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Michael Sipser Edition 3 Exercise 1 Question 22 (Page No. 87)
In certain programming languages, comments appear between delimiters such as $\text{/#}$ and $\text{#/}.$ Let $C$ be the language of all valid delimited comment strings. A member of $C$ must begin with $\text{/#}$ ... $DFA$ that recognizes $C.$ b. Give a regular expression that generates $C.$
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Michael Sipser Edition 3 Exercise 1 Question 21 (Page No. 86)
Use the procedure described in $\text{Lemma 1.60}$ to convert the following finite automata to regular expressions.
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Michael Sipser Edition 3 Exercise 1 Question 20 (Page No. 86)
For each of the following languages, give two strings that are members and two strings that are not membersa total of four strings for each part. Assume the alphabet $Σ = \{a,b\}$ in all parts. $a^{*}b^{*}$ $a(ba)^{*}b$ $a^{*}\cup b^{*}$ ... $aba\cup bab$ $(\epsilon\cup a)b$ $(a\cup ba\cup bb)\sum^{*}$
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Michael Sipser Edition 3 Exercise 1 Question 19 (Page No. 86)
Use the procedure described in $\text{Lemma 1.55}$ to convert the following regular expressions to nondeterministic finite automata. $(0\cup 1)^{*}000(0\cup 1)^{*}$ $(((00)^{*}(11))\cup 01)^{*}$ $\phi^{*}$
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Michael Sipser Edition 3 Exercise 1 Question 18 (Page No. 86)
Give regular expressions generating the languages of the alphabet is $\{0,1\}.$ $\text{\{w w begins with a 1 and ends with a 0\}}$ $\text{\{w w contains at least three 1s\}}$ ... $\text{The empty set}$ $\text{All strings except the empty string}$
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Peter Linz Edition 4 Exercise 5.2 Question 10 (Page No. 145)
Give an unambiguous grammar that generates the set of all regular expressions on $Σ =$ {$a,b$}.
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Ullman (TOC) Edition 3 Exercise 5.1 Question 5 (Page No. 182)
Let $T=\{0,1(,),+,*,\phi,e\}.$ We may think of $T$ as the set of symbols used by regular expressions over alphabet $\{0,1\};$ the only difference is that we use $e$ for symbol $\in,$ to avoid potential ... Your task is to design a CFG with set of terminals $T$ that generates exactly the regular expressions with alphabet $\{0,1\}.$
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Ullman (TOC) Edition 3 Exercise 5.1 Question 2 (Page No. 182)
The following grammar generates the language of regular expression $0^{*}1(0+1)^{*}:$ $S\rightarrow A1B$ $A\rightarrow 0A\in$ $B\rightarrow B1B\in$ Give leftmost and rightmost derivations of the following strings$:$ $00101$ $1001$ $00011$
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Ullman (TOC) Edition 3 Exercise 4.2 Question 14 (Page No. 149  150)
We described the $"$product construction$"$ that took two DFA's and constructed one DFA whose language is intersection of the languages of the first two. Show how to perform the product construction on NFA's ... the product construction so the resulting DFA accepts the union of the languages of the two given DFA's.
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Ullman (TOC) Edition 3 Exercise 4.2 Question 13 (Page No. 149)
We can use closure properties to help prove certain languages are not regular. Start with the fact that the language $L_{0n1n}=\{0^{n}1^{n}n\geq 0\}$ is not a regular set. Prove the following languages not to be regular by transforming them using operations known to ... $\{0^{n}1^{m}2^{nm}n\geq m\geq 0\}$
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Ullman (TOC) Edition 3 Exercise 4.2 Question 12 (Page No. 149)
Let $w_{1}=a_{0}a_{0}a_{1},$ and $w_{i}=w_{i1}w_{i1}a_{i}$ for all $i>1.$For instance,$w_{3}=a_{0}a_{0}a_{1}a_{0}a_{0}a_{1}a_{2}a_{0}a_{0}a_{1}a_{0}a_{0}a_{1}a_{2}a_{3}.$ The shortest ... is $O(n^{2}).$ Find such an expression. Hint $:$ Find $n$ languages, each with regular expressions of length $O(n).$ Whose intersection is $L.$
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Ullman (TOC) Edition 3 Exercise 4.2 Question 11 (Page No. 149)
Show that the regular languages are closed under the following operation$:$ $\text{cycle(L)={we can write w as w = xy,such that yx is in L}}.$ For example if $L=\{01,011\},$then $cycle(L)=\{01,10,011,110,101\}.$ Hint$:$ Start with a DFA for $L$ and construct an $\inNFA$ for $cycle(L).$
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Ullman (TOC) Edition 3 Exercise 4.2 Question 10 (Page No. 149)
Suppose that $L$ is any language not necessarily regular whose alphabet is $\{0\};$i.e. the strings of $L$ consist of $0's$ only. Prove that $L^{*}$ is regular. Hint$:$ At first this theorem sounds preposterous. However, an example will help you see why ... ,use copy of $000$ and $(j3)/2$ copies of $00.$ Thus ,$L^{*}=$ $\in + 000^{*}.$
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Ullman (TOC) Edition 3 Exercise 4.2 Question 9 (Page No. 148  149)
We can generalize question $8$ to a number of functions that determine how much of the string we take.If $f$ is a function of integers, define $f(L)$ to be $\text{\{w for some $x,$ with $x=f(w)$,we have $ ... what we do not take. $f(n)=2^{n}(i.e,$ what we take has length equal to the logarithm of what we leave$).$
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Ullman (TOC) Edition 3 Exercise 4.2 Question 8 (Page No. 148)
Let $L$ be a language.Define $\text{half(L)}$ to be the set of first halves of strings in $L,$ that is,$\{w\text{for some x such that x=w,we have wx in L}\}.$For example,if $L=\{\in,0010,011,010110\}$ then ... that oddlength strings do not contribute to $\text{half(L)}$Prove that if $L$ is a regular language,so is $\text{half(L)}.$
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Ullman (TOC) Edition 3 Exercise 4.2 Question 7 (Page No. 148)
If $w=a_{1}a_{2}.....a_{n}$ and $x=b_{1}b_{2}....b_{n}$ are strings of the same length, define $alt(w,x)$ to be the string in which the symbols of $w$ and $x$ alternate starting with $w$ ... in $L$ and $x$ is any string in $M$ of the same length. Prove that if $L$ and $M$ are regular,so is $alt(L,M).$
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Peter Linz Edition 4 Exercise 4.1 Question 4 (Page No. 109)
Theorem 4.3 Let h be a homomorphism. If L is a regular language, then its homomorphic image h (L) is also regular. The family of regular languages is therefore closed under arbitrary homomorphisms. Proof: Let L be a regular language denoted by some ... 3, show that $h (r)$ is a regular expression. Then show that $h (r)$ denotes $h (L)$.
