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Powered by Question2AnswerMichael Sipser Edition 3 Exercise 5 Question 34 (Page No. 241)
https://gateoverflow.in/324085/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 $w$ } \}$<br />
<br />
Is $X$ decidable? Prove your answer.Theory of Computationhttps://gateoverflow.in/324085/michael-sipser-edition-3-exercise-5-question-34-page-no-241Sun, 20 Oct 2019 12:30:35 +0000Michael Sipser Edition 3 Exercise 5 Question 32 (Page No. 241)
https://gateoverflow.in/324083/michael-sipser-edition-3-exercise-5-question-32-page-no-241
<p>Prove that the following two languages are undecidable.</p>
<ol start="1" style="list-style-type:lower-alpha">
<li>$OVERLAP_{CFG} = \{\langle G, H\rangle \mid \text{G and H are CFGs where}\: L(G) \cap L(H) \neq \emptyset\}$.</li>
<li>$PREFIX-FREE_{CFG} = \{\langle G \rangle \mid \text{G is a CFG where L(G) is prefix-free}\}$.</li>
</ol>Theory of Computationhttps://gateoverflow.in/324083/michael-sipser-edition-3-exercise-5-question-32-page-no-241Sun, 20 Oct 2019 12:24:55 +0000Michael Sipser Edition 3 Exercise 5 Question 31 (Page No. 241)
https://gateoverflow.in/324080/michael-sipser-edition-3-exercise-5-question-31-page-no-241
Let<br />
<br />
$f(x)=\left\{\begin{matrix}3x+1 & \text{for odd}\: x& \\ \dfrac{x}{2} & \text{for even}\: x & \end{matrix}\right.$<br />
<br />
for any natural number $x$. If you start with an integer $x$ and iterate $f$, you obtain a sequence, $x, f(x), f(f(x)), \dots$ Stop if you ever hit $1$. For example, if $x = 17$, you get the sequence $17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1$. Extensive computer tests have shown that every starting point between $1$ and a large positive integer gives a sequence that ends in $1$. But the question of whether all positive starting points end up at $1$ is unsolved; it is called the $3x + 1$ problem. Suppose that $ATM$ were decidable by a $TM\: H$. Use $H$ to describe a $TM$ that is guaranteed to state the answer to the $3x + 1$ problem.Theory of Computationhttps://gateoverflow.in/324080/michael-sipser-edition-3-exercise-5-question-31-page-no-241Sun, 20 Oct 2019 12:20:24 +0000Michael Sipser Edition 3 Exercise 5 Question 30 (Page No. 241)
https://gateoverflow.in/324077/michael-sipser-edition-3-exercise-5-question-30-page-no-241
<p>Use Rice’s theorem, to prove the undecidability of each of the following languages.</p>
<ol start="1" style="list-style-type:lower-alpha">
<li>$INFINITE_{TM} = \{\langle M \rangle \mid \text{M is a TM and L(M) is an infinite language}\}$.</li>
<li>$\{\langle M \rangle \mid \text{M is a TM and }\:1011 \in L(M)\}$.</li>
<li>$ ALL_{TM} = \{\langle M \rangle \mid \text{ M is a TM and}\: L(M) = Σ^{\ast} \}$.</li>
</ol>Theory of Computationhttps://gateoverflow.in/324077/michael-sipser-edition-3-exercise-5-question-30-page-no-241Sun, 20 Oct 2019 12:11:47 +0000Michael Sipser Edition 3 Exercise 5 Question 29 (Page No. 241)
https://gateoverflow.in/324076/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 terms, let $P$ be a language consisting of Turing machine descriptions where $P$ fulfills two conditions. First, $P$ is nontrivial—it contains some, but not all, $TM$ descriptions. Second, $P$ is a property of the $TM’s$ language—whenever $L(M_{1}) = L(M_{2})$, we have $\langle M_{1}\rangle \in P$ iff $\langle M_{2}\rangle \in P$ . Here, $M_{1}$ and $M_{2}$ are any $TMs$. Prove that $P$ is an undecidable language.<br />
<br />
Show that both conditions are necessary for proving that $P$ is undecidable.Theory of Computationhttps://gateoverflow.in/324076/michael-sipser-edition-3-exercise-5-question-29-page-no-241Sun, 20 Oct 2019 12:05:26 +0000Michael Sipser Edition 3 Exercise 5 Question 28 (Page No. 241)
