Is there anything outside of NP?

• The Halting Problem
• Given a program P and an input I, will P halt with I?
• This turns out to be an undecidable problem.
  • Assume an algorithm A(P,I) exists which will determine if a program P will halt on input I.

  • A(P,I)
    
      flag ← SOME_MAGIC_FUNCTION(P,I)
      if flag then
           return true
      else 
           return false
    
    	 

  • Construct program Q(P,I)

  • Q(P,I) 
    
     flag ← A(P,I)
     if flag then 
         loop forever
     else
         return true
    
        

• Q is somewhat problematic.
  • If an algorithm is predicted to exit, Q will loop.
  • If an algorithm is predicted to loop, Q will exit.
  • Sort of a contrary algorithm.
• Now execute Q(Q,Q)
  • If A predicts Q will not terminate, it will.
  • If A predicts Q will terminate it will not.
  • Therefore A does not correctly predict the behavior of Q
  • Which is a contradiction.
  • Therefore A can not exist.
• But here is a far more elegant argument.
• The halting problem, then is undecidable.
• One impact of this is
  • We can prove a specific algorithm (or program) will halt (By hand)
  • But we can never develop a program that will do this for us automatically.
• I found some discussion if the human brain can solve the halting problem.
• There are other such problems
  • decide whether a given polynomial equation in two or more unknowns with integer coefficients has a solution in integers.