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Ullman (TOC) Edition 3 Exercise 4.2 Question 6 (Page No. 148)
Show that the regular languages are closed under the following operations$:$ $ min(L)=\big\{\text{ww is in L, but no proper prefix of w is in L}\big\}.$ $ min(L)=\big\{\text{ww is in L, and for no x other than $\in$ is wx in L}\big\}.$ $ init(L)=\big\{\text{w for some x, wx is in L}\big\}.$
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Ullman (TOC) Edition 3 Exercise 4.2 Question 5 (Page No. 148)
The operation of Problem $4.2.3$ is sometimes viewed as a $"$derivative and $a/L$ is written $\frac{dL}{da}.$ These derivatives apply to regular expressions in a manner similar to the way ordinary derivatives apply to arithmetic ... $\frac{dL}{d0}=\phi$ Characterize those languages $L$ for which $\frac{dL}{d0}=L$
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Peter Linz Edition 4 Exercise 3.2 Question 17 (Page No. 89)
Analogous to the previous exercise, consider all words that can be formed from $L$ by dropping a single symbol of the string. Formally define this operation drop for languages. Construct an nfa for $drop (L)$, given an nfa for $L$.
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Peter Linz Edition 4 Exercise 3.2 Question 16 (Page No. 89)
In some applications, such as programs that check spelling, we may not need an exact match of the pattern, only an approximate one. Once the notion of an approximate match has been made precise, automata theory can be applied to ... this to write a patternrecognition program for $insert (L)$, using as input a regular expression for $L$.
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Peter Linz Edition 4 Exercise 3.2 Question 15 (Page No. 89)
Write a regular expression for the set of all $C$ real numbers.
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Peter Linz Edition 4 Exercise 3.2 Question 13 (Page No. 88)
Find a regular expression for the following languages on {$a, b$}. (a) $L =$ {$w : n_a (w)$ and $n_b (w)$ are both even}. (b) $L =$ {$w :(n_a (w)  n_b (w))$ mod $3 = 1$}. (c) $L =$ {$w :(n_a (w)  n_b (w))$ mod $3 = 0$}. (d) $L =$ {$w :2n_a (w)+3n_b (w)$ is even}.
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Ullman (TOC) Edition 3 Exercise 3.4 Question 4 (Page No. 123)
Prove that $(L^{*}M^{*})^{*}=(L+M)^{*}.$Complete the proof by showing that strings in $(L^{*}M^{*})^{*}$ are also in $(L+M)^{*}.$
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Ullman (TOC) Edition 3 Exercise 3.4 Question 3 (Page No. 122)
We developed the regular expression $(0+1)^{*}1(0+1)+(0+1)^{*}1(0+1)(0+1)$ Use the distributive laws to develop two different,simpler,equivalent expressions.
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Ullman (TOC) Edition 3 Exercise 3.4 Question 2 (Page No. 122)
Prove or disprove each of the following statements about regular expressions. $(R+S)^{*}=R^{*}+S^{*}$ $(RS+R)^{*}R=R(SR+R)^{*}$ $(RS+R)^{*}RS=(RR^{*}S)^{*}$ $(R+S)^{*}S=(R^{*}S)^{*}$ $S(RS+S)^{*}R=RR^{*}S(RR^{*}S)^{*}$
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Ullman (TOC) Edition 3 Exercise 3.4 Question 1 (Page No. 121  122)
Verify the following identities involving regular expressions. $R+S=S+R$ $(R+S)+T=R+(S+T)$ $(RS)T=R(ST)$ $R(S+T)=RS+RT$ $(R+S)T=RT+ST$ $(R^{*})^{*}=R^{*}$ $(\in+R)^{*}=R^{*}$ $(R^{*}S^{*})^{*}=(R+S)^{*}$
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Ullman (TOC) Edition 3 Exercise 3.3 Question 2 (Page No. 114)
Give a regular expression to represent salaries as they might appear in employment advertising. Consider that salaries might be given on a per hour, week, month or year basis. They may or may not appear with a dollar sign or ... classified ads in a newspaper, or online jobs listings to get an idea of what patterns might be useful.
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Ullman (TOC) Edition 3 Exercise 3.3 Question 1 (Page No. 114)
Give a regular expression to describe phone numbers in all the various forms you can think of. Consider international numbers as well as the fact that different countries have different numbers of digits in area codes and in local phone numbers.
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Ullman (TOC) Edition 3 Exercise 3.2 Question 5 (Page No. 108)
Convert the following regular expressions to NFA's with $\in$transactions. $01^{*}$ $(0+1)01$ $00(0+1)^{*}$ Eliminate $\in$transactions from your $\inNFA’s$
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