2. Consider the DFA defined by its transition graph as below. Given a DFA, we define the notion of a regular expression R) for the language L consisting of all strings that go from state q, to state q; without visiting a state numbered larger than k along the way (visiting the first and last state is not considered). (b) For the DFA given above, specify the following regular expressions. You do not need to justify your answer. (a) 440) (c) (d) (e) R12 F RO q2 R

2. Consider the DFA defined by its transition graph as below. Given a DFA, we define the notion of a regular expression R) for the language L consisting of all strings that go from state q, to state q; without visiting a state numbered larger than k along the way (visiting the first and last state is not considered). (b) For the DFA given above, specify the following regular expressions. You do not need to justify your answer. (a) 440) (c) (d) (e) R12 F RO q2 R

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Given: Language L2 = {w ϵ {a,b}* | w contains substring aab} Construct the DFA representing the given language L2. Assume the layout of the DFA is as follows: ->A->B->C->D*. That is, the number of states are four (4) and A is the starting state and D is the accepting state. How many self-loop edge/s in the constructed DFA representing the given language L2?
Giventhegrammar B-> BB | (B) | ε 1.Describe the language it generates 2.Show that it is ambiguous
4. Given a language L, the language L is defined as the language consisting of the reverse of all the strings in L. A language is said to be palindromic, if L = LR. Given a regular language L, how would you check if it is palindromic?
2. Prove that the language L(G) is not regular where G is the following cfg: G= ({S,A,B}, {a,b,c}, {S→Abbb|B, A→aS, B c}, S). Note: You must first determine L(G).
Question 2. Find a regular expression that describes the given language:(2.a) Language of strings over Σ = {a, b, c} in which every character c is followed by the character a.(2.b) Language of strings over Σ = {a, b} which contain both substrings abb and bab. Note that thesubstrings can occur in any order and can overlap, so for example the string babb is in this language.(2.c) Let Σ = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, −, .}. The language L contains all strings that form a properdecimal representation of some number.Some examples of strings in the language: 42, -999, 0, 12.333, -0.14, 5.000.Some examples of strings that are not in the language: 028 (leading zero), 75. (decimal point but nodecimal part), -.5 (no integer part), -0 (there is no negative zero), 4..2-6 (malformed nonsense).You can use the shorthand [0−9] to stand for (0 + 1 + 2 + ···+ 9).
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.
1. Given L₁ = {a, abc, ac} and L₂ = {ab, c ba} are the languages over Σ = {a,b,c}, determine L₁ L₂ Obtain the regular expressions for the set (b, b, b,..), using the appropriate Kleene's closure.
Q.No.4: What is K-Map Method, for which purpose K-Map is used? What are its advantages and disadvantages? Also explain grouping rules while making a K-Map for a Boolean expression along with examples for each rule?
Give both formulations (traditional and 3-set) of the LRP corresponding to the language L = the set of strings beginning with aaa over {a,b}.
Draw and formalize a DFA for the language over Σ = {a,b,c} with strings containing an odd number of as, of bs, and of c Hint: try using the intersection construction we discussed in class and label each state with a tuple.
Use the Pumping Lemma to prove that the language L (a"b2n :n >= 0} is NOT context free.
2. [Based on Problem 2.26, page 157 in Sipser's 3rd Edition] Suppose G is a context-free grammar in Chomsky normal form. This exercise will help you prove that for any string w e L(G) of length n 2 1, exactly 2n - 1 steps are required for any derivation of w. To make the proof simple, we define: a variable rule is any rule of the form A BC, where A, B,C are variables, • a terminal rule is any rule of the form A → a, where A is a variable and a is a terminal. Now consider a derivation in G that starts with the start variable and applies apply variable rules s times (may not be consecutive) and applies terminal rules t times, and let r be the string generated by the derivation (z can contain both variables and terminals). (a) Prove that the string r contains t terminals. (b) Prove that the string r contains (1+s – t) variables. (c) Prove that if r E L(G) and |2| = n, then s +t = 2n – 1. %3D