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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.
asked
Oct 20
in
Theory of Computation
by
Lakshman Patel RJIT
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54.8k
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55
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michaelsipser
theoryofcomputation
contextfreegrammars
turingrecognizablelanguages
decidability
proof
+1
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0
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2
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 Turingrecognizable. Show that $NECESSARY_{CFG} $is undecidable.
asked
Oct 20
in
Theory of Computation
by
Lakshman Patel RJIT
Veteran
(
54.8k
points)

16
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michaelsipser
theoryofcomputation
turingrecognizablelanguages
decidability
proof
0
votes
1
answer
3
Michael Sipser Edition 3 Exercise 5 Question 34 (Page No. 241)
Let $X = \{\langle M, w \rangle \mid \text{M is a singletape TM that never modifies the portion of the tape that contains the input $w$ } \}$ Is $X$ decidable? Prove your answer.
asked
Oct 20
in
Theory of Computation
by
Lakshman Patel RJIT
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(
54.8k
points)

36
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michaelsipser
theoryofcomputation
turingmachine
decidability
proof
0
votes
0
answers
4
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.
asked
Oct 20
in
Theory of Computation
by
Lakshman Patel RJIT
Veteran
(
54.8k
points)

7
views
michaelsipser
theoryofcomputation
pushdownautomata
decidability
proof
0
votes
0
answers
5
Michael Sipser Edition 3 Exercise 5 Question 32 (Page No. 241)
Prove that the following two languages are undecidable. $OVERLAP_{CFG} = \{\langle G, H\rangle \mid \text{G and H are CFGs where}\: L(G) \cap L(H) \neq \emptyset\}$. $PREFIXFREE_{CFG} = \{\langle G \rangle \mid \text{G is a CFG where L(G) is prefixfree}\}$.
asked
Oct 20
in
Theory of Computation
by
Lakshman Patel RJIT
Veteran
(
54.8k
points)

10
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michaelsipser
theoryofcomputation
contextfreegrammars
turingmachine
decidability
proof
0
votes
0
answers
6
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.
asked
Oct 20
in
Theory of Computation
by
Lakshman Patel RJIT
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54.8k
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15
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michaelsipser
theoryofcomputation
turingmachine
decidability
proof
+1
vote
0
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7
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} \}$.
asked
Oct 20
in
Theory of Computation
by
Lakshman Patel RJIT
Veteran
(
54.8k
points)

14
views
michaelsipser
theoryofcomputation
turingmachine
decidability
ricetheorem
proof
0
votes
0
answers
8
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.
asked
Oct 20
in
Theory of Computation
by
Lakshman Patel RJIT
Veteran
(
54.8k
points)

6
views
michaelsipser
theoryofcomputation
turingmachine
decidability
ricetheorem
proof
0
votes
0
answers
9
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.
asked
Oct 20
in
Theory of Computation
by
Lakshman Patel RJIT
Veteran
(
54.8k
points)

10
views
michaelsipser
theoryofcomputation
turingmachine
decidability
ricetheorem
proof
0
votes
0
answers
10
Michael Sipser Edition 3 Exercise 5 Question 27 (Page No. 241)
A twodimensional finite automaton $(2DIMDFA)$ 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.
asked
Oct 20
in
Theory of Computation
by
Lakshman Patel RJIT
Veteran
(
54.8k
points)

8
views
michaelsipser
theoryofcomputation
finiteautomata
turingmachine
decidability
proof
0
votes
0
answers
11
Michael Sipser Edition 3 Exercise 5 Question 26 (Page No. 240)
Define a twoheaded finite automaton $(2DFA)$ to be a deterministic finite automaton that has two readonly, bidirectional heads that start at the lefthand end of the input tape and can be independently controlled to move in either direction. The tape ... $E_{2DFA}$ is not decidable.
asked
Oct 20
in
Theory of Computation
by
Lakshman Patel RJIT
Veteran
(
54.8k
points)

5
views
michaelsipser
theoryofcomputation
finiteautomata
turingmachine
decidability
proof
0
votes
0
answers
12
Michael Sipser Edition 3 Exercise 5 Question 25 (Page No. 240)
Give an example of an undecidable language $B$, where $B \leq_{m} \overline{B}$.
asked
Oct 19
in
Theory of Computation
by
Lakshman Patel RJIT
Veteran
(
54.8k
points)

10
views
michaelsipser
theoryofcomputation
turingmachine
decidability
reduction
proof
0
votes
0
answers
13
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 Turingrecognizable.
asked
Oct 19
in
Theory of Computation
by
Lakshman Patel RJIT
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(
54.8k
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3
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michaelsipser
theoryofcomputation
turingmachine
turingrecognizablelanguages
proof
0
votes
0
answers
14
Michael Sipser Edition 3 Exercise 5 Question 23 (Page No. 240)
Show that $A$ is decidable iff $A \leq_{m} 0 ^{\ast} 1^{\ast}$ .
asked
Oct 19
in
Theory of Computation
by
Lakshman Patel RJIT
Veteran
(
54.8k
points)

