Tech
Are computers ready to solve this notoriously unwieldy math problem?
In a sense, the computer and the Collatz conjecture are a perfect match. For one, as Jeremy Avigad, a logician and professor of philosophy at Carnegie Mellon notes, the notion of an iterative algorithm is at the foundation of computer science—and Collatz sequences are an example of an iterative algorithm, proceeding step-by-step according to a deterministic rule. Similarly, showing that a process terminates is a common problem in computer science. “Computer scientists generally want to know that their algorithms terminate, which is to say, that they always return an answer,” Avigad says. Heule and his collaborators are leveraging that technology in tackling the Collatz conjecture, which is really just a termination problem.
“The beauty of this automated method is that you can turn on the computer, and wait.”
Jeffrey Lagarias
Heule’s expertise is with a computational tool called a “SAT solver”—or a “satisfiability” solver, a computer program that determines whether there is a solution for a formula or problem given a set of constraints. Though crucially, in the case of a mathematical challenge, a SAT solver first needs the problem translated, or represented, in terms that the computer understands. And as Yolcu, a PhD student with Heule, puts it: “Representation matters, a lot.”
A longshot, but worth a try
When Heule first mentioned tackling Collatz with a SAT solver, Aaronson thought, “There is no way in hell this is going to work.” But he was easily convinced it was worth a try, since Heule saw subtle ways to transform this old problem that might make it pliable. He’d noticed that a community of computer scientists were using SAT solvers to successfully find termination proofs for an abstract representation of computation called a “rewrite system.” It was a longshot, but he suggested to Aaronson that transforming the Collatz conjecture into a rewrite system might make it possible to get a termination proof for Collatz (Aaronson had previously helped transform the Riemann hypothesis into a computational system, encoding it in a small Turing machine). That evening, Aaronson designed the system. “It was like a homework assignment, a fun exercise,” he says.
“In a very literal sense I was battling a Terminator—at least a termination theorem prover.”
Scott Aaronson
Aaronson’s system captured the Collatz problem with 11 rules. If the researchers could get a termination proof for this analogous system, applying those 11 rules in any order, that would prove the Collatz conjecture true.
Heule tried with state-of-the-art tools for proving the termination of rewrite systems, which didn’t work—it was disappointing if not so surprising. “These tools are optimized for problems that can be solved in a minute, while any approach to solve Collatz likely requires days if not years of computation,” says Heule. This provided motivation to hone their approach and implement their own tools to transform the rewrite problem into a SAT problem.
MARIJN HEULE
Aaronson figured it would be much easier to solve the system minus one of the 11 rules—leaving a “Collatz-like” system, a litmus test for the larger goal. He issued a human-versus-computer challenge: The first to solve all subsystems with 10 rules wins. Aaronson tried by hand. Heule tried by SAT solver: He encoded the system as a satisfiability problem—with yet another clever layer of representation, translating the system into the computer’s lingo of variables that can be either 0s and 1s—and then let his SAT solver run on the cores, searching for evidence of termination.
MARIJN HEULE
They both succeeded in proving that the system terminates with the various sets of 10 rules. Sometimes it was a trivial undertaking, for both the human and the program. Heule’s automated approach took at most 24 hours. Aaronson’s approach required significant intellectual effort, taking a few hours or even a day—one set of 10 rules he never managed to prove, though he firmly believes he could have, with more effort. “In a very literal sense I was battling a Terminator,” Aaronson says—“at least a termination theorem prover.”
Yolcu has since fine-tuned the SAT solver, calibrating the tool to better fit the nature of the Collatz problem. These tricks made all the difference—speeding up the termination proofs for the 10-rule subsystems and reducing runtimes to mere seconds.
“The main question that remains,” says Aaronson, “is, What about the full set of 11? You try running the system on the full set and it just runs forever, which maybe shouldn’t shock us, because that is the Collatz problem.”
As Heule sees it, most research in automated reasoning has a blind eye for problems that require lots of computation. But based on his previous breakthroughs he believes these problems can be solved. Others have transformed Collatz as a rewrite system, but it’s the strategy of wielding a fine-tuned SAT solver at scale with formidable compute power that might gain traction toward a proof.
So far, Heule has run the Collatz investigation using about 5,000 cores (the processing units powering computers; consumer computers have four or eight cores). As an Amazon Scholar, he has an open invitation from Amazon Web Services to access “practically unlimited” resources—as many as one million cores. But he’s reluctant to use significantly more.
“I want some indication that this is a realistic attempt,” he says. Otherwise, Heule feels he’d be wasting resources and trust. “I don’t need 100% confidence, but I really would like to have some evidence that there’s a reasonable chance that it’s going to succeed.”