https://gateoverflow.in/324075/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 consisting of Turing machine descriptions where $P$ fulfills two conditions. First, $P$ is nontrivial—it contains some, but not all, $TM$ descriptions. Second, $P$ is a property of the $TM’s$ language—whenever $L(M_{1}) = L(M_{2})$, we have $\langle M_{1}\rangle \in P$ iff $\langle M_{2}\rangle \in P$ . Here, $M_{1}$ and $M_{2}$ are any $TMs$. Prove that $P$ is an undecidable language.Theory of Computationhttps://gateoverflow.in/324075/michael-sipser-edition-3-exercise-5-question-28-page-no-241Sun, 20 Oct 2019 12:03:03 +0000Michael Sipser Edition 3 Exercise 5 Question 27 (Page No. 241)
https://gateoverflow.in/324074/michael-sipser-edition-3-exercise-5-question-27-page-no-241
<p>A <strong>two-dimensional finite automaton</strong> $(2DIM-DFA)$ is defined as follows. The input is an $m \times n$ rectangle, for any $m, n \geq 2$. The squares along the boundary of the rectangle contain the symbol $\#$ and the internal squares contain symbols over the input alphabet $\Sigma$. The transition function $ \delta: Q \times (\Sigma \cup {\#})\rightarrow Q × \{L, R, U, D\}$ indicates the next state and the new head position (Left, Right, Up, Down). The machine accepts when it enters one of the designated accept states. It rejects if it tries to move off the input rectangle or if it never halts. Two such machines are equivalent if they accept the same rectangles. Consider the problem of determining whether two of these machines are equivalent. Formulate this problem as a language and show that it is undecidable.</p>Theory of Computationhttps://gateoverflow.in/324074/michael-sipser-edition-3-exercise-5-question-27-page-no-241Sun, 20 Oct 2019 11:57:40 +0000Michael Sipser Edition 3 Exercise 5 Question 26 (Page No. 240)
https://gateoverflow.in/324073/michael-sipser-edition-3-exercise-5-question-26-page-no-240
<p>Define a <strong>two-headed finite automaton</strong> $(2DFA)$ 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 of a $2DFA$ is finite and is just large enough to contain the input plus two additional blank tape cells, one on the left-hand end and one on the right-hand end, that serve as delimiters. A $2DFA$ accepts its input by entering a special accept state. For example, a $2DFA$ can recognize the language $\{a^{n}b^{n}c^{n}\mid n\geq 0\}$.</p>
<ol start="1" style="list-style-type:lower-alpha">
<li>Let $A_{2DFA} = \{ \langle M, x \rangle \mid \text {M is a 2DFA and M accepts}\: x\}$. Show that $A_{2DFA}$ is decidable.</li>
<li>Let $E_{2DFA} = \{\langle M \rangle \mid \text{M is a 2DFA and} \:L(M) = \emptyset\}$. Show that $E_{2DFA}$ is not decidable.</li>
</ol>Theory of Computationhttps://gateoverflow.in/324073/michael-sipser-edition-3-exercise-5-question-26-page-no-240Sun, 20 Oct 2019 11:51:36 +0000Michael Sipser Edition 3 Exercise 5 Question 25 (Page No. 240)
https://gateoverflow.in/323990/michael-sipser-edition-3-exercise-5-question-25-page-no-240
Give an example of an undecidable language $B$, where $B \leq_{m} \overline{B}$.Theory of Computationhttps://gateoverflow.in/323990/michael-sipser-edition-3-exercise-5-question-25-page-no-240Sat, 19 Oct 2019 12:38:49 +0000Michael Sipser Edition 3 Exercise 5 Question 24 (Page No. 240)
https://gateoverflow.in/323989/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.Theory of Computationhttps://gateoverflow.in/323989/michael-sipser-edition-3-exercise-5-question-24-page-no-240Sat, 19 Oct 2019 12:36:13 +0000Michael Sipser Edition 3 Exercise 5 Question 16 (Page No. 240)
