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Recent questions tagged proof
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31
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
214
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
32
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.
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...
admin
316
views
admin
asked
Oct 19, 2019
Theory of Computation
michael-sipser
theory-of-computation
turing-machine
post-correspondence-problem
proof
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0
votes
0
answers
33
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
238
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
34
Michael Sipser Edition 3 Exercise 5 Question 6 (Page No. 239)
Show that $\leq_{m}$ is a transitive relation.
Show that $\leq_{m}$ is a transitive relation.
admin
170
views
admin
asked
Oct 19, 2019
Theory of Computation
michael-sipser
theory-of-computation
turing-machine
reduction
proof
+
–
0
votes
0
answers
35
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 $E_{TM}$.)
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 contradictio...
admin
192
views
admin
asked
Oct 19, 2019
Theory of Computation
michael-sipser
theory-of-computation
turing-machine
reduction
proof
+
–
0
votes
0
answers
36
Michael Sipser Edition 3 Exercise 5 Question 4 (Page No. 239)
If $A \leq_{m} B$ and $B$ is a regular language, does that imply that $A$ is a regular language? Why or why not?
If $A \leq_{m} B$ and $B$ is a regular language, does that imply that $A$ is a regular language? Why or why not?
admin
190
views
admin
asked
Oct 19, 2019
Theory of Computation
michael-sipser
theory-of-computation
regular-language
reduction
proof
+
–
0
votes
0
answers
37
Michael Sipser Edition 3 Exercise 5 Question 3 (Page No. 239)
Find a match in the following instance of the Post Correspondence Problem. $\begin{Bmatrix} \bigg[\dfrac{ab}{abab}\bigg],&\bigg[\dfrac{b}{a}\bigg],&\bigg[\dfrac{aba}{b}\bigg], & \bigg[\dfrac{aa}{a}\bigg] \end{Bmatrix}$
Find a match in the following instance of the Post Correspondence Problem.$\begin{Bmatrix} \bigg[\dfrac{ab}{abab}\bigg],&\bigg[\dfrac{b}{a}\bigg],&\bigg[\dfrac{aba}{b}\bi...
admin
254
views
admin
asked
Oct 19, 2019
Theory of Computation
michael-sipser
theory-of-computation
turing-machine
post-correspondence-problem
proof
+
–
0
votes
0
answers
38
Michael Sipser Edition 3 Exercise 5 Question 2 (Page No. 239)
Show that $EQ_{CFG}$ is co-Turing-recognizable.
Show that $EQ_{CFG}$ is co-Turing-recognizable.
admin
176
views
admin
asked
Oct 17, 2019
Theory of Computation
michael-sipser
theory-of-computation
context-free-grammar
recursive-and-recursively-enumerable-languages
proof
+
–
0
votes
0
answers
39
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
+
–
0
votes
0
answers
40
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
325
views
admin
asked
Oct 17, 2019
Theory of Computation
michael-sipser
theory-of-computation
dpda
decidability
proof
+
–
1
votes
0
answers
41
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
457
views
admin
asked
Oct 17, 2019
Theory of Computation
michael-sipser
theory-of-computation
context-free-language
context-free-grammar
decidability
proof
+
–
0
votes
0
answers
42
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
313
views
admin
asked
Oct 17, 2019
Theory of Computation
michael-sipser
theory-of-computation
turing-machine
recursive-and-recursively-enumerable-languages
decidability
proof
+
–
0
votes
0
answers
43
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
262
views
admin
asked
Oct 17, 2019
Theory of Computation
michael-sipser
theory-of-computation
context-free-grammar
decidability
proof
+
–
0
votes
0
answers
44
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
182
views
admin
asked
Oct 17, 2019
Theory of Computation
michael-sipser
theory-of-computation
context-free-grammar
decidability
proof
+
–
0
votes
0
answers
45
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
234
views
admin
asked
Oct 17, 2019
Theory of Computation
michael-sipser
theory-of-computation
finite-automata
decidability
proof
+
–
0
votes
1
answer
46
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
+
–
0
votes
0
answers
47
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
+
–
0
votes
0
answers
48
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
506
views
admin
asked
Oct 17, 2019
Theory of Computation
michael-sipser
theory-of-computation
pushdown-automata
decidability
proof
+
–
0
votes
0
answers
49
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
363
views
admin
asked
Oct 17, 2019
Theory of Computation
michael-sipser
theory-of-computation
finite-automata
decidability
proof
+
–
0
votes
0
answers
50
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
400
views
admin
asked
Oct 17, 2019
Theory of Computation
michael-sipser
theory-of-computation
regular-expression
decidability
proof
+
–
0
votes
0
answers
51
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
128
views
admin
asked
Oct 17, 2019
Theory of Computation
michael-sipser
theory-of-computation
decidability
proof
+
–
0
votes
0
answers
52
Michael Sipser Edition 3 Exercise 4 Question 20 (Page No. 212)
Let $A$ and $B$ be two disjoint languages. Say that language $C$ separates $A$ and $B$ if $A \subseteq C$ and $B \subseteq \overline{C}$. Show that any two disjoint co-Turing-recognizable languages are separable by some decidable language.
Let $A$ and $B$ be two disjoint languages. Say that language $C$ separates $A$ and $B$ if $A \subseteq C$ and $B \subseteq \overline{C}$. Show that any two disjoint co-Tu...
admin
304
views
admin
asked
Oct 17, 2019
Theory of Computation
michael-sipser
theory-of-computation
recursive-and-recursively-enumerable-languages
proof
+
–
0
votes
0
answers
53
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
123
views
admin
asked
Oct 17, 2019
Theory of Computation
michael-sipser
theory-of-computation
decidability
proof
+
–
0
votes
0
answers
54
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
175
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
55
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
257
views
admin
asked
Oct 17, 2019
Theory of Computation
michael-sipser
theory-of-computation
finite-automata
decidability
proof
+
–
0
votes
0
answers
56
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
190
views
admin
asked
Oct 17, 2019
Theory of Computation
michael-sipser
theory-of-computation
decidability
proof
+
–
0
votes
0
answers
57
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
564
views
admin
asked
Oct 17, 2019
Theory of Computation
michael-sipser
theory-of-computation
context-free-grammar
decidability
proof
+
–
0
votes
0
answers
58
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
192
views
admin
asked
Oct 17, 2019
Theory of Computation
michael-sipser
theory-of-computation
context-free-grammar
decidability
proof
+
–
0
votes
0
answers
59
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
125
views
admin
asked
Oct 17, 2019
Theory of Computation
michael-sipser
theory-of-computation
decidability
proof
+
–
0
votes
0
answers
60
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
151
views
admin
asked
Oct 17, 2019
Theory of Computation
michael-sipser
theory-of-computation
decidability
proof
+
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