4
views
michaelsipser
theoryofcomputation
decidability
reduction
proof
0
votes
0
answers
15
Michael Sipser Edition 3 Exercise 5 Question 22 (Page No. 240)
Show that $A$ is Turingrecognizable iff $A \leq_{m} A_{TM}$.
asked
Oct 19
in
Theory of Computation
by
Lakshman Patel RJIT
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(
54.8k
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1
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michaelsipser
theoryofcomputation
turingrecognizablelanguages
reduction
proof
0
votes
0
answers
16
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.)
asked
Oct 19
in
Theory of Computation
by
Lakshman Patel RJIT
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(
54.8k
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2
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michaelsipser
theoryofcomputation
contextfreegrammars
reduction
postcorrespondenceproblem
decidability
proof
0
votes
0
answers
17
Michael Sipser Edition 3 Exercise 5 Question 20 (Page No. 240)
Prove that there exists an undecidable subset of $\{1\}^{\ast}$ .
asked
Oct 19
in
Theory of Computation
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Lakshman Patel RJIT
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54.8k
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3
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michaelsipser
theoryofcomputation
decidability
proof
0
votes
0
answers
18
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.
asked
Oct 19
in
Theory of Computation
by
Lakshman Patel RJIT
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(
54.8k
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3
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michaelsipser
theoryofcomputation
postcorrespondenceproblem
decidability
proof
0
votes
0
answers
19
Michael Sipser Edition 3 Exercise 5 Question 18 (Page No. 240)
Show that the Post Correspondence Problem is undecidable over the binary alphabet $\Sigma = \{0,1\}$.
asked
Oct 19
in
Theory of Computation
by
Lakshman Patel RJIT
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(
54.8k
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3
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michaelsipser
theoryofcomputation
postcorrespondenceproblem
decidability
proof
0
votes
0
answers
20
Michael Sipser Edition 3 Exercise 5 Question 17 (Page No. 240)
Show that the Post Correspondence Problem is decidable over the unary alphabet $\Sigma = \{1\}$.
asked
Oct 19
in
Theory of Computation
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Lakshman Patel RJIT
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54.8k
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3
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michaelsipser
theoryofcomputation
postcorrespondenceproblem
decidability
proof
0
votes
0
answers
21
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.
asked
Oct 19
in
Theory of Computation
by
Lakshman Patel RJIT
Veteran
(
54.8k
points)

1
view
michaelsipser
theoryofcomputation
turingmachine
computability
proof
+1
vote
0
answers
22
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.
asked
Oct 19
in
Theory of Computation
by
Lakshman Patel RJIT
Veteran
(
54.8k
points)

34
views
michaelsipser
theoryofcomputation
turingmachine
decidability
proof
0
votes
0
answers
23
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 leftmost tape cell. Formulate this problem as a language and show that it is undecidable.
asked
Oct 19
in
Theory of Computation
by
Lakshman Patel RJIT
Veteran
(
54.8k
points)

1
view
michaelsipser
theoryofcomputation
turingmachine
decidability
proof
0
votes
0
answers
24
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.
asked
Oct 19
in
Theory of Computation
by
Lakshman Patel RJIT
Veteran
(
54.8k
points)

4
views
michaelsipser
theoryofcomputation
turingmachine
decidability
proof
0
votes
0
answers
25
Michael Sipser Edition 3 Exercise 5 Question 12 (Page No. 239)
Consider the problem of determining whether a singletape 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.
asked
Oct 19
in
Theory of Computation
by
Lakshman Patel RJIT
Veteran
(
54.8k
points)

8
views
michaelsipser
theoryofcomputation
turingmachine
decidability
proof
0
votes
0
answers
26
Michael Sipser Edition 3 Exercise 5 Question 11 (Page No. 239)
Consider the problem of determining whether a twotape 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.
asked
Oct 19
in
Theory of Computation
by
Lakshman Patel RJIT
Veteran
(
54.8k
points)

6
views
michaelsipser
theoryofcomputation
turingmachine
decidability
proof
0
votes
0
answers
27
Michael Sipser Edition 3 Exercise 5 Question 10 (Page No. 239)
Consider the problem of determining whether a twotape 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.
asked
Oct 19
in
Theory of Computation
by
Lakshman Patel RJIT
Veteran
(
54.8k
points)

2
views
michaelsipser
theoryofcomputation
turingmachine
decidability
proof
0
votes
0
answers
28
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.
asked
Oct 19
in
Theory of Computation
by
Lakshman Patel RJIT
Veteran
(
54.8k
points)

4
views
michaelsipser
theoryofcomputation
turingmachine
decidability
proof
0
votes
0
answers
29
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 lefthand end of the tape. Suppose that we did not make this modification to $M$. Modify the $PCP$ construction to handle this case.
asked
Oct 19
in
Theory of Computation
by
Lakshman Patel RJIT
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54.8k
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2
views
michaelsipser
theoryofcomputation
turingmachine
postcorrespondingproblem
proof
0
votes
0
answers
30
Michael Sipser Edition 3 Exercise 5 Question 7 (Page No. 239)
Show that if $A$ is Turingrecognizable and $A\leq_{m} \overline{A},$ then $A$ is decidable.
asked
Oct 19
in
Theory of Computation
by
Lakshman Patel RJIT
Veteran
(
54.8k
points)

7
views
michaelsipser
theoryofcomputation
turingrecognizablelanguages
decidability
reduction
proof
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