https://gateoverflow.in/323980/michael-sipser-edition-3-exercise-5-question-16-page-no-240
<p>Let $\Gamma = \{0, 1, \sqcup\}$ be the tape alphabet for all TMs in this problem. Define the busy <strong>beaver function</strong> $BB: N \rightarrow N$ as follows. For each value of $k$, consider all $k-$state TMs that halt when started with a blank tape. Let $BB(k)$ be the maximum number of $1s$ that remain on the tape among all of these machines. Show that $BB$ is not a computable function.</p>Theory of Computationhttps://gateoverflow.in/323980/michael-sipser-edition-3-exercise-5-question-16-page-no-240Sat, 19 Oct 2019 12:02:27 +0000Michael Sipser Edition 3 Exercise 5 Question 15 (Page No. 240)
https://gateoverflow.in/323978/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.Theory of Computationhttps://gateoverflow.in/323978/michael-sipser-edition-3-exercise-5-question-15-page-no-240Sat, 19 Oct 2019 11:58:40 +0000Michael Sipser Edition 3 Exercise 5 Question 14 (Page No. 240)
https://gateoverflow.in/323977/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.Theory of Computationhttps://gateoverflow.in/323977/michael-sipser-edition-3-exercise-5-question-14-page-no-240Sat, 19 Oct 2019 11:57:47 +0000Michael Sipser Edition 3 Exercise 5 Question 13 (Page No. 239)
https://gateoverflow.in/323976/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.Theory of Computationhttps://gateoverflow.in/323976/michael-sipser-edition-3-exercise-5-question-13-page-no-239Sat, 19 Oct 2019 11:55:54 +0000Michael Sipser Edition 3 Exercise 5 Question 12 (Page No. 239)
https://gateoverflow.in/323975/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.Theory of Computationhttps://gateoverflow.in/323975/michael-sipser-edition-3-exercise-5-question-12-page-no-239Sat, 19 Oct 2019 11:54:19 +0000Michael Sipser Edition 3 Exercise 5 Question 11 (Page No. 239)
https://gateoverflow.in/323974/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.Theory of Computationhttps://gateoverflow.in/323974/michael-sipser-edition-3-exercise-5-question-11-page-no-239Sat, 19 Oct 2019 11:52:38 +0000Michael Sipser Edition 3 Exercise 5 Question 10 (Page No. 239)
https://gateoverflow.in/323973/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.Theory of Computationhttps://gateoverflow.in/323973/michael-sipser-edition-3-exercise-5-question-10-page-no-239Sat, 19 Oct 2019 11:50:31 +0000Michael Sipser Edition 3 Exercise 5 Question 9 (Page No. 239)
https://gateoverflow.in/323972/michael-sipser-edition-3-exercise-5-question-9-page-no-239
Let $T = \{\langle M \rangle \mid \text{M is a TM that accepts $w^{R}$ whenever it accepts} \:w\}$. Show that $T$ is undecidable.Theory of Computationhttps://gateoverflow.in/323972/michael-sipser-edition-3-exercise-5-question-9-page-no-239Sat, 19 Oct 2019 11:49:12 +0000Michael Sipser Edition 3 Exercise 5 Question 8 (Page No. 239)
https://gateoverflow.in/323970/michael-sipser-edition-3-exercise-5-question-8-page-no-239
In the proof of Theorem $5.15$, we modified the Turing machine $M$ so that it never tries to move its head off the left-hand end of the tape. Suppose that we did not make this modification to $M$. Modify the $PCP$ construction to handle this case.Theory of Computationhttps://gateoverflow.in/323970/michael-sipser-edition-3-exercise-5-question-8-page-no-239Sat, 19 Oct 2019 11:46:38 +0000Michael Sipser Edition 3 Exercise 5 Question 6 (Page No. 239)
https://gateoverflow.in/323968/michael-sipser-edition-3-exercise-5-question-6-page-no-239
