Continuation

To start out.

What is it that’s undecidable?

We cannot have a general algorithm that will tell you whether, for any string (data), whether it is in this set, or not in this set.

Steps. Assume (towards contradiction) that the set is decidable.

  • We have shown that TM can run any program, including an encoding of some other TM, on it. We could install VirtualBox on our VirtualBox image, right?

  • Therefore, then (toward contradiction), there must be a TM that computes that function (1,0) that recognizes the set.

  • Call H the algorithm that reads <M,w>, and outputs “Accept” if M accepts w, Right before it would reject, we go into an infinite loop. “Reject” if M rejects w.

Q1. Under what conditions will M accept w?

When the set M computes includes w.

Q2. Under what conditions will M reject w?

When the set M computes does not include w.

Q3. How does that manifest on the machine?

Either the machine would have rejected w or it go into an
 infinite loop. We ensured that it does.)

(Again, it doesn’t exist. Assume it does.)

Let’s construct a machine.

D. (for “diagonalization”, in honest.

Let’s build the machine D. D is a contrarian kind of thing.

Input to D: description of a machine, “M”.

D(“M”) if (M[“M”]) “REJECT” else “ACCEPT”

We could very well do the ones that do accept themselves as input.

That is: D is a collection of machine descriptions that do not accept themselves as input.

Run D on itself.

D(“D”) : the collection of machines that don’t accept themselves as input.

What does it mean to say that D doesn’t accept itself as input?

D is one of the machines that accepts itself as input.

What does it mean to say that D accepts itself as input?

D is one of the machines that doesn’t accept itself as input.

The theorem: that the halting problem is unsolvable.

History

  • In 1900 David Hilbert publishes his 23 problems
    • No. 2 is proving the consistency of the Peano axioms
  • ~1930 Hilbert has rephrased the problem to 3 parts:
    • Is mathematics complete?
    • Is mathematics consistent?
    • Is mathematics decidable?
  • Turing is not the first. Alonzo Church w/λ-calculus proof.
    • Turing submits his manuscript a month later, it’s published a year later.
    • However, before that people are working in recursive functions on number theory
    • Turing introduces the computer inter alia.
      • Introducing a mechanizable computer wasn’t the point of the work! Just incidental!
  • By 1940s, the British govt. is asking him to assist w/war-time decoding efforts, and they are actually using some of these kinds of machines that were before just theoretical inventions.
  • Jailed. Homosexual. His work was secret, the govt. People got credit for his prior work, or for things that would have been consequences, that he could have done, but for government classification.

Alan Turing

  • Bletchley Park is now home of the big computer museum.
  • The highest prize in CS is called the “Turing award” in his honor.

Left off

http://www.lel.ed.ac.uk/~gpullum/loopsnoop.html

A Third View:

Diagonalization POV

So What?

– It gets worse!

What else can we not decide?

Rice’s theorem:

Decision problem for any non-trivial extensional property of Turing machines is undecidable.

It states that for any non-trivial property, there is no general decision procedure that, for all programs, decides whether the partial function implemented by the input program has that property.

  • Non-trivial – not “all of them” or “none of them”
  • Extensional – not about “does it have 3 “0”s in its source code, about how it behaves

You can’t tell much of anything about every program.

Updated: