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Materials:
Decidability Problems for Grammars
Some Reduction Inferences
Example reductions
Recent questions tagged decidability
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61
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.
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Theory of Computation
Oct 19, 2019
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admin
328
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michael-sipser
theory-of-computation
turing-machine
decidability
proof
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votes
0
answers
62
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.
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Theory of Computation
Oct 19, 2019
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admin
391
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michael-sipser
theory-of-computation
turing-machine
decidability
proof
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63
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.
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asked
in
Theory of Computation
Oct 19, 2019
by
admin
277
views
michael-sipser
theory-of-computation
turing-machine
decidability
proof
1
vote
0
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64
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.
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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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65
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.
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in
Theory of Computation
Oct 19, 2019
by
admin
207
views
michael-sipser
theory-of-computation
turing-machine
decidability
proof
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0
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66
Michael Sipser Edition 3 Exercise 5 Question 7 (Page No. 239)
Show that if $A$ is Turing-recognizable and $A\leq_{m} \overline{A},$ then $A$ is decidable.
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asked
in
Theory of Computation
Oct 19, 2019
by
admin
241
views
michael-sipser
theory-of-computation
recursive-and-recursively-enumerable-languages
decidability
reduction
proof
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67
Michael Sipser Edition 3 Exercise 5 Question 1 (Page No. 239)
Show that $EQ_{CFG}$ is undecidable.
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Theory of Computation
Oct 17, 2019
by
admin
166
views
michael-sipser
theory-of-computation
context-free-grammar
decidability
proof
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0
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68
Michael Sipser Edition 3 Exercise 4 Question 32 (Page No. 213)
The proof of Lemma $2.41$ says that $(q, x)$ is a looping situation for a $DPDA \:P$ if when $P$ is started in state $q$ with $x \in \Gamma$ on the top of the stack, it never pops anything below $x$ and it never reads an input ... decidable, where $F = \{ \langle P, q, x \rangle \mid (q, x)\: \text{is a looping situation for P}\}$.
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asked
in
Theory of Computation
Oct 17, 2019
by
admin
305
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michael-sipser
theory-of-computation
dpda
decidability
proof
1
vote
0
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69
Michael Sipser Edition 3 Exercise 4 Question 31 (Page No. 212)
Say that a variable $A$ in $CFL\: G$ is usable if it appears in some derivation of some string $w \in G$. Given a $CFG\: G$ and a variable $A$, consider the problem of testing whether $A$ is usable. Formulate this problem as a language and show that it is decidable.
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in
Theory of Computation
Oct 17, 2019
by
admin
450
views
michael-sipser
theory-of-computation
context-free-language
context-free-grammar
decidability
proof
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0
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70
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 ... $A$. (Hint: You may find it helpful to consider an enumerator for $A$.)
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asked
in
Theory of Computation
Oct 17, 2019
by
admin
310
views
michael-sipser
theory-of-computation
turing-machine
recursive-and-recursively-enumerable-languages
decidability
proof
0
votes
0
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71
Michael Sipser Edition 3 Exercise 4 Question 29 (Page No. 212)
Let $C_{CFG} = \{\langle G, k \rangle \mid \text{ G is a CFG and L(G) contains exactly $k$ strings where $k \geq 0$ or $k = \infty$}\}$. Show that $C_{CFG}$ is decidable.
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asked
in
Theory of Computation
Oct 17, 2019
by
admin
251
views
michael-sipser
theory-of-computation
context-free-grammar
decidability
proof
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0
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72
Michael Sipser Edition 3 Exercise 4 Question 28 (Page No. 212)
Let $C = \{ \langle G, x \rangle \mid \text{G is a CFG $x$ is a substring of some $y \in L(G)$}\}$. Show that $C$ is decidable. (Hint: An elegant solution to this problem uses the decider for $E_{CFG}$.)