Show that $\leq_{m}$ is a transitive relation.Theory of Computationhttps://gateoverflow.in/323968/michael-sipser-edition-3-exercise-5-question-6-page-no-239Sat, 19 Oct 2019 11:41:04 +0000Michael Sipser Edition 3 Exercise 5 Question 5 (Page No. 239)
https://gateoverflow.in/323966/michael-sipser-edition-3-exercise-5-question-5-page-no-239
Show that $A_{TM}$ is not mapping reducible to $E_{TM}$. In other words, show that no computable function reduces $A_{TM}$ to $E_{TM}$. (Hint: Use a proof by contradiction, and facts you already know about $A_{TM}$ and $E_{TM}$.)Theory of Computationhttps://gateoverflow.in/323966/michael-sipser-edition-3-exercise-5-question-5-page-no-239Sat, 19 Oct 2019 11:31:26 +0000Michael Sipser Edition 3 Exercise 5 Question 3 (Page No. 239)
https://gateoverflow.in/323943/michael-sipser-edition-3-exercise-5-question-3-page-no-239
Find a match in the following instance of the Post Correspondence Problem.<br />
$\begin{Bmatrix} \bigg[\dfrac{ab}{abab}\bigg],&\bigg[\dfrac{b}{a}\bigg],&\bigg[\dfrac{aba}{b}\bigg], & \bigg[\dfrac{aa}{a}\bigg] \end{Bmatrix}$Theory of Computationhttps://gateoverflow.in/323943/michael-sipser-edition-3-exercise-5-question-3-page-no-239Sat, 19 Oct 2019 09:54:38 +0000Michael Sipser Edition 3 Exercise 4 Question 30 (Page No. 212)
https://gateoverflow.in/323826/michael-sipser-edition-3-exercise-4-question-30-page-no-212
Let $A$ be a Turing-recognizable language consisting of descriptions of Turing machines, $\{ \langle M_{1}\rangle,\langle M_{2}\rangle,\dots\}$, where every $M_{i}$ is a decider. Prove that some decidable language $D$ is not decided by any decider $M_{i}$ whose description appears in $A$. (Hint: You may find it helpful to consider an enumerator for $A$.)Theory of Computationhttps://gateoverflow.in/323826/michael-sipser-edition-3-exercise-4-question-30-page-no-212Thu, 17 Oct 2019 16:25:06 +0000Michael Sipser Edition 3 Exercise 4 Question 11 (Page No. 211)
https://gateoverflow.in/323772/michael-sipser-edition-3-exercise-4-question-11-page-no-211
Let $INFINITE_{PDA} = \{\langle{ M \rangle} \mid \text{M is a PDA and L(M) is an infinite language}\}$. Show that $INFINITE_{PDA}$ is decidable.Theory of Computationhttps://gateoverflow.in/323772/michael-sipser-edition-3-exercise-4-question-11-page-no-211Thu, 17 Oct 2019 09:18:05 +0000Michael Sipser Edition 3 Exercise 4 Question 10 (Page No. 211)
https://gateoverflow.in/323771/michael-sipser-edition-3-exercise-4-question-10-page-no-211
Let $INFINITE_{DFA} = \{\langle{ A \rangle} \mid \text{ A is a DFA and L(A) is an infinite language}\}$. Show that $INFINITE_{DFA}$ is decidable.Theory of Computationhttps://gateoverflow.in/323771/michael-sipser-edition-3-exercise-4-question-10-page-no-211Thu, 17 Oct 2019 09:12:39 +0000Michael Sipser Edition 3 Exercise 4 Question 9 (Page No. 211)
https://gateoverflow.in/323770/michael-sipser-edition-3-exercise-4-question-9-page-no-211
Review the way that we define sets to be the same size in Definition $4.12$ (page $203$). Show that “is the same size” is an equivalence relation.Theory of Computationhttps://gateoverflow.in/323770/michael-sipser-edition-3-exercise-4-question-9-page-no-211Thu, 17 Oct 2019 09:05:00 +0000Michael Sipser Edition 3 Exercise 4 Question 8 (Page No. 211)
https://gateoverflow.in/323769/michael-sipser-edition-3-exercise-4-question-8-page-no-211
Let $T = \{(i, j, k)\mid i, j, k \in N \}$. Show that $T$ is countable.Theory of Computationhttps://gateoverflow.in/323769/michael-sipser-edition-3-exercise-4-question-8-page-no-211Thu, 17 Oct 2019 09:00:22 +0000Michael Sipser Edition 3 Exercise 4 Question 7 (Page No. 211)
https://gateoverflow.in/323768/michael-sipser-edition-3-exercise-4-question-7-page-no-211
