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Recent questions tagged proof
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91
Michael Sipser Edition 3 Exercise 2 Question 23 (Page No. 157)
Let $D = \{xy\mid x, y\in \{0,1\}^{*}$ $\text{and}$ $\mid x\mid = \mid y\mid$ $\text{but}$ $x\neq y\}.$ Show that $D$ is a context-free language$.$
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Theory of Computation
May 4, 2019
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michael-sipser
theory-of-computation
context-free-language
proof
0
votes
0
answers
92
Michael Sipser Edition 3 Exercise 1 Question 73 (Page No. 93)
Let $\sum = \{0,1, \#\}.$ Let $C = \{x\#x^{R}\#x| x\in\{0,1\}^{*}\}.$Show that $\overline{C}$ is a $\text{CFL}.$
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Theory of Computation
Apr 30, 2019
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184
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michael-sipser
theory-of-computation
context-free-language
proof
descriptive
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93
Michael Sipser Edition 3 Exercise 1 Question 72 (Page No. 93)
Let $M_{1}$ and $M_{2}$ be $\text{DFA's}$ that have $k_{1}$ and $k_{2}$ states, respectively, and then let $U = L(M_{1})\cup L(M_{2}).$ Show that if $U\neq\phi$ then $U$ contains some string $s,$ where $|s| < max(k1, k2).$ Show that if $U\neq\sum^{*},$ then $U$ excludes some string $s,$ where $|s| < k1k2.$
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Theory of Computation
Apr 30, 2019
by
admin
323
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michael-sipser
theory-of-computation
finite-automata
regular-language
proof
descriptive
2
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94
Michael Sipser Edition 3 Exercise 1 Question 71 (Page No. 93)
Let $\sum = \{0,1\}$ Let $A=\{0^{k}u0^{k}|k\geq 1$ $\text{and}$ $u\in \sum^{*}\}.$ Show that $A$ is regular. Let $B=\{0^{k}1u0^{k}|k\geq 1$ $\text{and}$ $u\in \sum^{*}\}.$Show that $B$ is not regular.
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in
Theory of Computation
Apr 30, 2019
by
admin
500
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michael-sipser
theory-of-computation
finite-automata
regular-language
proof
descriptive
0
votes
1
answer
95
Michael Sipser Edition 3 Exercise 1 Question 70 (Page No. 93)
We define the $\text{avoids}$ operation for languages $A$ and $B$ to be $\text{A avoids B = {w| w ∈ A and w doesn’t contain any string in B as a substring}.}$ Prove that the class of regular languages is closed under the ${avoids}$ operation.
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Theory of Computation
Apr 30, 2019
by
admin
488
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michael-sipser
theory-of-computation
finite-automata
regular-language
proof
descriptive
1
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96
Michael Sipser Edition 3 Exercise 1 Question 69 (Page No. 93)
Let $\sum=\{0,1\}.$ Let $WW_{k}=\{ww|w\in \sum^{*}$ and $w$ is of length $k\}.$ Show that for each $k,$ no DFA can recognize $WW_{k}$ with fewer than $2^{k}$ states. Describe a much smaller $NFA$ for $\overline{WW_{k}},$ the complement of $WW_{k}.$
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in
Theory of Computation
Apr 30, 2019
by
admin
338
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michael-sipser
theory-of-computation
finite-automata
proof
descriptive
0
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97
Michael Sipser Edition 3 Exercise 1 Question 68 (Page No. 93)
In the traditional method for cutting a deck of playing cards, the deck is arbitrarily split two parts, which are exchanged beforereassembling the deck. In a more complex cut, called $\text{Scarne's cut,}$ the deck is broken into three parts ... $ CUT(CUT(B)).}$ Show that the class of regular languages is closed under $\text{CUT}.$
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Theory of Computation
Apr 30, 2019
by
admin
523
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michael-sipser
theory-of-computation
regular-language
scarnes-cut
proof
descriptive
0
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98
Michael Sipser Edition 3 Exercise 1 Question 61 (Page No. 92)
Let $Σ = \{a, b\}.$ For each $k\geq 1,$ let $C_{k}$ be the language consisting of all strings that contain an a exactly $k$ places from the right-hand end$.$ Thus $C_{k}=\sum^{*}a\sum^{k-1}.$ Prove that for each $k,$ $\text{no DFA}$ can recognize $C_{k}$ with fewer than $2^{k}$ states.
