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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.
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....
admin
337
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admin
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Oct 19, 2019
Theory of Computation
michael-sipser
theory-of-computation
turing-machine
decidability
proof
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0
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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.
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 inp...
admin
396
views
admin
asked
Oct 19, 2019
Theory of Computation
michael-sipser
theory-of-computation
turing-machine
decidability
proof
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0
votes
0
answers
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.
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 s...
admin
281
views
admin
asked
Oct 19, 2019
Theory of Computation
michael-sipser
theory-of-computation
turing-machine
decidability
proof
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1
votes
0
answers
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.
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...
admin
247
views
admin
asked
Oct 19, 2019
Theory of Computation
michael-sipser
theory-of-computation
turing-machine
decidability
proof
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0
votes
0
answers
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.
Let $T = \{\langle M \rangle \mid \text{M is a TM that accepts $w^{R}$ whenever it accepts} \:w\}$. Show that $T$ is undecidable.
admin
213
views
admin
asked
Oct 19, 2019
Theory of Computation
michael-sipser
theory-of-computation
turing-machine
decidability
proof
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0
votes
0
answers
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.
Show that if $A$ is Turing-recognizable and $A\leq_{m} \overline{A},$ then $A$ is decidable.
admin
237
views
admin
asked
Oct 19, 2019
Theory of Computation
michael-sipser
theory-of-computation
recursive-and-recursively-enumerable-languages
decidability
reduction
proof
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0
votes
0
answers
67
Michael Sipser Edition 3 Exercise 5 Question 1 (Page No. 239)
Show that $EQ_{CFG}$ is undecidable.
Show that $EQ_{CFG}$ is undecidable.
admin
164
views
admin
asked
Oct 17, 2019
Theory of Computation
michael-sipser
theory-of-computation
context-free-grammar
decidability
proof
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0
votes
0
answers
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 $F = \{ \langle P, q, x \rangle \mid (q, x)\: \text{is a looping situation for P}\}$.
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 n...
admin
319
views
admin
asked
Oct 17, 2019
Theory of Computation
michael-sipser
theory-of-computation
dpda
decidability
proof
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1
votes
0
answers
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.
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 te...
admin
454
views
admin
asked
Oct 17, 2019
Theory of Computation
michael-sipser
theory-of-computation
context-free-language
context-free-grammar
decidability
proof
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0
votes
0
answers
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 $A$.)
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 ...
admin
311
views
admin
asked
Oct 17, 2019
Theory of Computation
michael-sipser
theory-of-computation
turing-machine
recursive-and-recursively-enumerable-languages
decidability
proof
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0
votes
0
answers
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.
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...
admin
259
views
admin
asked
Oct 17, 2019
Theory of Computation
michael-sipser
theory-of-computation
context-free-grammar
decidability
proof
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0
votes
0
answers
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 $E_{CFG}$.)
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...
admin
181
views
admin
asked
Oct 17, 2019
Theory of Computation
michael-sipser
theory-of-computation
context-free-grammar
decidability
proof
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0
votes
0
answers
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.)
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 helpfu...
admin
232
views
admin
asked
Oct 17, 2019
Theory of Computation
michael-sipser
theory-of-computation
finite-automata
decidability
proof
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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.)
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 helpfu...
admin
320
views
admin
asked
Oct 17, 2019
Theory of Computation
michael-sipser
theory-of-computation
finite-automata
decidability
proof
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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.)
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. (Hin...
admin
236
views
admin
asked
Oct 17, 2019
Theory of Computation
michael-sipser
theory-of-computation
finite-automata
decidability
proof
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0
votes
0
answers
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.
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. For...
admin
498
views
admin
asked
Oct 17, 2019
Theory of Computation
michael-sipser
theory-of-computation
pushdown-automata
decidability
proof
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0
votes
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.)
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 N...
admin
353
views
admin
asked
Oct 17, 2019
Theory of Computation
michael-sipser
theory-of-computation
finite-automata
decidability
proof
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0
votes
0
answers
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?
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 simi...
admin
397
views
admin
asked
Oct 17, 2019
Theory of Computation
michael-sipser
theory-of-computation
regular-expression
decidability
proof
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0
votes
0
answers
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.
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.
admin
127
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admin
asked
Oct 17, 2019
Theory of Computation
michael-sipser
theory-of-computation
decidability
proof
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0
votes
0
answers
80
Michael Sipser Edition 3 Exercise 4 Question 19 (Page No. 212)
Prove that the class of decidable languages is not closed under homomorphism.
Prove that the class of decidable languages is not closed under homomorphism.
admin
122
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admin
asked
Oct 17, 2019
Theory of Computation
michael-sipser
theory-of-computation
decidability
proof
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0
votes
0
answers
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 $C = \{x \mid \exists y (\langle{ x, y \rangle} \in D)\}$.
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
172
views
admin
asked
Oct 17, 2019
Theory of Computation
michael-sipser
theory-of-computation
recursive-and-recursively-enumerable-languages
decidability
proof
+
–
0
votes
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.
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
256
views
admin
asked
Oct 17, 2019
Theory of Computation
michael-sipser
theory-of-computation
finite-automata
decidability
proof
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–
0
votes
0
answers
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.
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 = x...
admin
189
views
admin
asked
Oct 17, 2019
Theory of Computation
michael-sipser
theory-of-computation
decidability
proof
+
–
0
votes
0
answers
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.
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 C...
admin
554
views
admin
asked
Oct 17, 2019
Theory of Computation
michael-sipser
theory-of-computation
context-free-grammar
decidability
proof
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–
0
votes
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.
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 \ra...
admin
190
views
admin
asked
Oct 17, 2019
Theory of Computation
michael-sipser
theory-of-computation
context-free-grammar
decidability
proof
+
–
0
votes
0
answers
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.
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
123
views
admin
asked
Oct 17, 2019
Theory of Computation
michael-sipser
theory-of-computation
decidability
proof
+
–
0
votes
0
answers
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.
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
148
views
admin
asked
Oct 17, 2019
Theory of Computation
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.
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.
admin
172
views
admin
asked
Oct 17, 2019
Theory of Computation
michael-sipser
theory-of-computation
turing-machine
decidability
proof
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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.
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
185
views
admin
asked
Oct 17, 2019
Theory of Computation
michael-sipser
theory-of-computation
turing-machine
decidability
proof
+
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0
votes
0
answers
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.
Let $B$ be the set of all infinite sequences over $\{0,1\}$. Show that $B$ is uncountable using a proof by diagonalization.
admin
147
views
admin
asked
Oct 17, 2019
Theory of Computation
michael-sipser
theory-of-computation
turing-machine
decidability
proof
+
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