Let $B$ be the set of all infinite sequences over $\{0,1\}$. Show that $B$ is uncountable using a proof by diagonalization.Theory of Computationhttps://gateoverflow.in/323768/michael-sipser-edition-3-exercise-4-question-7-page-no-211Thu, 17 Oct 2019 08:57:01 +0000Michael Sipser Edition 3 Exercise 4 Question 6 (Page No. 211)
https://gateoverflow.in/323767/michael-sipser-edition-3-exercise-4-question-6-page-no-211
<p>Let $X$ be the set $\{1, 2, 3, 4, 5\}$ and $Y$ be the set $\{6, 7, 8, 9, 10\}$. We describe the functions $f : X\rightarrow Y$ and $g : X\rightarrow Y$ in the following tables. Answer each part and give a reason for each negative answer.</p>
<p> </p>
<table border="1" cellpadding="1" style="width:100px">
<tbody>
<tr>
<td>$n$</td>
<td>$f(n)$</td>
</tr>
<tr>
<td>1</td>
<td>6</td>
</tr>
<tr>
<td>2</td>
<td>7</td>
</tr>
<tr>
<td>3</td>
<td>6</td>
</tr>
<tr>
<td>4</td>
<td>7</td>
</tr>
<tr>
<td>5</td>
<td>6</td>
</tr>
</tbody>
</table>
<p> </p>
<table border="1" cellpadding="1" style="width:100px">
<tbody>
<tr>
<td>$n$</td>
<td>$g(n)$</td>
</tr>
<tr>
<td>1</td>
<td>10</td>
</tr>
<tr>
<td>2</td>
<td>9</td>
</tr>
<tr>
<td>3</td>
<td>8</td>
</tr>
<tr>
<td>4</td>
<td>7</td>
</tr>
<tr>
<td>5</td>
<td>6</td>
</tr>
</tbody>
</table>
<p> </p>
<ol start="1" style="list-style-type:lower-alpha">
<li>Is $f$ one-to-one?</li>
<li>Is $f$ onto?</li>
<li>Is $f$ a correspondence?</li>
<li>Is $g$ one-to-one?</li>
<li>Is $g$ onto?</li>
<li>Is $g$ a correspondence?</li>
</ol>Theory of Computationhttps://gateoverflow.in/323767/michael-sipser-edition-3-exercise-4-question-6-page-no-211Thu, 17 Oct 2019 08:47:04 +0000Michael Sipser Edition 3 Exercise 4 Question 5 (Page No. 211)
https://gateoverflow.in/323766/michael-sipser-edition-3-exercise-4-question-5-page-no-211
Let $E_{TM} = \{\langle{ M \rangle } \mid M\: \text{is a TM}\: \text{and}\: L(M) = \phi\}$. Show that $E_{TM}$, the complement of $E_{TM}$, is Turing-recognizable.Theory of Computationhttps://gateoverflow.in/323766/michael-sipser-edition-3-exercise-4-question-5-page-no-211Thu, 17 Oct 2019 08:39:00 +0000Michael Sipser Edition 3 Exercise 4 Question 4 (Page No. 211)
https://gateoverflow.in/323655/michael-sipser-edition-3-exercise-4-question-4-page-no-211
Let $A\varepsilon_{CFG} = \{ \langle{ G }\rangle \mid G\: \text{is a CFG that generates}\: \epsilon \}.$Show that $A\varepsilon_{CFG}$ is decidable.Theory of Computationhttps://gateoverflow.in/323655/michael-sipser-edition-3-exercise-4-question-4-page-no-211Tue, 15 Oct 2019 18:45:49 +0000Michael Sipser Edition 3 Exercise 4 Question 3 (Page No. 211)
https://gateoverflow.in/323654/michael-sipser-edition-3-exercise-4-question-3-page-no-211
Let $ALL_{DFA} = \{ \langle{ A }\rangle \mid A \text{ is a DFA and}\: L(A) = \Sigma^{\ast}\}.$ Show that $ALL_{DFA}$ is decidable.Theory of Computationhttps://gateoverflow.in/323654/michael-sipser-edition-3-exercise-4-question-3-page-no-211Tue, 15 Oct 2019 18:40:44 +0000Michael Sipser Edition 3 Exercise 4 Question 1 (Page No. 210)
https://gateoverflow.in/323651/michael-sipser-edition-3-exercise-4-question-1-page-no-210
<p><img alt="" src="https://gateoverflow.in/?qa=blob&qa_blobid=5193292118754934893"></p>Theory of Computationhttps://gateoverflow.in/323651/michael-sipser-edition-3-exercise-4-question-1-page-no-210Tue, 15 Oct 2019 18:33:59 +0000Michael Sipser Edition 3 Exercise 3 Question 22 (Page No. 190)
https://gateoverflow.in/323650/michael-sipser-edition-3-exercise-3-question-22-page-no-190
Let $A$ be the language containing only the single string $s$, where<br />
<br />
$s = \left\{\begin{matrix} \text{0 if life never will be found on Mars} \\ \:\: \text{1 if life will be found on Mars someday} \end{matrix}\right.$<br />
