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How easy would it be to write a program that takes a string

Trivial examples

def main(input):
  return true

Not too complicated. Basic static analysis can do it.

def main(input):
  while(true): print(1)
  return false

This one is trivial to prove that it will never halt. Modern code linters could even warn you that this function will never return.

But there’s more complex examples

def main(input):

Pretty much the same example, but with recursion.

Halting problem

the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running, or continue to run forever.


1900: David Hilbert had some fundamental questions about peano arithmatic, upon which most of mathematics depended: Was mathematics complete? (ie. every true statement can be represented within that system) Was mathematics consistent? (ie. there are no contradications) Was mathematics decideable? (ie. theorem are provable in a finite amount of time)

Today we’re dealing with 3. Next episode, we’ll deal with 1 and 2.

Alfonso Church published the first proof of undecideability of first-order logic a year before Alan Turing did, and in a completely different way, but Turing’s proof is more widely known because of the unique way he solved it. It was later determined that both systems express the same “computing power”. Why can’t you just wait until the program halts?

Alan Turing proved in 1936 that a general algorithm to solve the halting problem for all possible program-input pairs cannot exist

To do this proof, Alan Turing created a mathematical definition of a computer and a program. This is now known as the Turing machine, which is the conceptual basis for all modern computers.

Turing’s proof

Alan Turing proof was a proof by contradiction. Let’s say we have a function f(g) that returns if g halts.

def h():
  if f(h):

This is impossible. In the if, if f(h) returns true (so the program will halt), it calls while(true) which will prevent the program from ever halting. And same for the inverse. If f(h) returns false (because the program will never halt), the program will halt.

Conclusion: There is a fundamental limitation to the power and expressibility of computing, and therefore logic.