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in
Theory of Computation
Apr 30, 2019
by
admin
295
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michael-sipser
theory-of-computation
finite-automata
proof
descriptive
0
votes
0
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99
Michael Sipser Edition 3 Exercise 1 Question 58 (Page No. 92)
If $A$ is any language,let $A_{\frac{1}{2}-\frac{1}{3}}$ be the set of all strings in $A$ with their ,middle thirds removed so that $A_{\frac{1}{2}-\frac{1}{3}}=\{\text{xz|for some y,|x|=|y|=|z| and xyz $\in$ A\}}.$ Show that if $A$ is regular,then $A_{\frac{1}{2}-\frac{1}{3}}$ is not necessarily regular.
admin
asked
in
Theory of Computation
Apr 30, 2019
by
admin
221
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michael-sipser
theory-of-computation
regular-language
proof
descriptive
0
votes
0
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100
Michael Sipser Edition 3 Exercise 1 Question 57 (Page No. 92)
If $A$ is any language,let $A_{\frac{1}{2}-}$ be the set of all first halves of strings in $A$ so that $A_{\frac{1}{2}-}=\{\text{x|for some y,|x|=|y| and xy $\in$ A\}}.$ Show that if $A$ is regular,then so is $A_{\frac{1}{2}-}.$
admin
asked
in
Theory of Computation
Apr 30, 2019
by
admin
264
views
michael-sipser
theory-of-computation
regular-language
proof
descriptive
0
votes
0
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101
Michael Sipser Edition 3 Exercise 1 Question 56 (Page No. 91)
If $A$ is a set of natural numbers and $k$ is a natural number greater than $1,$ let $B_{k}(A)=\{\text{w| w is the representation in base k of some number in A\}}.$ Here, we do not allow leading $0's$ in the representation ... a set $A$ for which $B_{2}(A)$ is regular but $B_{3}(A)$ is not regular$.$ Prove that your example works.
admin
asked
in
Theory of Computation
Apr 30, 2019
by
admin
333
views
michael-sipser
theory-of-computation
finite-automata
regular-language
proof
descriptive
1
vote
1
answer
102
Michael Sipser Edition 3 Exercise 1 Question 55 (Page No. 91)
The pumping lemma says that every regular language has a pumping length $p,$ such that every string in the language can be pumped if it has length $p$ or more. If $p$ is a pumping length for language $A,$ so is any length $p^{'}\geq p.$ The minimum pumping ... $\epsilon$ $1^{*}01^{*}01^{*}$ $10(11^{*}0)^{*}0$ $1011$ $\sum^{*}$
admin
asked
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Theory of Computation
Apr 30, 2019
by
admin
2.2k
views
michael-sipser
theory-of-computation
regular-language
pumping-lemma
proof
descriptive
0
votes
1
answer
103
Michael Sipser Edition 3 Exercise 1 Question 54 (Page No. 91)
Consider the language $F=\{a^{i}b^{j}c^{k}|i,j,k\geq 0$ $\text{and if}$ $ i = 1$ $\text{then} $ $ j=k\}.$ Show that $F$ is not regular. Show that $F$ acts like a regular language in the pumping lemma. ... three conditions of the pumping lemma for this value of $p.$ Explain why parts $(a)$ and $(b)$ do not contradict the pumping lemma.
admin
asked
in
Theory of Computation
Apr 30, 2019
by
admin
921
views
michael-sipser
theory-of-computation
finite-automata
regular-language
pumping-lemma
proof
descriptive
0
votes
0
answers
104
Michael Sipser Edition 3 Exercise 1 Question 53 (Page No. 91)
Let $\sum=\{0,1,+,=\}$ and $ADD=\{x=y+z|x,y,z$ $\text{are binary integers,and}$ $x$ $\text{is the sum of}$ $y$ $\text{and}$ $z\}.$ Show that $\text{ADD}$ is not a regular.