<br />
Is $A$ decidable? Why or why not? For the purposes of this problem, assume that the question of whether life will be found on Mars has an unambiguous $YES$ or $NO$ answer.Theory of Computationhttps://gateoverflow.in/323650/michael-sipser-edition-3-exercise-3-question-22-page-no-190Tue, 15 Oct 2019 18:28:10 +0000Michael Sipser Edition 3 Exercise 3 Question 21 (Page No. 190)
https://gateoverflow.in/323649/michael-sipser-edition-3-exercise-3-question-21-page-no-190
Let $c_{1}x^{n} + c_{2}x^{n-1} + \dots + c_{n}x + c_{n+1}$ be a polynomial with a root at $x = x_{0}.$ Let $c_{max}$ be the largest absolute value of a $c_{i}.$ Show that $\mid x_{0} \mid < (n+1)\frac{c_{max}}{\mid c_{1} \mid}.$Theory of Computationhttps://gateoverflow.in/323649/michael-sipser-edition-3-exercise-3-question-21-page-no-190Tue, 15 Oct 2019 18:23:54 +0000Michael Sipser Edition 3 Exercise 3 Question 20 (Page No. 190)
https://gateoverflow.in/323647/michael-sipser-edition-3-exercise-3-question-20-page-no-190
Show that single-tape $TMs$ that cannot write on the portion of the tape containing the input string recognize only regular languages.Theory of Computationhttps://gateoverflow.in/323647/michael-sipser-edition-3-exercise-3-question-20-page-no-190Tue, 15 Oct 2019 18:18:54 +0000Michael Sipser Edition 3 Exercise 3 Question 19 (Page No. 190)
https://gateoverflow.in/323646/michael-sipser-edition-3-exercise-3-question-19-page-no-190
Show that every infinite Turing-recognizable language has an infinite decidable subset.Theory of Computationhttps://gateoverflow.in/323646/michael-sipser-edition-3-exercise-3-question-19-page-no-190Tue, 15 Oct 2019 18:17:05 +0000Michael Sipser Edition 3 Exercise 3 Question 17 (Page No. 189)
https://gateoverflow.in/323643/michael-sipser-edition-3-exercise-3-question-17-page-no-189
Let $B = \{\langle {M_{1}\rangle},\langle{ M_{1}\rangle} , \dots \}$ be a Turing-recognizable language consisting of $TM$ descriptions. Show that there is a decidable language $C$ consisting of $TM$ descriptions such that every machine described in $B$ has an equivalent machine in $C$ and vice versa.Theory of Computationhttps://gateoverflow.in/323643/michael-sipser-edition-3-exercise-3-question-17-page-no-189Tue, 15 Oct 2019 18:12:31 +0000Michael Sipser Edition 3 Exercise 3 Question 16 (Page No. 189)
https://gateoverflow.in/323642/michael-sipser-edition-3-exercise-3-question-16-page-no-189
<p>Show that the collection of Turing-recognizable languages is closed under the operation of</p>
<ol start="1" style="list-style-type:lower-alpha">
<li>union.</li>
<li>concatenation.</li>
<li>star.</li>
<li>intersection.</li>
<li>homomorphism. </li>
</ol>Theory of Computationhttps://gateoverflow.in/323642/michael-sipser-edition-3-exercise-3-question-16-page-no-189Tue, 15 Oct 2019 18:03:56 +0000Michael Sipser Edition 3 Exercise 3 Question 15 (Page No. 189)
https://gateoverflow.in/323640/michael-sipser-edition-3-exercise-3-question-15-page-no-189
<p>Show that the collection of decidable languages is closed under the operation of</p>
<ol start="1" style="list-style-type:lower-alpha">
<li>union.</li>
<li>concatenation.</li>
<li>star.</li>
<li>complementation.</li>
<li>intersection.</li>
</ol>Theory of Computationhttps://gateoverflow.in/323640/michael-sipser-edition-3-exercise-3-question-15-page-no-189Tue, 15 Oct 2019 18:01:09 +0000Michael Sipser Edition 3 Exercise 3 Question 14 (Page No. 189)
https://gateoverflow.in/323638/michael-sipser-edition-3-exercise-3-question-14-page-no-189