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asked
in
Theory of Computation
Oct 17, 2019
by
admin
178
views
michael-sipser
theory-of-computation
context-free-grammar
decidability
proof
0
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0
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73
Michael Sipser Edition 3 Exercise 4 Question 27 (Page No. 212)
Let $E = \{\langle M \rangle \mid \text{ M is a DFA that accepts some string with more 1s than 0s}\}$. Show that $E$ is decidable. (Hint: Theorems about $CFLs$ are helpful here.)
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asked
in
Theory of Computation
Oct 17, 2019
by
admin
233
views
michael-sipser
theory-of-computation
finite-automata
decidability
proof
0
votes
1
answer
74
Michael Sipser Edition 3 Exercise 4 Question 26 (Page No. 212)
Let $PAL_{DFA} = \{ \langle M \rangle \mid \text{ M is a DFA that accepts some palindrome}\}$. Show that $PAL_{DFA}$ is decidable. (Hint: Theorems about $CFLs$ are helpful here.)
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asked
in
Theory of Computation
Oct 17, 2019
by
admin
323
views
michael-sipser
theory-of-computation
finite-automata
decidability
proof
0
votes
0
answers
75
Michael Sipser Edition 3 Exercise 4 Question 25 (Page No. 212)
Let $BAL_{DFA} = \{ \langle M \rangle \mid \text{ M is a DFA that accepts some string containing an equal number of 0s and 1s}\}$. Show that $BAL_{DFA}$ is decidable. (Hint: Theorems about $CFLs$ are helpful here.)
admin
asked
in
Theory of Computation
Oct 17, 2019
by
admin
236
views
michael-sipser
theory-of-computation
finite-automata
decidability
proof
0
votes
0
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76
Michael Sipser Edition 3 Exercise 4 Question 24 (Page No. 212)
A useless state in a pushdown automaton is never entered on any input string. Consider the problem of determining whether a pushdown automaton has any useless states. Formulate this problem as a language and show that it is decidable.
admin
asked
in
Theory of Computation
Oct 17, 2019
by
admin
487
views
michael-sipser
theory-of-computation
pushdown-automata
decidability
proof
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0
answers
77
Michael Sipser Edition 3 Exercise 4 Question 23 (Page No. 212)
Say that an $NFA$ is ambiguous if it accepts some string along two different computation branches. Let $AMBIG_{NFA} = \{ \langle N \rangle \mid \text{ N is an ambiguous NFA}\}$. Show that $AMBIG_{NFA}$ is decidable. (Suggestion: One elegant way to solve this problem is to construct a suitable $DFA$ and then run $E_{DFA}$ on it.)
admin
asked
in
Theory of Computation
Oct 17, 2019
by
admin
335
views
michael-sipser
theory-of-computation
finite-automata
decidability
proof
0
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0
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78
Michael Sipser Edition 3 Exercise 4 Question 22 (Page No. 212)
Let $PREFIX-FREE_{REX} = \{\langle R \rangle \mid \text{R is a regular expression and L(R) is prefix-free}\}$. Show that $PREFIX FREE_{REX}$ is decidable. Why does a similar approach fail to show that $PREFIX-FREE_{CFG}$ is decidable?
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asked
in
Theory of Computation
Oct 17, 2019
by
admin
397
views
michael-sipser
theory-of-computation
regular-expression
decidability
proof
0
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0
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79
Michael Sipser Edition 3 Exercise 4 Question 21 (Page No. 212)
Let $S = \{\langle M \rangle \mid \text{M is a DFA that accepts}\: \text{ $w^{R}$ whenever it accepts $w$}\}$. Show that $S$ is decidable.
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asked
in
Theory of Computation
Oct 17, 2019
by
admin
122
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michael-sipser
theory-of-computation
decidability
proof
0
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0
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80
Michael Sipser Edition 3 Exercise 4 Question 19 (Page No. 212)
Prove that the class of decidable languages is not closed under homomorphism.
