Overview

Last class:

• Uncomputability
• Toddler, Halt
• HALTONZERO
• Reductions

Today:

• Different proof that HALT is uncomputable
• Reduction example
• Rice’s theorem

Proof HALT is uncomputable

Suppose there is a machine $H$ that can compute HALT.

def CANTSOLVEME(<M>):
	if H(<M>, <M>) == 1:
		while True:
			pass
	else:
		return 0

What does CANTSOLVEME do on <CANTSOLVEME>?

If CANTSOLVEME(<CANTSOLVEME>) halts:

→ H(<CANTSOLVEME>, <CANTSOLVEME>) = 1

→ But, we enter an infinite loop meaning it does not halt (a contradiction).

If CANTSOLVEME(<CANTSOLVEME>) does not halt:

→ H(<CANTSOLVEME>, <CANTSOLVEME>) = 0

→ But, we enter do indeed HALT.

Hence, HALT is uncomputable.