Homework 8
Last updated: Mon, 29 Mar 2021 13:45:06 -0400
Out: Mon April 5, 00:00 EST Due: Sun April 11, 23:59 EST
This homework covers material from Chapters 4 and 5 of the textbook.
Homework Problems
Uncountability and Diagonalization (6 points)
DFAs That Accept Everything (6 points)
Turing Machines That Accept Everything (6 points)
EQCFG is not Decidable (6 points)
EQCFG is not Recognizable (6 points)
README (2 points)
Total: 32 points
Submitting
Submit this assignment at Gradescope hw8.
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and each file must be assigned to the correct problem in Gradescope.
1 Uncountability and Diagonalization
Prove that the set of all infinite strings made up of characters from the alphabet \Sigma = \{0,1\} is uncountable. Use diagonalization.
2 DFAs That Accept Everything
Show that the following language is decidable:
\textrm{ACCEPTEVERYTHING}_{DFA} = \{\left\langle D\right\rangle\mid D \textrm{ is a DFA that accepts any string in } \Sigma^*\}
You may use the theorem proved in the The Complement Operation for Regular Languages (Bonus) (even if you didn’t do it), and of course any other theorems we’ve covered in this course so far.
3 Turing Machines That Accept Everything
Show that the following language is undecidable:
\textrm{ACCEPTEVERYTHING}_{TM} = \{\left\langle M\right\rangle\mid M \textrm{ is a TM that accepts any string in } \Sigma^*\}
Prove it via contradiction by constructing a decider for A_{TM}.
4 EQCFG is not Decidable
We’ve been speculating that:
EQ_{CFG} = \{\left\langle G,H\right\rangle\mid G \textrm{ and }H\textrm{ are CFGs and } L(G) = L(H)\}
is an undecidable language (page 200 of Sipser).
Now that we’ve learned about reducibility, we can finally prove it.
Prove that EQ_{CFG} is undecidable.
You may find it helpful to use the Theorem 5.13 from the book, that \{\left\langle G\right\rangle\mid G\textrm{ is a CFG and } L(G) = \Sigma^*\} is an undecidable language.
5 EQCFG is not Recognizable
Show that EQ_{CFG} is not recognizable either.
You may assume that the theorem from EQCFG is not Decidable has already been proved.
Hint: Show that EQ_{CFG} is co-Turing-recognizable first.