, e, f} accepting all strings so that: 1.There are precisely two b’s in the string. 2.Every a is immediately followed by an odd number of e’s. 3.Every d is immediately followed by an even number of f’s. 4.e’s and f’s don’t occur except as provided in rules 2 and 3. 5.All a’s occur after the first b. 6.All d’s occur before the second b. 7.In between the two b’s there are exactly twice as many a’s as d’s.HINT: Don’t be intimidated by the number of rules. The way they interact actually helps you simplify the number of cases you have to deal with while constructing this grammar. a. Construct a context-free grammar generating C.Explain how your construction accounts for each rule. b. We could eliminate one rule from Cand make it regular. Which rule would that be, and why would it work? You do not need a formal proof, but you do need an explanation.Given: DFA M1 Transition Table Describe the Language L represented by DFA M1 over the alphabet = {a, b}. w contains baba as a substring a’s immediately follow b’s an even number of times OR b’s immediately follow a’s an even number of times a’s immediately follow b’s an even number of times a’s immediately follow b’s an even number of times AND b’s immediately follow a’s an even number of times w does not contain baba as a substring b’s immediately follow a’s an even number of timesGive a state transition diagram of a DFA that recognizes the following language over the alphabet{0,1}): The set of all strings such that every occurrence of 1 is followed by at least two Os, e.g., 01000100100, 10010000001000000, 00001000 are in this language, but 001 is not. The set of all strings that does not contain pattern 0101. The set of all strings except 101 and 11. The set of all string such that every even position contains 0. For example, 00000,101,100010 are in this language, but 010 and 1101 are not.