Computability (Solvability)

1
Computability (Solvability)

• Can all functions be computed?

2
Clock example

• Your Job
• Let me know as soon as the clock stops.
• Let me know as soon as the clock never stops.
• Which one of these is not possible?

3
Does this loop end?

• Repeat as long as X is not equal to 0
• Add one to X
• ?  If the initial value of X is 0. Since X is
  already 0, the whole loop is skipped (no
  statements are repeated).
• ?  If the initial value of X is a negative
  integer. The loop will end when X becomes 0.
• ?  If the initial value of X is greater than 0
  the loop will never end since there are an
  infinite number of integers greater than 0.

4
Halting problem

• Can we construct an algorithm A that would take
  any program P (and P's input) as input. A should
  determine if P halts
• If P stops print P halts.
• If P does not stop (infinite loop)
• print P does not halt.

5
How about these?

• If P stops print hello
• If P does not stop print see you later
• If P does not stop do nothing (undefined)
• If P stops print see you later
• If P stops do nothing
• If P does not stop print see you later

6
continued

• The non computable example above is known as the
  Halting Problem and it cannot be solved.
• The Church-Turing thesis stated conditions which
  a function must satisfy in order to be
  computable. It is named after Alonzo Church and
  Alan Turing.
• This question concerned mathematicians even
  before digital computers were in developed. In
  particular, during the 1930's much work was
  devoted to this.

7
Another Unsolvable Problem

• Problem Construct an algorithm that would
  figure out the steps needed to perform any task
  asked of it.
• This is also unsolvable. It is not possible to
  construct an algorithm that would, on its own,
  construct an algorithm to solve a random problem.