Supercharging a transformation
“The beauty of this automated method is that you can turn on the computer, and wait,” says the mathematician Jeffrey Lagarias, of the University of Michigan. He’s toyed with Collatz for about fifty years and become keeper of the knowledge, compiling annotated bibliographies and editing a book on the subject, “The Ultimate Challenge.” For Lagarias, the automated approach brought to mind a 2013 paper by the Princeton mathematician John Horton Conway, who mused that the Collatz problem might be among an elusive class of problems that are true and “undecidable”—but at once not provably undecidable. As Conway noted: “… it might even be that the assertion that they are not provable is not itself provable, and so on.”
“If Conway is right,” Lagarias says, “there will be no proof, automated or not, and we will never know the answer.”
My senior spring in high school, I decided to defer my MIT enrollment by a year. I had always planned to take a gap year, but after receiving the silver tube in the mail and seeing all my college-bound friends plan out their classes and dorm decor, I got cold feet. Every time I mentioned my plans, I was met with questions like “But what about school?” and “MIT is cool with this?”
Yeah. MIT totally is. Postponing your MIT start date is as simple as clicking a checkbox.
COURTESY PHOTO
Now, having finished my first year of classes, I’m really grateful that I stuck with my decision to delay MIT, as I realized that having a full year of unstructured time is a gift. I could let my creative juices run. Pick up hobbies for fun. Do cool things like work at an AI startup and teach myself how to create latte art. My favorite part of the year, however, was backpacking across Europe. I traveled through Austria, Slovakia, Russia, Spain, France, the UK, Greece, Italy, Germany, Poland, Romania, and Hungary.
Moreover, despite my fear that I’d be losing a valuable year, traveling turned out to be the most productive thing I could have done with my time. I got to explore different cultures, meet new people from all over the world, and gain unique perspectives that I couldn’t have gotten otherwise. My travels throughout Europe allowed me to leave my comfort zone and expand my understanding of the greater human experience.
“In Iceland there’s less focus on hustle culture, and this relaxed approach to work-life balance ends up fostering creativity. This was a wild revelation to a bunch of MIT students.”
When I became a full-time student last fall, I realized that StartLabs, the premier undergraduate entrepreneurship club on campus, gives MIT undergrads a similar opportunity to expand their horizons and experience new things. I immediately signed up. At StartLabs, we host fireside chats and ideathons throughout the year. But our flagship event is our annual TechTrek over spring break. In previous years, StartLabs has gone on TechTrek trips to Germany, Switzerland, and Israel. On these fully funded trips, StartLabs members have visited and collaborated with industry leaders, incubators, startups, and academic institutions. They take these treks both to connect with the global startup sphere and to build closer relationships within the club itself.
Most important, however, the process of organizing the TechTrek is itself an expedited introduction to entrepreneurship. The trip is entirely planned by StartLabs members; we figure out travel logistics, find sponsors, and then discover ways to optimize our funding.
COURTESY PHOTO
In organizing this year’s trip to Iceland, we had to learn how to delegate roles to all the planners and how to maintain morale when making this trip a reality seemed to be an impossible task. We woke up extra early to take 6 a.m. calls with Icelandic founders and sponsors. We came up with options for different levels of sponsorship, used pattern recognition to deduce the email addresses of hundreds of potential contacts at organizations we wanted to visit, and all got scrappy with utilizing our LinkedIn connections.
And as any good entrepreneur must, we had to learn how to be lean and maximize our resources. To stretch our food budget, we planned all our incubator and company visits around lunchtime in hopes of getting fed, played human Tetris as we fit 16 people into a six-person Airbnb, and emailed grocery stores to get their nearly expired foods for a discount. We even made a deal with the local bus company to give us free tickets in exchange for a story post on our Instagram account.
This is today’s edition of The Download, our weekday newsletter that provides a daily dose of what’s going on in the world of technology.
How ubiquitous keyboard software puts hundreds of millions of Chinese users at risk
For millions of Chinese people, the first software they download onto devices is always the same: a keyboard app. Yet few of them are aware that it may make everything they type vulnerable to spying eyes.
QWERTY keyboards are inefficient as many Chinese characters share the same latinized spelling. As a result, many switch to smart, localized keyboard apps to save time and frustration. Today, over 800 million Chinese people use third-party keyboard apps on their PCs, laptops, and mobile phones.
But a recent report by the Citizen Lab, a University of Toronto–affiliated research group, revealed that Sogou, one of the most popular Chinese keyboard apps, had a massive security loophole. Read the full story.
—Zeyi Yang
Why we should all be rooting for boring AI
Earlier this month, the US Department of Defense announced it is setting up a Generative AI Task Force, aimed at “analyzing and integrating” AI tools such as large language models across the department. It hopes they could improve intelligence and operational planning.
But those might not be the right use cases, writes our senior AI reporter Melissa Heikkila. Generative AI tools, such as language models, are glitchy and unpredictable, and they make things up. They also have massive security vulnerabilities, privacy problems, and deeply ingrained biases.
