Department of Computer Science and Engineering Year & Section: II & E Academic Year: 2022-23 Subject: Theory of Computation 1. For each of the following languages, construct an DFA and NFA, that accepts the language. In all cases, the alphabet is {0,1}. • {๐ค โถ ๐ค contains the substring 1011}. • {๐ค โถ ๐ค contains at least two 1s and at most two 0s}. • {๐ค โถ ๐ค contains an odd number of 1s or exactly two 0s}. • {๐ค โถ ๐ค begins with 1 and ends with 0}. • {๐ค โถ ๐๐ฃ๐๐๐ฆ ๐๐๐ ๐๐๐ ๐๐ก๐๐๐ ๐๐ ๐ค is 1}. • {๐ค โถ ๐ค has length at least 3 and its third symbol is 0}. • {∈ ,0} i. Construct RE for the above Languages by constructing minimized DFA. ii. Convert and compare the NFA to DFA and write your comments. 2. Translate the Melay machine into its equivalent Moore machine. Construct equivalent DFA for the Moore Machine by considering output 1 as accepted and 0 as rejected. Write a RE for the constructed DFA. 3. Construct NFA for the given ฯต-NFA i. Convert the NFA to DFA ii. Reduce the number of states for the above constructed DFA and design Minimized DFA. 4. Construct Mealy Machine and Moore machine that takes all binary numbers as input and produces Y as output if last two symbols are same. Otherwise, produces N as output. Convert the above machines from Mealy to Moore and Moore to Mealy. 5. Consider a Deterministic Finite Automaton (DFA) which takes N states numbered from 0 to N-1, with alphabets 0 and 1. Now consider a string of 0s and 1s, let the decimal value of this string as M. For e.g., if string be "010" then the decimal value or M is 2. The speciality in our DFA is that when we simulate the string through this DFA, it finally takes us to the state with the same number as of M%N, where % is the modulus operator. For e.g. if our string is "1100" i.e. M is 12 and N is 5, then after simulating this string through DFA, it will reach to state 12%5 i.e. state no. 2. Since M%N can be any number from 0 to N-1, and the states in our DFA are also from 0 to N-1, there must always be some state we will reach. Assume the starting state of DFA is 0. You are given N, the no. of states in our DFA. Print a transition table for this DFA. Write the regular expression for the above language as well as construct the minimized DFA. 6. Consider the Finite Automata given by the transition table: a b *c d e f g h i. ii. iii. iv. 0 1 b g a c h c g g f c c g f g e c Construct minimized FA for the give FA Write the regular expression for the above transition table Construct the DFA for the above RE Construct the minimum state equivalent DFA. 7. Recall (part of) the Roman numeral system: I = 1, V = 5, X = 10, L = 50, C = 100. Recall also that in a valid Roman numeral, strings such as IIIII or VV are not allowed, since there are alternative representations using larger values, such as V and X, respectively. Draw a DFA that accepts exactly the set of valid Roman numerals between 1 and 100 that are strictly ordered: that is, no letter may appear after a letter that has a smaller value. Thus 9 must be represented by VIII rather than by IX. The alphabet for your DFA should be {I, V, X, L, C}. You should adopt the convention that edges not drawn lead to a unique "dead" or "trap" state, which is nonaccepting, and which has edges leading back to itself on any input. Also, be sure that your DFA doesn't accept any numeral with value 101 or more. 8. Consider the language L which accepts all strings of a’s and b’s where each string contains 5th symbol from RHS is a. i. Derive the strings which belong to the above language. ii. Design NFA for the above language. iii. Design equivalent DFA for the obtained NFA. iv. Design minimized DFA for the obtained DFA. v. Derive the Regular Expression accepted by minimized DFA. Consider from LHS, 5th Symbol as a and construct the finite automata from i-v. 9. For each of the following languages, construct an NFA that accepts the language. In all cases, the alphabet is {0,1}. • {๐ค โถ ๐ค ๐๐๐๐ก๐๐๐๐ ๐กโ๐ ๐ ๐ข๐๐ ๐ก๐๐๐๐ 11001}. • {๐ค โถ ๐ค โ๐๐ ๐๐๐๐๐กโ ๐๐ก ๐๐๐๐ ๐ก 2 ๐๐๐ ๐๐๐๐ ๐๐๐ก ๐๐๐ ๐ค๐๐กโ 10}. • {๐ค โถ ๐ค ๐๐๐๐๐๐ ๐ค๐๐กโ 1 ๐๐ ๐๐๐๐ ๐ค๐๐กโ 0}. i. Construct DFA for the above designed NFA ii. Mention the transition table for the above DFA iii. Design the minimized DFA for the above DFA with reduced no. of states 10. For each of the following regular expressions, give two strings that are memberes and two strings that are not members of the language described by the expression. The alphabet is ๐ด = {๐, ๐}. • a(ba)*b. • (a∪ ๐)*a(a∪ ๐)*b(a∪ ๐)*a(a∪ ๐)*. • (a ∪ ba ∪ bb)(a ∪ b)*. i. Construct DFA for the above RE ii. Mention the transition table for the above DFA iii. Design the minimized DFA for the above DFA with reduced no. of states