Evan A. Sultanik, Ph.D.

My friend, colleague, and homonymous intellectual doppelgänger Evan Teran always likes to remind me of Betteridge’s law of headlines:

The answer to virtually every clickbait title that asks a yes/no question is: “No.”

Mark Cuban, in an interview on Bloomberg TV, claimed that the next wave of automation will be “the automation of automation.”

By that, Cuban means software will soon begin writing itself, which will ultimately eliminate those lucrative software development jobs. About writing software, Cuban said: “It’s just math, right?” Humans will no longer be needed.

Hold my beer, I’ma explain some math.

Remember The Imitation Game? The Academy Award winning movie from a few years back, starring Buttercup Cummerbund and British Natalie Portman? It was loosely based on Alan Turing’s groundbreaking work cracking the Nazis’ cryptographic codes, helping the Allies win the war. That very same dude is much more famous among mathematicians and computer scientists for his work on the Halting Problem.

The Halting Problem dates back to the seventeenth century, from none other than Gottfried Leibniz—the same dude who independently discovered Calculus, even though Isaac Newton usually gets all the credit. Leibniz dreamt of building a machine that could automatically check whether mathematical formulas were correct or not. If you’ve gathered by this point that mathematical formulas are effectively the same as computer programs, then congratulations! Take a sip of my beer as a reward.

Fast forward to the early 20^th century, and a sort of revolution was going on in the world of mathematics: New systems were being developed to formally encode logic. Think of it like the system of algebra you learned in grade school, except that instead of numbers you’re working with logical statements. David Hilbert was at the forefront of this research, and in 1928 posed the question: Is it possible to devise an algorithm (vi&., a computer program) that can automatically determine whether a given logical statement is universally valid? This became known with the extraordinarily Deutschtastic name „Entscheidungsproblem“. A few years later, Austrian expat Kurt Gödel—who is also famous for discovering a “bug” in the US constitution at an immigration court appearance with Albert Einstein—proved that the answer to the entscheidungsproblem is: No, it is provably impossible. In his seminal paper titled „Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I“, he shattered the entscheidungsproblem both in theory and in Deutschtasticness.

So, back to Turing. A few years later, but before the events that inspired The Imitation Game, Turing created a mathematical definition of a computer which we now call a Turing Machine, and on which practically all modern computers and computer science is based. Turing realized that the logical statements and algorithms Hilbert and Gödel were working with were in fact no different than the programs that could be computed by his Turing Machine. He then restated the entscheidungsproblem in this context: Is it possible to write a program for a Turing Machine that can take another program and that other program’s input and automatically determine whether that other program will terminate? This became known as the Halting Problem and, just like the entscheidungsproblem, Turing was able to prove that the answer is “No”: It is impossible to create a program that can automatically determine whether another program will terminate.

Please indulge me as I briefly delve into metaphysics and, dare I say, even digital physics. Can our universe be described using logic and/or computation? In other words, if there were some infinitely powerful Turing machine, could one write a program to run on it such that it could accurately simulate the entire universe? If so, then we humans are bound by the same theorems that prove mathematical logic is incomplete and that the Halting Problem is undecidable. In other words, it’d be impossible for even humans to prove that a given program is correct.

The Halting Problem is part of what makes programming computers hard: Any program with even a modicum of real-world complexity cannot have provably correct behavior. Those logical flaws and unintended behaviors are precisely what hackers exploit to make programs do things that the programmer never intended. Human programmers are only able to do a passable job of quality control because we have the benefit of heuristics and instincts that are guided by our general intelligence.

Let’s say we develop a computer program that can take a human’s high-level description of a task and automatically generate a computer program that can complete that task. There are two possibilities:

1. the human is specifying all of the complex logic, and the computer is simply translating that specification into a program by rote; or
2. the human is not specifying any complex logic, and the computer needs to determine the logic that needs to be in the program to accomplish the task.

Could I write a program that accepts inputs like, “Create a program that displays pictures of cats,” and then automatically generates a program to do so? Sure. Could I create a program that automatically creates a program that is able to interface with the highly formally specified interface of another automatically generated program? Probably. But the vast majority of software projects are for solving complex human problems, and typically involve integration with numerous legacy programs and software that were written by crazies, idiots, and crazy idiots. Professional programming sucks. Don’t believe me? If you take away nothing else from this essay, please read this article. I’m serious, read it. Any professional-grade program that can automatically generate other programs will have to grok how to interface with the undocumented source code of other humans. That’s impossible unless the computer is intellectually indistinguishable from a human.

In order for a computer to automatically generate a program that solves a human’s problem, there either needs to be a human deciding the logic (i.e., a programmer), or the computer’s intelligence needs to be indistinguishable from that of a human. And if we have computers that are intellectually indistinguishable from humans, we’ll have more issues to deal with than simply losing software jobs.

The absurd realization that it has almost exactly been a decade since I defended my master’s thesis took me by surprise. It seems like yesterday. That work has had some decent exposure over the years, but, like most creative works produced toward the beginning of one’s career, I do not look back on it as a paragon.

Moshe Kam, upon reading a preliminary draft, summarized it very well:

[It] reads like Tristram Shandy, but devoid of humor.

Despite somehow managing to get that chapter of my thesis published in a book, I feel like it always suffered from my cumbersome presentation. Therefore, perhaps inspired by Stetson hats and hip flasks, I have since devised an analogy that I hope at least makes the problem (if not my solution) more accessible. That’s what I am going to describe in the remainder of this post.

Consider, for a moment, that you are piloting a car in a huge bumper car arena.

CC BY 2.0, adapted from here

You have some control over your own movement, but there are so many others driving around in the arena that there are constant, unavoidable collisions that throw you off course. From afar, everyone’s movement seems random.

The challenge is that you need to know the time, but you do not have a watch. In fact, there are very few others that have a watch. How do you find the time?

The naïve solution is to simply yell out, “Who has the time‽” The problems with this solution are:

1. If everyone needs to know the time at once, there will be very many yelling people.
2. What if no one with a watch can hear you? Bumper car arenas tend to be loud.

A slightly more intelligent approach might be to:

1. Take a piece of paper and write a request on it for the current time.
2. Pass the piece of paper to the next person who bumps into you.
3. If one who is watch-less receives such a piece of paper, he or she passes it on to the next person that bumps into him or her.
4. When the note eventually reaches someone with a watch, he or she will write the current time on the piece of paper and send it back to you.

The obvious shortcomings to this approach, however, are that:

1. When the note eventually gets back to you, the time will be incorrect!
2. What if, in fact, no one has a watch? How long do you have to wait without receiving a response before you can be sure that no one has a watch?
3. What if not everyone speaks and/or reads English?

My realization is that: If one roughly knows the topology of the network (i.e., the locations of all of the cars), and if the messages are truly passed randomly through the network, one can use ergodic theory and random graph theory to accurately predict the frequency of message arrivals. Assuming knowledge of the network topology is reasonable, since many ad hoc networking algorithms already provide it. Even if it is not available, an approximation of the topological properties is often sufficient. So, what this provides is a model for predicting how long it will take for one’s message to eventually reach someone with a watch and, subsequently, how long it will take for the response to be returned. What I discovered was that the variance is often small enough such that this estimate can be used to correctly adjust for the delay in returning the time. Furthermore, this model provides a probability distribution for how likely it is that one would have received at least one response as a function of waiting time. Therefore, if n seconds have passed and the model says that with probability 0.99 one should have received a response to the time query, yet no response has been received, one can conclude with 99% certainty that there is no one in the network with a watch!

The last challenge question, “What if not everyone speaks and/or reads English?” will have to wait for another post…