Exam: How’d you all go?

PollE

Termination Results, Problems

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

ACL2 cannot do it for us!

Today, the Proof

  • BTW: If you want to see this recapitulated, Harry Porter @ PSU has a nice video

Poll: Turing Machines

Do you know what a Turing machine is?

Turing Machines—Why do they matter?

A TM is a most-general model of computation (according to the /Church-Turing thesis/).

If we want to understand ‘algorithm,’ we do it with respect to some, particular model of computation. Here’s an analogy: what do you mean when you say “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. It’s kind of like that for (descriptions of) Turing machines. Which particular one we choose matters in the details.

The purpose is kind of like that. It’s /an/ adequate, general model of computation. However you formally describe them (there’s lots of different ways)—or for choosing your models of computation more generally—more important than the particulars of any one model is that we have an adequate model; the details matter less.

By the way, there are any number of good common choices. Turing Machines are not even my favorite. Point is that we have such a model, and a description of it.

Why are they useful models.

Put yourself in this historical time and place.

DYK: “computer” used to be a job they’d hire you for? If you were looking for a computer, you’d put an ad up in the “help wanted” section of a newspaper: “Looking to hire..”

This is coming out of work in applied mathematics and areas of philosophy, really. There was no computer science, there was no computer. Everyone else is doing applied mathematics and working on questions in recursive function theory.

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

– They make marks on paper. If you were going to mechanize a mathematician, what would the least job of it be?

  1. This is self-evidently the task of computation.

  2. Minimalist models of computation. Importantly, they are also /programmable/. Shown though that they are powerful enough to /simulate/ other computers.

“Universal” Turing machine.

/Stored-program/ concept

Early computers weren’t utilizing some of the important capacities. John Von Neumann gets credit for this at the time, IIRC. These early computers were missing the concept of “stored program”. Data as code. Running data. Turing had that idea, with the “Universal” Turing machine.

Code-as-data, data-as-code.

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.

This was essential to Turing’s work.

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. Imagine if you lived in a world where you could run one program at a time, and that program was loaded on boot.

“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. This is, by the way, an inessential detail, but it is also very cool and neat and fun to think about how /mathematicians/ would think of “encoding” data.

   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.

The Decision Problem

How do you even phrase halting on input 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
```

Paradox questions.

– Autological – Heterological

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