admin
asked
in
Theory of Computation
Apr 30, 2019
by
admin
289
views
michael-sipser
theory-of-computation
finite-automata
regular-language
proof
descriptive
1
vote
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105
Michael Sipser Edition 3 Exercise 1 Question 52 (Page No. 91)
$\text{Myhill-Nerode theorem.}$ Refer to $\text{Question 51}.$Let $L$ be a language and let $X$ be a set of strings. Say that $X$ is $\text{pairwise distinguishable}$ by $L$ if every two distinct strings in $X$ are ... $\text{DFA}$ recognizing it$.$
admin
asked
in
Theory of Computation
Apr 30, 2019
by
admin
394
views
michael-sipser
theory-of-computation
finite-automata
finite-automata
proof
descriptive
0
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answers
106
Michael Sipser Edition 3 Exercise 1 Question 51 (Page No. 90)
Let $x$ and $y$ be strings and let $L$ be any language. We say that $x$ and $y$ are $\text{distinguishable}$ by $L$ if some string $z$ exists whereby exactly one of the strings $xz$ and $yz$ ... $≡L$ is an equivalence relation. A $\text{palindrome}$ is a string that reads the same forward and backward.
admin
asked
in
Theory of Computation
Apr 30, 2019
by
admin
302
views
michael-sipser
theory-of-computation
finite-automata
proof
descriptive
1
vote
0
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107
Michael Sipser Edition 3 Exercise 1 Question 50 (Page No. 90)
Read the informal definition of the finite state transducer given in Question $24.$ Prove that $\text{no FST}$ can output $w^{R}$ for every input $w$ if the input and output alphabets are $\{0,1\}.$
admin
asked
in
Theory of Computation
Apr 30, 2019
by
admin
197
views
michael-sipser
theory-of-computation
finite-automata
finite-state-transducer
proof
descriptive
0
votes
1
answer
108
Michael Sipser Edition 3 Exercise 1 Question 49 (Page No. 90)
Let $B=\{1^{k}y|y\in\{0,1\}^{*}$ $\text{ and y contains at least}$ $k$ $1's,$ $\text{for every}$ $k\geq 1\}.$ Show that $B$ is a regular language. Let $C=\{1^{k}y|y\in\{0,1\}^{*}$ $\text{ and y contains at most}$ $k$ $1's,$ $\text{for every}$ $k\geq 1\}.$ Show that $C$ isn’t a regular language.
admin
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in
Theory of Computation
Apr 30, 2019
by
admin
433
views
michael-sipser
theory-of-computation
finite-automata
regular-language
proof
descriptive
1
vote
0
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109
Michael Sipser Edition 3 Exercise 1 Question 48 (Page No. 90)
Let $\sum = \{0,1\}$ and let $D = \{w|w$ $\text{contains an equal number of occurrences of the sub strings 01 and 10}\}.$ Thus $101\in D$ because $101$ contains a single $01$ and a single $10,$ but $1010\notin D$ because $1010$ contains two $10's$ and one $01.$ Show that $D$ is a regular language.
admin
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in
Theory of Computation
Apr 30, 2019
by
admin
255
views
michael-sipser
theory-of-computation
finite-automata
regular-language
proof
descriptive
0
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0
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110
Michael Sipser Edition 3 Exercise 1 Question 47 (Page No. 90)
Let $\sum=\{1,\#\}$ and let $Y=\{w|w=x_{1}\#x_{2}\#...\#x_{k}$ $\text{for}$ $k\geq 0,$ $\text{each}$ $ x_{i}\in 1^{*},$ $\text{and}$ $x_{i}\neq x_{j}$ $\text{for}$ $i\neq j\}.$ Prove that $Y$ is not regular.
