Homework 8

Last updated: Sat, 2 Apr 2022 10:55:16 -0400

Out: Mon Mar 29, 00:00 EST Due: Sun Apr 03 06, 23:59 EST

This assignment explores undecidability.

Homework Problems

1. Sequences of Prime Numbers (7 points)

2. An Algorithm About TMs? (10 points)

3. Number of Strings in a Language (10 points)

4. A Variation of Rice’s Theorem (2 + 8 + 2 = 12 points)

5. README (1 point)

Total: 40 points

Submitting

Submit your solution to this assignment in Gradescope hw8. 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 Sequences of Prime Numbers

Show that the set of (possibly infinite) sequences of prime numbers is uncountable.

Use proof by contradiction and diagonalization.

2 An Algorithm About TMs?

Prove that the following language is undecidable:

\mathrm{\textit{CS}}_{\textsf{TM}} = \left\{\left\langle M\right\rangle\mid M\textrm{ is a \textsf{TM} where }\texttt{CS420}\notin L(M)\right\}

You may not use Rice’s Theorem in the proof.

3 Number of Strings in a Language

Prove that the following language is undecidable:

L_{> 0} = \left\{\left\langle M\right\rangle\mid M\textrm{ is a \textsf{TM} where }|L(M)|> 0\right\}

Use a reduction from E_\textsf{TM} for your proof.

4 A Variation of Rice’s Theorem

1. Prove that the following operation is closed for decidable languages: \mathrm{NOTIN}(A)=\{w\mid w \notin A\}

2. Prove that all languages P that satisfy the following criteria are undecidable:

   • P is not empty

   • P consists of only valid TM descriptions

   • P is not the set of all TM descriptions

   • whether a TM description \left\langle M\right\rangle\in P is determined by computing something about L(M), i.e., P looks like: \{\left\langle M\right\rangle\mid \ldots M \textrm{ is a \textsf{TM} and }\ldots\textrm{ something about }L(M) \ldots\}

   • we know that some \left\langle M\right\rangle\in P where L(M) = \emptyset

3. Prove that the following language is undecidable: \{\left\langle M\right\rangle\mid M \textrm{ is a \textsf{TM} and }w\in L(M)\textrm{ only if }w\textrm{ is a valid JavaScript program}\}