Recent questions tagged proof

Recent questions tagged proof

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1

Kenneth Rosen Edition 7th Exercise 8.2 Question 10 (Page No. 525)
Prove Theorem $2:$ Let $c_{1}$ and $c_{2}$ be real numbers with $c_{2}\neq 0.$ Suppose that $r^{2}-c_{1}r-c_{2} = 0$ has only one root $r_{0}.$ A sequence $\{a_{n}\}$ ... $n = 0,1,2,\dots,$ where $\alpha_{1}$ and $\alpha_{2}$ are constants.

Lakshman Patel RJIT asked in Combinatory May 3, 2020

by Lakshman Patel RJIT

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2

Kenneth Rosen Edition 7th Exercise 6.5 Question 63 (Page No. 434)
Prove the Multinomial Theorem: If $n$ ... $C(n:n_{1},n_{2},\dots,n_{m}) = \dfrac{n!}{n_{1}!n_{2}!\dots n_{m}!}$ is a multinomial coefficient.

Lakshman Patel RJIT asked in Combinatory May 1, 2020

by Lakshman Patel RJIT

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3

Kenneth Rosen Edition 7th Exercise 6.4 Question 32 (Page No. 422)
Prove the binomial theorem using mathematical induction.

Lakshman Patel RJIT asked in Combinatory Apr 30, 2020

by Lakshman Patel RJIT

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4

Michael Sipser Edition 3 Exercise 5 Question 36 (Page No. 242)
Say that a $CFG$ is minimal if none of its rules can be removed without changing the language generated. Let $MIN_{CFG} = \{\langle G \rangle \mid \text{G is a minimal CFG}\}$. Show that $MIN_{CFG}$ is $T-$recognizable. Show that $MIN_{CFG}$ is undecidable.

Lakshman Patel RJIT asked in Theory of Computation Oct 20, 2019

by Lakshman Patel RJIT

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5

Michael Sipser Edition 3 Exercise 5 Question 35 (Page No. 242)
Say that a variable $A$ in $CFG \:G$ is necessary if it appears in every derivation of some string $w \in G$. Let $NECESSARY_{CFG} = \{\langle G, A\rangle \mid \text{A is a necessary variable in G}\}$. Show that $NECESSARY_{CFG}$ is Turing-recognizable. Show that $NECESSARY_{CFG} $is undecidable.

Lakshman Patel RJIT asked in Theory of Computation Oct 20, 2019

by Lakshman Patel RJIT

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6

Michael Sipser Edition 3 Exercise 5 Question 34 (Page No. 241)
Let $X = \{\langle M, w \rangle \mid \text{M is a single-tape TM that never modifies the portion of the tape that contains the input $w$ } \}$ Is $X$ decidable? Prove your answer.

Lakshman Patel RJIT asked in Theory of Computation Oct 20, 2019

by Lakshman Patel RJIT

389 views

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7

Michael Sipser Edition 3 Exercise 5 Question 33 (Page No. 241)
Consider the problem of determining whether a $PDA$ accepts some string of the form $\{ww \mid w \in \{0,1\}^{\ast} \}$ . Use the computation history method to show that this problem is undecidable.

Lakshman Patel RJIT asked in Theory of Computation Oct 20, 2019

by Lakshman Patel RJIT

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8

Michael Sipser Edition 3 Exercise 5 Question 32 (Page No. 241)
Prove that the following two languages are undecidable. $OVERLAP_{CFG} = \{\langle G, H\rangle \mid \text{G and H are CFGs where}\: L(G) \cap L(H) \neq \emptyset\}$. $PREFIX-FREE_{CFG} = \{\langle G \rangle \mid \text{G is a CFG where L(G) is prefix-free}\}$.

Lakshman Patel RJIT asked in Theory of Computation Oct 20, 2019

by Lakshman Patel RJIT

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9

Michael Sipser Edition 3 Exercise 5 Question 31 (Page No. 241)
Let $f(x)=\left\{\begin{matrix}3x+1 & \text{for odd}\: x& \\ \dfrac{x}{2} & \text{for even}\: x & \end{matrix}\right.$ for any natural number $x$. If you start with an integer $x$ and iterate $f$, you ... decidable by a $TM\: H$. Use $H$ to describe a $TM$ that is guaranteed to state the answer to the $3x + 1$ problem.

