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Recent questions tagged michael-sipser
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Michael Sipser Decidability Problem
Is the following language decidable or not? If you deem it decidable, you need to give an algorithm and analyse its running time. If not decidable, you need to prove it. L = {M | M is a Turing machine and L(M) ∈ TIME(2^n)}
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Theory of Computation
Nov 14, 2023
by
baofbuiafbi
144
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theory-of-computation
time-complexity
michael-sipser
0
votes
0
answers
2
Michael Sipster Decidability Problems
Prove L = {F | F is a boolean formula and F evaluates to true on every asignment" is decidable (include algorithm and running time in big o notation)
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in
Theory of Computation
Nov 14, 2023
by
baofbuiafbi
144
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theory-of-computation
michael-sipser
algorithms
0
votes
0
answers
3
Michael Sipster Theory of Computation
Prove the language L={(G,H)|G is a CFG, H is a DFA, and L(G)∩L(H)=∅} is undecidable.
baofbuiafbi
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in
Theory of Computation
Nov 14, 2023
by
baofbuiafbi
154
views
theory-of-computation
number-of-dfa
michael-sipser
3
votes
0
answers
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.
admin
asked
in
Theory of Computation
Oct 20, 2019
by
admin
616
views
michael-sipser
theory-of-computation
context-free-grammar
recursive-and-recursively-enumerable-languages
decidability
proof
1
vote
0
answers
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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.
admin
asked
in
Theory of Computation
Oct 20, 2019
by
admin
335
views
michael-sipser
theory-of-computation
recursive-and-recursively-enumerable-languages
decidability
proof
0
votes
1
answer
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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.
admin
asked
in
Theory of Computation
Oct 20, 2019
by
admin
702
views
michael-sipser
theory-of-computation
turing-machine
decidability
proof
1
vote
2
answers
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.
admin
asked
in
Theory of Computation
Oct 20, 2019
by
admin
561
views
michael-sipser
theory-of-computation
pushdown-automata
decidability
proof
0
votes
0
answers
8
Michael Sipser Edition 3 Exercise 5 Question 32 (Page No. 241)
Prove that the following two languages are undecidable. $OVERLAP_{CFG} = \{\langle G, H\rangle \mid \text{G and H are CFGs where}\: L(G) \cap L(H) \neq \emptyset\}$. $PREFIX-FREE_{CFG} = \{\langle G \rangle \mid \text{G is a CFG where L(G) is prefix-free}\}$.
admin
asked
in
Theory of Computation
Oct 20, 2019
by
admin
448
views
michael-sipser
theory-of-computation
context-free-grammar
turing-machine
decidability
proof
0
votes
0
answers
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.
admin
asked
in
Theory of Computation
Oct 20, 2019
by
admin
354
views
michael-sipser
theory-of-computation
turing-machine
decidability
proof
1
vote
0
answers
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} \}$.
admin
asked
in
Theory of Computation
Oct 20, 2019
by
admin
388
views
michael-sipser
theory-of-computation
turing-machine
decidability
rice-theorem
proof
0
votes
0
answers
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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.
admin
asked
in
Theory of Computation
Oct 20, 2019
by
admin
434
views
michael-sipser
theory-of-computation
turing-machine
decidability
rice-theorem
proof
0
votes
0
answers
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.
admin
asked
in
Theory of Computation
Oct 20, 2019
by
admin
406
views
michael-sipser
theory-of-computation
turing-machine
decidability
rice-theorem
proof
1
vote
0
answers
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.
admin
asked
in
Theory of Computation
Oct 20, 2019
by
admin
461
views
michael-sipser
theory-of-computation
finite-automata
turing-machine
decidability
proof
0
votes
0
answers
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.
admin
asked
in
Theory of Computation
Oct 20, 2019
by
admin
388
views
michael-sipser
theory-of-computation
finite-automata
turing-machine
decidability
proof
0
votes
0
answers
15
Michael Sipser Edition 3 Exercise 5 Question 25 (Page No. 240)
Give an example of an undecidable language $B$, where $B \leq_{m} \overline{B}$.