admin
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in
Theory of Computation
Apr 30, 2019
by
admin
317
views
michael-sipser
theory-of-computation
finite-automata
regular-language
proof
0
votes
0
answers
111
Michael Sipser Edition 3 Exercise 1 Question 46 (Page No. 90)
Prove that the following languages are not regular. You may use the pumping lemma and the closure of the class of regular languages under union, intersection,and complement. $\{0^{n}1^{m}0^{n}|m,n\geq 0\}$ $\{0^{m}1^{n}|m\neq n\}$ $\{w|w\in\{0,1\}^{*} \text{is not a palindrome}\}$ $\{wtw|w,t\in\{0,1\}^{+}\}$
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Theory of Computation
Apr 30, 2019
by
admin
493
views
michael-sipser
theory-of-computation
finite-automata
regular-language
proof
0
votes
0
answers
112
Michael Sipser Edition 3 Exercise 1 Question 43 (Page No. 90)
Let $A$ be any language. Define $\text{DROP-OUT(A)}$ to be the language containing all strings that can be obtained by removing one symbol from a string in $A.$ ... $\text{Theorem 1.47.}$
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Theory of Computation
Apr 30, 2019
by
admin
666
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michael-sipser
theory-of-computation
finite-automata
regular-language
proof
descriptive
0
votes
0
answers
113
Michael Sipser Edition 3 Exercise 1 Question 39 (Page No. 89)
The construction in $\text{Theorem 1.54}$ shows that every $\text{GNFA}$ is equivalent to a $\text{GNFA}$ with only two states$.$ We can show that an opposite phenomenon occurs for $\text{DFAs.}$ Prove that for every $k > 1,$ a ... exists that is recognized by a $\text{DFA}$ with $k$ states but not by one with only $k − 1$ states$.$
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Theory of Computation
Apr 28, 2019
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admin
370
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michael-sipser
theory-of-computation
finite-automata
proof
0
votes
1
answer
114
Michael Sipser Edition 3 Exercise 1 Question 30 (Page No. 88)
Describe the error in the following $ $proof$"$ that $0^{*}1^{*}$ is not a regular language. $($An error must exist because $0^{*}1^{*}$ is regular.$)$ The proof is by contradiction. Assume that $0^{*}1^{*}$ is regular ... example $1.73$ shows that $s$ cannot be pumped. Thus you have a contradiction. So $0^{*}1^{*}$ is not regular.
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in
Theory of Computation
Apr 21, 2019
by
admin
896
views
michael-sipser
theory-of-computation
finite-automata
pumping-lemma
proof
0
votes
0
answers
115
Michael Sipser Edition 3 Exercise 1 Question 23 (Page No. 87)
Let $B$ be any language over the alphabet $Σ.$ Prove that $B = B^{+}$ iff $BB ⊆ B.$
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asked
in
Theory of Computation
Apr 21, 2019
by
admin
534
views
michael-sipser
theory-of-computation
regular-language
proof
0
votes
1
answer
116
Michael Sipser Edition 3 Exercise 1 Question 11 (Page No. 85)
Prove that every $\text{NFA}$ can be converted to an equivalent one that has a single accept state.
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Theory of Computation
Apr 21, 2019
by
admin
1.1k
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michael-sipser
theory-of-computation
finite-automata
proof
0
votes
0
answers
117
Peter Linz Edition 5 Exercise 8.1 Question 7(k) (Page No. 212)
Show that the following languages on $\Sigma = \{a,b,c\}$ are not context-free $L = \{a^nb^m: \text{n is prime and m is not prime}\}$.
Rishi yadav
asked
in
Theory of Computation
Apr 15, 2019
by
Rishi yadav
352
views
peter-linz
peter-linz-edition5
theory-of-computation
pumping-lemma
proof
context-free-language
0
votes
1
answer
118
Peter Linz Edition 5 Exercise 8.1 Question 7(j) (Page No. 212)
Show that the following languages on $\Sigma = \{a,b,c\}$ are not context-free $L = \{a^nb^m:\text{n is prime or m is prime}\}$.
Rishi yadav
asked
in
Theory of Computation
Apr 15, 2019
by
Rishi yadav
342
views
peter-linz
peter-linz-edition5
theory-of-computation
pumping-lemma
proof
context-free-language
0
votes
0
answers
119
Peter Linz Edition 5 Exercise 8.1 Question 7(i) (Page No. 212)
Show that the following languages on $\Sigma = \{a,b,c\}$ are not context-free $L = \{a^nb^m: \text{n and m are both prime}\}$.
Rishi yadav
asked
in
Theory of Computation
Apr 15, 2019
by
Rishi yadav
193
views
peter-linz
peter-linz-edition5
theory-of-computation
pumping-lemma
proof
context-free-language
0
votes
0
answers
120
Peter Linz Edition 5 Exercise 8.1 Question 7(h) (Page No. 212)
Show that the following languages on $\Sigma = \{a,b,c\}$ are not context-free. $L = \{w\in\{a,b,c\}^*:n_a(w)+n_b(w)=2n_c(w),n_a(w) = n_b(w)\}$.
Rishi yadav
asked
in
Theory of Computation
Apr 15, 2019
by
Rishi yadav
290
views
peter-linz
peter-linz-edition5
theory-of-computation
pumping-lemma
proof
context-free-language
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