Lie Detectors and the Story of Halting Turing Machines

 

Is it possible to design a machine that detects lies? Sure, people have already built devices called polygraphs that monitor blood pressure, respiration, pulse etc. to determine if a person is giving false information. However, that is not same as “detecting lies” because polygraphs merely measure physical effects of telling lies. On the top of that, they are hella inaccurate. I am talking about REAL lie detectors, ie. machines that can instantly validate the statements themselves. Can we actually make such machines? (*insert new startup idea*)

Well, turns out if we ever could make such machines, they would become incredibly handy tools for scientists and mathematicians. That’s because they could babble all sorts of scientific and mathematical conjectures to such a machine, if it beeped, they would know those conjectures were false. Their conversation might go like this:

Crazy Physicist: Higgs Boson exists.
Machine: *Says Nothing*
Crazy Physicist: Ureka! I was right all along.

Computer Scientist: P=NP #changemymind
Machine: *beep*
Computer Scientist: OMG. P=/=NP. I just became a millionaire.

Clearly, you can see how these lie detectors would trivialize almost every known problem of science and mathematics. Then, a very legit question would be why nobody even tried to build such a machine. Indeed, humankind has spared far more effort after making junks like philosopher stones and elixir of life.

Turns out there is something fundamentally wrong about the design of the lie detectors. Before I get to the details, let me tell you a cool life hack. If you ever meet people that claim to be oracles, do this to them:

You: Will I offer you $10?

Whatever the oracle says, do the opposite.

This clearly shows the oracles cannot predict the future. The same thing is true for our lie detectors. Here is why:

Suppose you brought your friend Joe to show him the lie detector you bought. Now do the following:

You: I’m gonna hand Joe $10.

Whatever the lie detector says, do the opposite.

Therefore, just as above, our precious lie detectors cannot exist. However, they did have a purpose. They helped me to show you a proof technique called Cantor’s Diagonal Argument. It is a really nifty tool to prove many advanced statements in computability theory. Here is the general idea:

  • First you assume a machine with utterly unbelievable functions exists.
  • Then you come up for a statement or input for this machine using its own functions.
  • Finally, show that existence of such an input is contradictory to the existence of the machine itself.

As I’m running out of time, I’ll give you just one example of this technique. First proven by Alan Turing, it’s still one of the most famous proofs in computability theory.

Halting Turing Machine (HTM) is a special type of Turing machine that can instantly determine whether another Turing Machine halts on a specific input. In English, this means HTM is a cool app that can tell you whether one of the apps in your smartphone is gonna become unresponsive forever for a specific environment condition inside your smartphone. HTM would be a handy app, right? If HTM tells you an app is gonna be unresponsive beforehand, you’ll save time by not installing that crappy app. However, like all good things in computability theory, turns out HTM cannot exist either. This is because what if an evil developer created an app like this?

  • Give my app’s code and current environment condition of the smartphone to the HTM
  • Whatever the HTM says about my app, do the opposite inside the app.

Hence HTM is unable to predict what this evil dev’s app will do, and that’s why HTM cannot exist.

That’s it for today. I hope you enjoyed this blogpost. If you want to learn Computability Theory for real, I highly recommend Michael Sipser’s book.

 

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