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.”
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.
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.
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.
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.”
2021 has broken the record for zero-day hacking attacks
“Part of the reason you’re seeing more now is because we’re finding more,” says Microsoft’s Doerr. “We’re better at shining a spotlight. Now you can learn from what’s happening at all your customers, which helps you get smarter faster. In the bad situation where you see something new, that will impact one customer instead of 10,000.”
The reality is a lot messier than the theory, however. Earlier this year, multiple hacking groups launched offensives against Microsoft Exchange email servers. What started as a critical zero-day attack briefly became even worse in the period after a fix became available but before it was actually applied to users. That gap is a sweet spot hackers love to hit.
As a rule, however, Doerr is spot on.
Exploits are getting harder—and more valuable
Even if zero-days are being seen more than ever, there is one fact that all the experts agree on: they are getting harder and more expensive to pull off.
Better defenses and more complicated systems mean hackers have to do more work to break into a target than they did a decade ago—attacks are costlier and require more resources. The payoff, however, is that with so many companies operating in the cloud, a vulnerability can open millions of customers up to attack.
“Ten years ago, when everything was on premises, a lot of the attacks only one company would see,” says Doerr, “and few companies were equipped to understand what was going on.”
Faced with improving defenses, hackers often must link together multiple exploits instead of using just one. These “exploit chains” require more zero-days. Success at spotting these chains is also part of the reason for the steep rise in numbers.
Today, says Dowd, attackers are “having to invest more and risk more by having these chains to achieve their goals.”
One important signal comes from the rising cost of the most valuable exploits. The limited data available, such as Zerodium’s public zero-day prices, shows as much as a 1,150% rise in the cost of the highest-end hacks over the last three years.
But even if zero-day attacks are harder, the demand has risen, and supply follows. The sky might not be falling—but neither is it a perfectly sunny day.
How these US schools reopened without sparking a covid outbreak
“Cleaning high-touch areas is very important in schools,” Cogan said. But mask-wearing, physical distancing, vaccinations, and other measures are “higher protective factors.”
8. Give agency to parents and teachers in protecting their kids.
Last school year, many districts used temperature checks and symptom screenings as an attempt to catch infected students before they gave the coronavirus to others. But in Austin, Indiana, such formalized screenings proved less useful than teachers’ and parents’ intuition. Instructors could identify when a student wasn’t feeling well and ask them to go see the nurse, even if that student passed a temperature check.
Jetelina said that teachers and parents can both act as a layer of protection, stopping a sick child from entering the classroom. “Parents are pretty good at understanding the symptoms of their kids and the health of their kids,” she said.
In Andrews, Texas, district administrators provided parents with information on covid symptoms and entrusted those parents to determine when a child may need to stay home from school. The Texas district may have “gone way overboard with giving parents agency,” though, Cogan said, in allowing students to opt out of quarantines and mask-wearing—echoing concerns from the Andrews County public health department.
9. We need more granular data to drive school policies.
Throughout the pandemic, I’ve consistently called out a lack of detailed public data on covid-19 cases in schools. The federal government still does not provide such data, and most states offer scattered numbers that don’t provide crucial context for cases (such as in-person enrollment or testing figures). Without these numbers, it is difficult to compare school districts and identify success stories.
My research on school reopenings illuminated another data issue: most states are not providing any covid-19 metrics down to the individual district, making it hard for school leaders to know when they must tighten down on or loosen safety protocols. At the tiny Port Orford–Langlois district in Oregon, for example, administrators had to rely on covid-19 numbers for their overall county. Even though the district had zero cases in fall 2020, it wasn’t able to bring older students back in person until the spring because outbreaks in another part of the county drove up case numbers. Cogan has observed similar issues in New Jersey.
At a local level, school districts may work with their local public health departments to get the data they need for more informed decision-making, Jetelina said. But at a larger, systemic level, getting granular covid-19 data is more difficult—a job for the federal government.
10. Invest in school staff and invite their contributions to safety strategies.
School staff described working long hours, familiarizing themselves with the science of covid-19, and exercising immense determination and creativity to provide their students with a decent school experience. Teaching is typically a challenging job, but in the last 18 months, it has become heroic—even though many people outside school environments take this work for granted, Jetelina said.