admin
asked
in
Theory of Computation
Oct 19, 2019
by
admin
294
views
michael-sipser
theory-of-computation
turing-machine
decidability
reduction
proof
0
votes
0
answers
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.
admin
asked
in
Theory of Computation
Oct 19, 2019
by
admin
203
views
michael-sipser
theory-of-computation
turing-machine
recursive-and-recursively-enumerable-languages
proof
0
votes
0
answers
17
Michael Sipser Edition 3 Exercise 5 Question 23 (Page No. 240)
Show that $A$ is decidable iff $A \leq_{m} 0 ^{\ast} 1^{\ast}$ .
admin
asked
in
Theory of Computation
Oct 19, 2019
by
admin
201
views
michael-sipser
theory-of-computation
decidability
reduction
proof
0
votes
0
answers
18
Michael Sipser Edition 3 Exercise 5 Question 22 (Page No. 240)
Show that $A$ is Turing-recognizable iff $A \leq_{m} A_{TM}$.
admin
asked
in
Theory of Computation
Oct 19, 2019
by
admin
166
views
michael-sipser
theory-of-computation
recursive-and-recursively-enumerable-languages
reduction
proof
0
votes
0
answers
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.)
admin
asked
in
Theory of Computation
Oct 19, 2019
by
admin
387
views
michael-sipser
theory-of-computation
context-free-grammar
reduction
post-correspondence-problem
decidability
proof
0
votes
0
answers
20
Michael Sipser Edition 3 Exercise 5 Question 20 (Page No. 240)
Prove that there exists an undecidable subset of $\{1\}^{\ast}$ .
admin
asked
in
Theory of Computation
Oct 19, 2019
by
admin
285
views
michael-sipser
theory-of-computation
decidability
proof
0
votes
0
answers
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.
admin
asked
in
Theory of Computation
Oct 19, 2019
by
admin
305
views
michael-sipser
theory-of-computation
post-correspondence-problem
decidability
proof
0
votes
0
answers
22
Michael Sipser Edition 3 Exercise 5 Question 18 (Page No. 240)
Show that the Post Correspondence Problem is undecidable over the binary alphabet $\Sigma = \{0,1\}$.
admin
asked
in
Theory of Computation
Oct 19, 2019
by
admin
480
views
michael-sipser
theory-of-computation
post-correspondence-problem
decidability
proof
0
votes
0
answers
23
Michael Sipser Edition 3 Exercise 5 Question 17 (Page No. 240)
Show that the Post Correspondence Problem is decidable over the unary alphabet $\Sigma = \{1\}$.
admin
asked
in
Theory of Computation
Oct 19, 2019
by
admin
254
views
michael-sipser
theory-of-computation
post-correspondence-problem
decidability
proof
0
votes
0
answers
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.
admin
asked
in
Theory of Computation
Oct 19, 2019
by
admin
405
views
michael-sipser
theory-of-computation
turing-machine
computability
proof
1
vote
0
answers
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.
admin
asked
in
Theory of Computation
Oct 19, 2019
by
admin
960
views
michael-sipser
theory-of-computation
turing-machine
decidability
proof
0
votes
0
answers
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.
admin
asked
in
Theory of Computation
Oct 19, 2019
by
admin
264
views
michael-sipser
theory-of-computation
turing-machine
decidability
proof
0
votes
0
answers
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.
admin
asked
in
Theory of Computation
Oct 19, 2019
by
admin
328
views
michael-sipser
theory-of-computation
turing-machine
decidability
proof
0
votes
0
answers
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.
admin
asked
in
Theory of Computation
Oct 19, 2019
by
admin
390
views
michael-sipser
theory-of-computation
turing-machine
decidability
proof
0
votes
0
answers
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.
admin
asked
in
Theory of Computation
Oct 19, 2019
by
admin
277
views
michael-sipser
theory-of-computation
turing-machine
decidability
proof
1
vote
0
answers
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.
admin
asked
in
Theory of Computation
Oct 19, 2019
by
admin
237
views
michael-sipser
theory-of-computation
turing-machine
decidability
proof
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