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in
Theory of Computation
Oct 17, 2019
by
admin
123
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michael-sipser
theory-of-computation
decidability
proof
0
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0
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81
Michael Sipser Edition 3 Exercise 4 Question 18 (Page No. 212)
Let $C$ be a language. Prove that $C$ is Turing-recognizable iff a decidable language $D$ exists such that $C = \{x \mid \exists y (\langle{ x, y \rangle} \in D)\}$.
admin
asked
in
Theory of Computation
Oct 17, 2019
by
admin
170
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michael-sipser
theory-of-computation
recursive-and-recursively-enumerable-languages
decidability
proof
0
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0
answers
82
Michael Sipser Edition 3 Exercise 4 Question 17 (Page No. 212)
Prove that $EQ_{DFA}$ is decidable by testing the two DFAs on all strings up to a certain size. Calculate a size that works.
admin
asked
in
Theory of Computation
Oct 17, 2019
by
admin
255
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michael-sipser
theory-of-computation
finite-automata
decidability
proof
0
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83
Michael Sipser Edition 3 Exercise 4 Question 16 (Page No. 212)
Let $A = \{ \langle R \rangle \mid \text{R is a regular expression describing a language containing at least one string w that has 111 as a substring} \text{(i.e., w = x111y for some x and y)\}}$. Show that $A$ is decidable.
admin
asked
in
Theory of Computation
Oct 17, 2019
by
admin
184
views
michael-sipser
theory-of-computation
decidability
proof
0
votes
0
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84
Michael Sipser Edition 3 Exercise 4 Question 15 (Page No. 212)
Show that the problem of determining whether a CFG generates all strings in $1^{\ast}$ is decidable. In other words, show that $\{\langle { G \rangle} \mid \text{G is a CFG over {0,1} and } 1^{\ast} \subseteq L(G) \}$ is a decidable language.
admin
asked
in
Theory of Computation
Oct 17, 2019
by
admin
536
views
michael-sipser
theory-of-computation
context-free-grammar
decidability
proof
0
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0
answers
85
Michael Sipser Edition 3 Exercise 4 Question 14 (Page No. 211)
Let $\Sigma = \{0,1\}$. Show that the problem of determining whether a $CFG$ generates some string in $1^{\ast}$ is decidable. In other words, show that $\{\langle {G \rangle}\mid \text{G is a CFG over {0,1} and } 1^{\ast} \cap L(G) \neq \phi \}$ is a decidable language.
admin
asked
in
Theory of Computation
Oct 17, 2019
by
admin
191
views
michael-sipser
theory-of-computation
context-free-grammar
decidability
proof
0
votes
0
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86
Michael Sipser Edition 3 Exercise 4 Question 13 (Page No. 211)
Let $A = \{ \langle{ R, S \rangle} \mid \text{R and S are regular expressions and} \: L(R) \subseteq L(S)\}$. Show that $A$ is decidable.
admin
asked
in
Theory of Computation
Oct 17, 2019
by
admin
123
views
michael-sipser
theory-of-computation
decidability
proof
0
votes
0
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87
Michael Sipser Edition 3 Exercise 4 Question 12 (Page No. 211)
Let $A = \{\langle{ M \rangle} \mid \text{M is a DFA that doesn’t accept any string containing an odd number of 1s}\}$.Show that $A$ is decidable.
admin
asked
in
Theory of Computation
Oct 17, 2019
by
admin
144
views
michael-sipser
theory-of-computation
decidability
proof
0
votes
0
answers
88
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.
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asked
in
Theory of Computation
Oct 17, 2019
by
admin
168
views
michael-sipser
theory-of-computation
turing-machine
decidability
proof
0
votes
0
answers
89
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.
admin
asked
in
Theory of Computation
Oct 17, 2019
by
admin
179
views
michael-sipser
theory-of-computation
turing-machine
decidability
proof
0
votes
0
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90
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.
admin
asked
in
Theory of Computation
Oct 17, 2019
by
admin
144
views
michael-sipser
theory-of-computation
turing-machine
decidability
proof
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