A queue automaton is like a push-down automaton except that the stack is replaced by a queue. A queue is a tape allowing symbols to be written only on the left-hand end and read only at the right-hand end. Each write operation (we’ll call it a push) adds a symbol to the left-hand end of the queue and each read operation (we’ll call it a pull) reads and removes a symbol at the right-hand end. As with a PDA, the input is placed on a separate read-only input tape, and the head on the input tape can move only from left to right. The input tape contains a cell with a blank symbol following the input, so that the end of the input can be detected. A queue automaton accepts its input by entering a special accept state at any time. Show that a language can be recognized by a deterministic queue automaton iff the language is Turing-recognizable.Theory of Computationhttps://gateoverflow.in/323638/michael-sipser-edition-3-exercise-3-question-14-page-no-189Tue, 15 Oct 2019 17:58:31 +0000Michael Sipser Edition 3 Exercise 3 Question 13 (Page No. 189)
https://gateoverflow.in/323637/michael-sipser-edition-3-exercise-3-question-13-page-no-189
A Turing machine with stay put instead of left is similar to an ordinary Turing machine, but the transition function has the form<br />
<br />
$$\delta: Q\times \Gamma \rightarrow Q\times \Gamma \times \{R,S\}$$<br />
<br />
At each point, the machine can move its head right or let it stay in the same position. Show that this Turing machine variant is not equivalent to the usual version. What class of languages do these machines recognize?Theory of Computationhttps://gateoverflow.in/323637/michael-sipser-edition-3-exercise-3-question-13-page-no-189Tue, 15 Oct 2019 17:56:01 +0000Michael Sipser Edition 3 Exercise 3 Question 12 (Page No. 189)
https://gateoverflow.in/323636/michael-sipser-edition-3-exercise-3-question-12-page-no-189
A Turing machine with left reset is similar to an ordinary Turing machine, but the transition function has the form<br />
<br />
$$\delta: Q\times \Gamma \rightarrow Q\times \Gamma \times \{R,RESET\}$$<br />
<br />
If $\delta(q, a) = (r, b, RESET),$ when the machine is in state $q$ reading an $a,$ the machine’s head jumps to the left-hand end of the tape after it writes $b$ on the tape and enters state $r$. Note that these machines do not have the usual ability to move the head one symbol left. Show that Turing machines with left reset recognize the class of Turing-recognizable languages.Theory of Computationhttps://gateoverflow.in/323636/michael-sipser-edition-3-exercise-3-question-12-page-no-189Tue, 15 Oct 2019 17:52:37 +0000Michael Sipser Edition 3 Exercise 3 Question 11 (Page No. 189)
https://gateoverflow.in/323634/michael-sipser-edition-3-exercise-3-question-11-page-no-189
A Turing machine with doubly infinite tape is similar to an ordinary Turing machine, but its tape is infinite to the left as well as to the right. The tape is initially filled with blanks except for the portion that contains the input. Computation is defined as usual except that the head never encounters an end to the tape as it moves leftward. Show that this type of Turing machine recognizes the class of Turing-recognizable languages.Theory of Computationhttps://gateoverflow.in/323634/michael-sipser-edition-3-exercise-3-question-11-page-no-189Tue, 15 Oct 2019 17:46:13 +0000Michael Sipser Edition 3 Exercise 3 Question 10 (Page No. 188)
https://gateoverflow.in/323633/michael-sipser-edition-3-exercise-3-question-10-page-no-188