Lakshman Patel RJIT asked in Theory of Computation Oct 20, 2019

by Lakshman Patel RJIT

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10

Michael Sipser Edition 3 Exercise 5 Question 30 (Page No. 241)
Use Rice’s theorem, to prove the undecidability of each of the following languages. $INFINITE_{TM} = \{\langle M \rangle \mid \text{M is a TM and L(M) is an infinite language}\}$. $\{\langle M \rangle \mid \text{M is a TM and }\:1011 \in L(M)\}$. $ ALL_{TM} = \{\langle M \rangle \mid \text{ M is a TM and}\: L(M) = Σ^{\ast} \}$.

Lakshman Patel RJIT asked in Theory of Computation Oct 20, 2019

by Lakshman Patel RJIT

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11

Michael Sipser Edition 3 Exercise 5 Question 29 (Page No. 241)
Rice's theorem. Let $P$ be any nontrivial property of the language of a Turing machine. Prove that the problem of determining whether a given Turing machine's language has property $P$ is undecidable. In more formal ... that $P$ is an undecidable language. Show that both conditions are necessary for proving that $P$ is undecidable.

Lakshman Patel RJIT asked in Theory of Computation Oct 20, 2019

by Lakshman Patel RJIT

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12

Michael Sipser Edition 3 Exercise 5 Question 28 (Page No. 241)
Rice's theorem. Let $P$ be any nontrivial property of the language of a Turing machine. Prove that the problem of determining whether a given Turing machine's language has property $P$ is undecidable. In more formal terms, let $P$ be a language ... $M_{1}$ and $M_{2}$ are any $TMs$. Prove that $P$ is an undecidable language.

Lakshman Patel RJIT asked in Theory of Computation Oct 20, 2019

by Lakshman Patel RJIT

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13

Michael Sipser Edition 3 Exercise 5 Question 27 (Page No. 241)
A two-dimensional finite automaton $(2DIM-DFA)$ is defined as follows. The input is an $m \times n$ rectangle, for any $m, n \geq 2$ ... of determining whether two of these machines are equivalent. Formulate this problem as a language and show that it is undecidable.

Lakshman Patel RJIT asked in Theory of Computation Oct 20, 2019

by Lakshman Patel RJIT

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14

Michael Sipser Edition 3 Exercise 5 Question 26 (Page No. 240)
Define a two-headed finite automaton $(2DFA)$ to be a deterministic finite automaton that has two read-only, bidirectional heads that start at the left-hand end of the input tape and can be independently controlled to move in either direction. The tape ... $E_{2DFA}$ is not decidable.

Lakshman Patel RJIT asked in Theory of Computation Oct 20, 2019

by Lakshman Patel RJIT

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15

Michael Sipser Edition 3 Exercise 5 Question 25 (Page No. 240)
Give an example of an undecidable language $B$, where $B \leq_{m} \overline{B}$.

Lakshman Patel RJIT asked in Theory of Computation Oct 19, 2019

by Lakshman Patel RJIT

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16

Michael Sipser Edition 3 Exercise 5 Question 24 (Page No. 240)
Let $J = \{w \mid \text{either $w = 0x$ for some $x \in A_{TM},$ or $w = 1y\:$ for some $y \in \overline{A_{TM}}\:\:$}\}$. Show that neither $J$ nor $\overline{J}$ is Turing-recognizable.

Lakshman Patel RJIT asked in Theory of Computation Oct 19, 2019

by Lakshman Patel RJIT

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17

Michael Sipser Edition 3 Exercise 5 Question 23 (Page No. 240)
Show that $A$ is decidable iff $A \leq_{m} 0 ^{\ast} 1^{\ast}$ .

Lakshman Patel RJIT asked in Theory of Computation Oct 19, 2019

by Lakshman Patel RJIT

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18

Michael Sipser Edition 3 Exercise 5 Question 22 (Page No. 240)
Show that $A$ is Turing-recognizable iff $A \leq_{m} A_{TM}$.

Lakshman Patel RJIT asked in Theory of Computation Oct 19, 2019

by Lakshman Patel RJIT

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19

Michael Sipser Edition 3 Exercise 5 Question 21 (Page No. 240)
Let $AMBIG_{CFG} = \{\langle G \rangle \mid \text{G is an ambiguous CFG}\}$. Show that $AMBIG_{CFG}$ is undecidable. (Hint: Use a reduction from $PCP$ ... $a_{1},\dots,a_{k}$ are new terminal symbols. Prove that this reduction works.)

