Overview

Last time:

• Code as Data
• Universal TMs
• General Purpose Computing

Today:

• Uncomputability
• Halting Problem
• Reductions

Code as Data

Can take any TM and make it into some binary string.

Can TMs compute every function?

Is there a TM to compute any $f$?

Turing 1936: There are functions that are not computable.

Theorem:

Input: some number of equations in variables.

Output: $1$ if has an integer valued solution, $0$ else.

Somehow, uncomputable.

$\mathrm{TODD}: \{0,1\}^*\to \{0,1\}$

$\mathrm{TODD}(<M>)$ is $1$ if $M(<M>) = 0$, $0$ otherwise.