Districts can thank their staff by giving them a say in school safety decisions, Cogan recommended. “Educators—they’ve had a God-awful time and had a lot more put on them,” she said. But “every single person that works in a school has as well.” That includes custodians, cafeteria workers, and—crucially—school nurses, who Cogan calls the “chief wellness officers” of the school.
11. Allow students and staff the space to process pandemic hardship.
About 117,000 children in the US have lost one or both parents during the pandemic, according to research from Imperial College London. Thousands more have lost other relatives, mentors, and friends—while millions of children have faced job loss in their families, food and housing insecurity, and other hardships. Even if a school district has all the right safety logistics, school staff cannot truly support students unless they allow time and space to process the trauma that they’ve faced.
P.S. 705 in Brooklyn may serve as a model for this practice. School staff preemptively reached out to families when a student missed class, offering support: “705 is just the kind of place where it is a ‘wrap your arms around the whole family’ kind of a school,” one parent said.
On the first day of school in September 2021—when many students returned in person for the first time since spring 2020—the school held a moment of silence for loved ones that the school community has lost.
New challenges ahead
These lessons are drawn from school communities that were successful in the 2020-2021 school year, before the delta variant hit the US. This highly transmissible strain of the virus poses new challenges for the fall 2021 semester. The data analysis underlying this project led me to profile primarily rural communities, which may have gotten lucky with low covid-19 case numbers in previous phases of the pandemic—but are now unable to escape delta. For example, the Oregon county including Port Orford–Langlois saw its highest case rates yet in August 2021.
The delta challenge is multiplied by increasing polarization over masks, vaccines, and other safety measures. Still, Jetelina pointed out that there are also “a ton of champions out there,” referring to parents, teachers, public health experts, and others who continue to learn from past school reopening experiences—and advocate for their communities to do a better job.
The Solutions Journalism Network supported this project with a reporting grant, as well as trainings and other guidance. Learn more about the five school communities I profiled in this project for the COVID-19 Data Dispatch.
This story is part of the Pandemic Technology Project, supported by The Rockefeller Foundation.
US unfairly targeting Chinese over industrial spying, says report
For years, civil rights groups have accused the US Department of Justice of racial profiling against scientists of Chinese descent. Today, a new report provides data that may quantify some of their claims.
The study, published by the Committee of 100, an association of prominent Chinese-American civic leaders, found that individuals of Chinese heritage were more likely than others to be charged under the Economic Espionage Act—and significantly less likely to be convicted.
“The basic question that this study tries to answer is whether Asian-Americans are treated differently with respect to suspicions of espionage,” said the report’s author, Andrew C. Kim, a lawyer and visiting scholar at the South Texas College of Law Houston. “The answer to that question is yes. “
The study, which looked at data from economic espionage cases brought by the US from 1996 to 2020, found that just under half of all defendants were accused of stealing secrets that would benefit China. This is far lower than the figures laid out by US officials to justify the Department of Justice’s flagship China Initiative.
According to the report, 46% of defendants charged under the Economic Espionage Act were accused of activity that would benefit Chinese people or entities, while 42% of defendants were accused of stealing secrets that would benefit American businesses.
The numbers directly contradict much of the Justice Department’s messaging around the China Initiative, which was launched in 2018 to combat economic espionage. The department has stated publicly—for example, in the first line of its home page for the China Initiative—that 80% of its prosecutions would benefit the Chinese state, reflecting “theft on a scale so massive that it represents one of the largest transfers of wealth in human history,” as FBI director Christopher Wray described it in 2020.
Since 2019, the program has largely targeted academic researchers.
“Strong evidence of charges with less evidence”
The report was based on an analysis of public court filings, as well as Department of Justice press releases, for all Economic Espionage Act prosecutions between 1996 and 2020. It’s an update of an earlier analysis, published in the Cardozo Law Review, which covered the period up to 2016.
Charges for “theft of trade secrets” and “economic espionage” were both included, with the “economic espionage” charge requiring proof of a “nexus to foreign entity” and accompanied by higher penalties. (These two categories make up only a portion of the charges under the China Initiative; Kim briefly mentions “false statements and process crimes,” and people have also been charged with grant fraud and lying on visa applications, among other crimes.)
Because demographic information and citizenship data is not included in court filings, Kim used names as proxies for race, and he used Google searches when names, like Lee and Park, were ethnically ambiguous. For citizenship, Kim noted that press releases often make prominent mention if a defendant is a “foreign national,” so he assumed that defendants were all citizens unless otherwise indicated.