Lakshman Patel RJIT asked in Theory of Computation Oct 19, 2019

by Lakshman Patel RJIT

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20

Michael Sipser Edition 3 Exercise 5 Question 20 (Page No. 240)
Prove that there exists an undecidable subset of $\{1\}^{\ast}$ .

Lakshman Patel RJIT asked in Theory of Computation Oct 19, 2019

by Lakshman Patel RJIT

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21

Michael Sipser Edition 3 Exercise 5 Question 19 (Page No. 240)
In the silly Post Correspondence Problem, $SPCP$, the top string in each pair has the same length as the bottom string. Show that the $SPCP$ is decidable.

Lakshman Patel RJIT asked in Theory of Computation Oct 19, 2019

by Lakshman Patel RJIT

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22

Michael Sipser Edition 3 Exercise 5 Question 18 (Page No. 240)
Show that the Post Correspondence Problem is undecidable over the binary alphabet $\Sigma = \{0,1\}$.

Lakshman Patel RJIT asked in Theory of Computation Oct 19, 2019

by Lakshman Patel RJIT

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23

Michael Sipser Edition 3 Exercise 5 Question 17 (Page No. 240)
Show that the Post Correspondence Problem is decidable over the unary alphabet $\Sigma = \{1\}$.

Lakshman Patel RJIT asked in Theory of Computation Oct 19, 2019

by Lakshman Patel RJIT

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24

Michael Sipser Edition 3 Exercise 5 Question 16 (Page No. 240)
Let $\Gamma = \{0, 1, \sqcup\}$ be the tape alphabet for all TMs in this problem. Define the busy beaver function $BB: N \rightarrow N$ as follows. For each value of $k$, consider all $k-$state TMs that halt when ... maximum number of $1s$ that remain on the tape among all of these machines. Show that $BB$ is not a computable function.

Lakshman Patel RJIT asked in Theory of Computation Oct 19, 2019

by Lakshman Patel RJIT

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25

Michael Sipser Edition 3 Exercise 5 Question 15 (Page No. 240)
Consider the problem of determining whether a Turing machine $M$ on an input w ever attempts to move its head left at any point during its computation on $w$. Formulate this problem as a language and show that it is decidable.

Lakshman Patel RJIT asked in Theory of Computation Oct 19, 2019

by Lakshman Patel RJIT

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26

Michael Sipser Edition 3 Exercise 5 Question 14 (Page No. 240)
Consider the problem of determining whether a Turing machine $M$ on an input $w$ ever attempts to move its head left when its head is on the left-most tape cell. Formulate this problem as a language and show that it is undecidable.

Lakshman Patel RJIT asked in Theory of Computation Oct 19, 2019

by Lakshman Patel RJIT

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27

Michael Sipser Edition 3 Exercise 5 Question 13 (Page No. 239)
A useless state in a Turing machine is one that is never entered on any input string. Consider the problem of determining whether a Turing machine has any useless states. Formulate this problem as a language and show that it is undecidable.

Lakshman Patel RJIT asked in Theory of Computation Oct 19, 2019

by Lakshman Patel RJIT

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28

Michael Sipser Edition 3 Exercise 5 Question 12 (Page No. 239)
Consider the problem of determining whether a single-tape Turing machine ever writes a blank symbol over a nonblank symbol during the course of its computation on any input string. Formulate this problem as a language and show that it is undecidable.

Lakshman Patel RJIT asked in Theory of Computation Oct 19, 2019

by Lakshman Patel RJIT

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29

Michael Sipser Edition 3 Exercise 5 Question 11 (Page No. 239)
Consider the problem of determining whether a two-tape Turing machine ever writes a nonblank symbol on its second tape during the course of its computation on any input string. Formulate this problem as a language and show that it is undecidable.

Lakshman Patel RJIT asked in Theory of Computation Oct 19, 2019

by Lakshman Patel RJIT

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30

Michael Sipser Edition 3 Exercise 5 Question 10 (Page No. 239)
Consider the problem of determining whether a two-tape Turing machine ever writes a nonblank symbol on its second tape when it is run on input $w$. Formulate this problem as a language and show that it is undecidable.

Lakshman Patel RJIT asked in Theory of Computation Oct 19, 2019

by Lakshman Patel RJIT

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