Undecidability Halting Problem Results

HWs

That’s not fair to you all.
Made this lighter; review.
Other sections had been on an exam review rn.

Termination Results, Problems

I have asserted at you that ACL2 cannot prove termination, and that it’s not just ACL2.

Today, the Proof

Turing Machines

Based upon results, I don’t need to explain to you all what a Turing machine is.

It’s a most-general model of computation (C-T thesis).

If we want to understand ‘algorithm,’ we do it with respect to some, particular model of computation. Analogy: play “poker”. You can play the game under pretty much any “good enough” set of rules. You need to decide on a set of rules. Kind of like that for descriptions of Turing machines, or for models of computation generally.

There are any number of good common choices. Which particular one we choose doesn’t matter.

It’s not even my favorite. Point is that we have one.

Why are they useful models.

Everyone else is doing applied mathematics and working on questions in recursive function theory. DYK: “computer” used to be a job they’d hire you for? Or advertise for in the newspaper? I want to find one, so that I have one, a picture of some newspaper article with an ad for a computer. Looking to hire one.

Turing lies out in a field for a month or so, and thinks about, “What is it that mathematicians /do/?”

Self-evidently the task of computation.

They are programmable. Early computers weren’t utilizing some of the important capacities.

“Universal” Turing machine.

Without the stored program concept, you’d have one wired up to do the work of a toaster, and then re-wire it to do the logic of a washing machine. That kind of stuff. These early computers were missing the concept of “stored program”. Data as code. Running data. Turing had that idea.

It was essential to his work.

Minimalist models of computation. Shown though that they can, and are powerful enough to, /simulate/ other computers.

Computers that can simulate other computers.

This was exciting and novel at the time! You all have it now, on your machines! The VM you’re running in VirtualBox Any OS. And you could install software on it. We installed CodeTogether. So it’s not nearly so foreign as it was. But would you have thought about virtual machines before you ever saw a computer? Before anyone had seen one?

“Codes for”. There is a program on your hard drive.

It’s data. When you run it, it looks and acts just like a computer.

Your VM boots up to an OS. You could imagine it was a dedicated computer, though, that did just one thing.

“Codes for”

Things “code for.” We know all about encoding systems. ASCII, etc. Programs as data are not foreign concepts. But you want to know that we aren’t cheating or getting by with anything with these “string” things. We can encode strings as numbers.

   54
    |\ 
   27 2
    |\ 
    9 3
    |\ 
	3 3
  
   2^1 3^3 5^0 ....

Here is a unique way of encoding and decoding data, sequences of numbers, as numbers themselves. Not the only way. Not especially short. But it works and that’s all we need. Point being, we can discuss sets of strings, and you’ll know I’m not cheating.

Decision Problem. How do you even phrase this as a decision problem?

A set. A set of strings. Which sets of strings? What we want is the set of strings, that /codes/ for,

H = {<M,w> | where M codes for a pair of a TM M and M-input w, and halts on input w.}

We have the ability to put programs in an infinite loop:

```
10: goto 10
```

To start out. Paradox questions.

– Autological – Heterological

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.

(Under what conditions will M accept w?

When the set M computes includes w. Under what conditions will M reject w?
When the set M computes does not include w. 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.

In class exercise. `http://www.lel.ed.ac.uk/~gpullum/loopsnoop.html`

14 groups. Each one have a representative, who’s going to explain their stanza to us.

It gets worse!

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.

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.
- Bombes. Great book on Colossus
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.

HWs