Applying these technologies in high-stakes settings could lead to deadly accidents where it’s unclear who or what should be held responsible, or even why the problem occurred. The DoD’s best bet is to apply generative AI to more mundane things like Excel, email, or word processing. Read the full story.
This story is from The Algorithm, Melissa’s weekly newsletter giving you the inside track on all things AI. Sign up to receive it in your inbox every Monday.
The ice cores that will let us look 1.5 million years into the past
To better understand the role atmospheric carbon dioxide plays in Earth’s climate cycles, scientists have long turned to ice cores drilled in Antarctica, where snow layers accumulate and compact over hundreds of thousands of years, trapping samples of ancient air in a lattice of bubbles that serve as tiny time capsules.
By analyzing those cores, scientists can connect greenhouse-gas concentrations with temperatures going back 800,000 years. Now, a new European-led initiative hopes to eventually retrieve the oldest core yet, dating back 1.5 million years. But that impressive feat is still only the first step. Once they’ve done that, they’ll have to figure out how they’re going to extract the air from the ice. Read the full story.
—Christian Elliott
This story is from the latest edition of our print magazine, set to go live tomorrow. Subscribe today for as low as $8/month to ensure you receive full access to the new Ethics issue and in-depth stories on experimental drugs, AI assisted warfare, microfinance, and more.
The must-reads
I’ve combed the internet to find you today’s most fun/important/scary/fascinating stories about technology.
1 How AI got dragged into the culture wars
Fears about ‘woke’ AI fundamentally misunderstand how it works. Yet they’re gaining traction. (The Guardian)
+ Why it’s impossible to build an unbiased AI language model. (MIT Technology Review)
2 Researchers are racing to understand a new coronavirus variant
It’s unlikely to be cause for concern, but it shows this virus still has plenty of tricks up its sleeve. (Nature)
+ Covid hasn’t entirely gone away—here’s where we stand. (MIT Technology Review)
+ Why we can’t afford to stop monitoring it. (Ars Technica)
3 How Hilary became such a monster storm
Much of it is down to unusually hot sea surface temperatures. (Wired $)
+ The era of simultaneous climate disasters is here to stay. (Axios)
+ People are donning cooling vests so they can work through the heat. (Wired $)
4 Brain privacy is set to become important
Scientists are getting better at decoding our brain data. It’s surely only a matter of time before others want a peek. (The Atlantic $)
+ How your brain data could be used against you. (MIT Technology Review)
5 How Nvidia built such a big competitive advantage in AI chips
Today it accounts for 70% of all AI chip sales—and an even greater share for training generative models. (NYT $)
+ The chips it’s selling to China are less effective due to US export controls. (Ars Technica)
+ These simple design rules could turn the chip industry on its head. (MIT Technology Review)
6 Inside the complex world of dissociative identity disorder on TikTok
Reducing stigma is great, but doctors fear people are self-diagnosing or even imitating the disorder. (The Verge)
7 What TikTok might have to give up to keep operating in the US
This shows just how hollow the authorities’ purported data-collection concerns really are. (Forbes)
8 Soldiers in Ukraine are playing World of Tanks on their phones
It’s eerily similar to the war they are themselves fighting, but they say it helps them to dissociate from the horror. (NYT $)
9 Conspiracy theorists are sharing mad ideas on what causes wildfires
But it’s all just a convoluted way to try to avoid having to tackle climate change. (Slate $)
10 Christie’s accidentally leaked the location of tons of valuable art 
Seemingly thanks to the metadata that often automatically attaches to smartphone photos. (WP $)
Quote of the day
“Is it going to take people dying for something to move forward?”
—An anonymous air traffic controller warns that staffing shortages in their industry, plus other factors, are starting to threaten passenger safety, the New York Times reports.
The big story
Inside effective altruism, where the far future counts a lot more than the present
October 2022
Since its birth in the late 2000s, effective altruism has aimed to answer the question “How can those with means have the most impact on the world in a quantifiable way?”—and supplied methods for calculating the answer.
It’s no surprise that effective altruisms’ ideas have long faced criticism for reflecting white Western saviorism, alongside an avoidance of structural problems in favor of abstract math. And as believers pour even greater amounts of money into the movement’s increasingly sci-fi ideals, such charges are only intensifying. Read the full story.
—Rebecca Ackermann
We can still have nice things
A place for comfort, fun and distraction in these weird times. (Got any ideas? Drop me a line or tweet ’em at me.)
+ Watch Andrew Scott’s electrifying reading of the 1965 commencement address ‘Choose One of Five’ by Edith Sampson.
+ Here’s how Metallica makes sure its live performances ROCK. ($)
+ Cannot deal with this utterly ludicrous wooden vehicle.
+ Learn about a weird and wonderful new instrument called a harpejji.