Say that a write-once Turing machine is a single-tape TM that can alter each tape square at most once (including the input portion of the tape). Show that this variant Turing machine model is equivalent to the ordinary Turing machine model. (Hint: As a first step, consider the case whereby the Turing machine may alter each tape square at most twice. Use lots of tape.)Theory of Computationhttps://gateoverflow.in/323633/michael-sipser-edition-3-exercise-3-question-10-page-no-188Tue, 15 Oct 2019 17:43:47 +0000Michael Sipser Edition 3 Exercise 3 Question 8 (Page No. 188)
https://gateoverflow.in/323631/michael-sipser-edition-3-exercise-3-question-8-page-no-188
<p>Give implementation-level descriptions of Turing machines that decide the following languages over the alphabet $\{0,1\}$.</p>
<ol start="1" style="list-style-type:lower-alpha">
<li> $\{w \mid w \text{contains an equal number of 0s and 1s}\}$</li>
<li> $\{w \mid w \text{contains twice as many 0s as 1s}\}$</li>
<li> $\{w \mid w \text{does not contain twice as many 0s as 1s}\}$</li>
</ol>Theory of Computationhttps://gateoverflow.in/323631/michael-sipser-edition-3-exercise-3-question-8-page-no-188Tue, 15 Oct 2019 17:35:39 +0000Michael Sipser Edition 3 Exercise 3 Question 7 (Page No. 188)
https://gateoverflow.in/323630/michael-sipser-edition-3-exercise-3-question-7-page-no-188
<p>Explain why the following is not a description of a legitimate Turing machine.</p>
<p>$M_{bad} = “$ On input $\langle p \rangle,$ a polynomial over variables $x_{1},\dots,x_{k}:$</p>
<ol>
<li>Try all possible settings of $x_{1},\dots, x_{k}$ to integer values.</li>
<li>Evaluate $p$ on all of these settings.</li>
<li> If any of these settings evaluates to $0$, accept; otherwise, reject.$”$</li>
</ol>Theory of Computationhttps://gateoverflow.in/323630/michael-sipser-edition-3-exercise-3-question-7-page-no-188Tue, 15 Oct 2019 17:29:26 +0000Michael Sipser Edition 3 Exercise 3 Question 5 (Page No. 188)
https://gateoverflow.in/323458/michael-sipser-edition-3-exercise-3-question-5-page-no-188
<p>Examine the formal definition of a Turing machine to answer the following questions, and explain your reasoning.</p>
<ol start="1" style="list-style-type:lower-alpha">
<li>Can a Turing machine ever write the blank symbol $\sqcup$ on its tape?</li>
<li>Can the tape alphabet $\Gamma$ be the same as the input alphabet $\Sigma$?</li>
<li>Can a Turing machine’s head ever be in the same location in two successive steps?</li>
<li>Can a Turing machine contain just a single state? </li>
</ol>Theory of Computationhttps://gateoverflow.in/323458/michael-sipser-edition-3-exercise-3-question-5-page-no-188Sat, 12 Oct 2019 20:22:14 +0000Michael Sipser Edition 3 Exercise 3 Question 4 (Page No. 187)
https://gateoverflow.in/323457/michael-sipser-edition-3-exercise-3-question-4-page-no-187
Give a formal definition of an enumerator. Consider it to be a type of two-tape Turing machine that uses its second tape as the printer. Include a definition of the enumerated language.Theory of Computationhttps://gateoverflow.in/323457/michael-sipser-edition-3-exercise-3-question-4-page-no-187Sat, 12 Oct 2019 20:15:33 +0000Michael Sipser Edition 3 Exercise 3 Question 3 (Page No. 187)
https://gateoverflow.in/323456/michael-sipser-edition-3-exercise-3-question-3-page-no-187
Modify the proof of Theorem $3.16$ to obtain Corollary $3.19$, showing that a language is decidable iff some nondeterministic Turing machine decides it. (You may assume the following theorem about trees. If every node in a tree has finitely many children and every branch of the tree has finitely many nodes, the tree itself has finitely many nodes.)Theory of Computationhttps://gateoverflow.in/323456/michael-sipser-edition-3-exercise-3-question-3-page-no-187Sat, 12 Oct 2019 20:13:30 +0000