Homework 9
Last updated: Tue, 12 Apr 2022 18:24:52 -0400
Out: Wed April 06, 00:00 EST Due: Sun Apr 17 Tue April 19, 23:59 EST
This assignment explores undecidability.
Homework Problems
An Undecidable Algorithm About CS 420 (10 points)
An Undecidable Algorithm About CFGs (10 points)
The Non-Halting Problem (10 points)
E_TM Is Unrecognizable (9 points)
README (1 point)
Total: 40 points
Submitting
Submit your solution to this assignment in Gradescope hw9. Please assign each page to the correct problem and make sure your solutions are legible.
A submission must also include a README containing the required information.
1 An Undecidable Algorithm About CS 420
Prove that the following language (from Homework 8)is undecidable:
\mathrm{\textit{NOCS}420}_{\textsf{TM}} = \left\{\left\langle M\right\rangle\mid M\textrm{ is a \textsf{TM} where }\texttt{CS420}\notin L(M)\right\}
This time, you must use mapping reducibility.
Remember that mapping reducibility has two parts: a computable function, and an if-and-only-if statement (which itself has two parts).
Your answer must be a full proof of undecidability, i.e., the proof should conclude with something like "therefore \mathrm{\textit{NOCS}420}_{\textsf{TM}} is undecidable". In particular, mapping reducibility by itself is insufficient for proving undecidability
2 An Undecidable Algorithm About CFGs
Prove that the following language is undecidable:
\mathrm{\textit{SUB}}_{\textsf{CFG}} = \left\{\left\langle G_1, G_2\right\rangle\mid G_1 \textrm{ and } G_2\textrm{ are CFGs where } L(G_1) \subseteq L(G_2)\right\}
You must use mapping reducibility.
3 The Non-Halting Problem
Prove that the complement to the halting problem not a Turing-recognizable language.
In other words, prove that the following language is unrecognizable:
\mathrm{\textit{NOHALT}}_{\textsf{TM}} = \left\{\left\langle M,w\right\rangle\mid M \textrm{ is a } \textsf{TM}\textrm{ and loops when run with }w\right\}
4 E_TM Is Unrecognizable
Prove that E_\textsf{TM} is not Turing-recognizable.
You must use mapping reducibility.
Make sure to include all the needed parts of the proof.