That classic result was a way to transform any algorithm with a given time budget into a new algorithm with a slightly smaller space budget. Williams saw that a simulation based on squishy pebbles would make the new algorithm’s space usage much smaller—roughly equal to the square root of the original algorithm’s time budget. That new space-efficient algorithm would also be much slower, so the simulation was not likely to have practical applications. But from a theoretical point of view, it was nothing short of revolutionary.
For 50 years, researchers had assumed it was impossible to improve Hopcroft, Paul and Valiant’s universal simulation. Williams’ idea—if it worked—wouldn’t just beat their record—it would demolish it.
“I thought about it, and I was like, ‘Well, that just simply can’t be true,’” Williams said. He set it aside and didn’t come back to it until that fateful day in July, when he tried to find the flaw in the argument and failed. After he realized that there was no flaw, he spent months writing and rewriting the proof to make it as clear as possible.
At the end of February, Williams finally put the finished paper online. Cook and Mertz were as surprised as everyone else. “I had to go take a long walk before doing anything else,” Mertz said.
Valiant got a sneak preview of Williams’ improvement on his decades-old result during his morning commute. For years, he’s taught at Harvard University, just down the road from Williams’ office at MIT. They’d met before, but they didn’t know they lived in the same neighborhood until they bumped into each other on the bus on a snowy February day, a few weeks before the result was public. Williams described his proof to the startled Valiant and promised to send along his paper.
“I was very, very impressed,” Valiant said. “If you get any mathematical result which is the best thing in 50 years, you must be doing something right.”
PSPACE: The Final Frontier
With his new simulation, Williams had proved a positive result about the computational power of space: Algorithms that use relatively little space can solve all problems that require a somewhat larger amount of time. Then, using just a few lines of math, he flipped that around and proved a negative result about the computational power of time: At least a few problems can’t be solved unless you use more time than space. That second, narrower result is in line with what researchers expected. The weird part is how Williams got there, by first proving a result that applies to all algorithms, no matter what problems they solve.
“I still have a hard time believing it,” Williams said. “It just seems too good to be true.”
Williams used Cook and Mertz’s technique to establish a stronger link between space and time—the first progress on that problem in 50 years.Photograph: Katherine Taylor for Quanta Magazine
Phrased in qualitative terms, Williams’ second result may sound like the long-sought solution to the P versus PSPACE problem. The difference is a matter of scale. P and PSPACE are very broad complexity classes, while Williams’ results work at a finer level. He established a quantitative gap between the power of space and the power of time, and to prove that PSPACE is larger than P, researchers will have to make that gap much, much wider.
That’s a daunting challenge, akin to prying apart a sidewalk crack with a crowbar until it’s as wide as the Grand Canyon. But it might be possible to get there by using a modified version of Williams’ simulation procedure that repeats the key step many times, saving a bit of space each time. It’s like a way to repeatedly ratchet up the length of your crowbar—make it big enough, and you can pry open anything. That repeated improvement doesn’t work with the current version of the algorithm, but researchers don’t know whether that’s a fundamental limitation.
“It could be an ultimate bottleneck, or it could be a 50-year bottleneck,” Valiant said. “Or it could be something which maybe someone can solve next week.”
If the problem is solved next week, Williams will be kicking himself. Before he wrote the paper, he spent months trying and failing to extend his result. But even if such an extension is not possible, Williams is confident that more space exploration is bound to lead somewhere interesting—perhaps progress on an entirely different problem.
“I can never prove precisely the things that I want to prove,” he said. “But often, the thing I prove is way better than what I wanted.”
Editor’s note: Scott Aaronson is a member of Quanta Magazine’s advisory board.
Original story reprinted with permission from Quanta Magazine, an editorially independent publication of the Simons Foundation whose mission is to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences.
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![‘Project Hail Mary’ Won’t Be Coming to Streaming Any Time Soon
With all the excitement of movies to come this week thanks to CinemaCon, it was almost easy to forget that MGM provided an interesting update on one of our favorite movies of the year that’s already out: Project Hail Mary will head back to IMAX theaters this weekend for an extended theatrical run. But that extension also means one thing: you’ll have to wait to stream it at home for a good while longer. During its presentation at CinemaCon this week MGM confirmed that Phil Lord and Chris Miller’s sci-fi hit would make its return to IMAX screens for a limited-time, one-week run starting this weekend, a move that will likely inch Project Hail Mary ever closer to crossing the $600 million box office mark. But to put a finer point on the news, Miller took to Twitter yesterday to confirm specifically that the extension means you won’t be able to watch the film at home for the forseeable future.
We announced yesterday that MGM is extending the exclusive theatrical window for PROJECT HAIL MARY so it won’t be on streaming anytime soon. This is a movie that needs to be seen on a big screen – and w a full return to IMAX screens for 1 week only starting this weekend, make… https://t.co/suK8NYpgWM — Christopher Miller (@chrizmillr) April 16, 2026 “It won’t be on streaming any time soon,” Miller’s tweet reads in part. “This is a movie that needs to be seen on a big screen […] Bring friends and loved ones. It’s an experience to share with others.” Project Hail Mary launched on March 20, so it’s not too surprising that it’s not headed home just yet—it’s just shy of a month into its theatrical window, which has now been extended by at least another week with the return to IMAX. But as studios begin to try realigning towards more theatrical releases with longer exclusivity windows again (one of the lingering aftereffects of covid’s impact on movie theaters), we should probably expect some of the biggest films of the year and beyond to try and hold off of hitting streaming for as long as they can.
At least in Project Hail Mary‘s case, you can still go and see it somewhere, even if it’s not at home. Good things come to those who wait, but for now, you can head to a movie theater to get your fix again. Want more io9 news? Check out when to expect the latest Marvel, Star Wars, and Star Trek releases, what’s next for the DC Universe on film and TV, and everything you need to know about the future of Doctor Who. #Project #Hail #Mary #Wont #Coming #Streaming #TimeAmazon MGM,Project Hail Mary,Streaming ‘Project Hail Mary’ Won’t Be Coming to Streaming Any Time Soon
With all the excitement of movies to come this week thanks to CinemaCon, it was almost easy to forget that MGM provided an interesting update on one of our favorite movies of the year that’s already out: Project Hail Mary will head back to IMAX theaters this weekend for an extended theatrical run. But that extension also means one thing: you’ll have to wait to stream it at home for a good while longer. During its presentation at CinemaCon this week MGM confirmed that Phil Lord and Chris Miller’s sci-fi hit would make its return to IMAX screens for a limited-time, one-week run starting this weekend, a move that will likely inch Project Hail Mary ever closer to crossing the $600 million box office mark. But to put a finer point on the news, Miller took to Twitter yesterday to confirm specifically that the extension means you won’t be able to watch the film at home for the forseeable future.
We announced yesterday that MGM is extending the exclusive theatrical window for PROJECT HAIL MARY so it won’t be on streaming anytime soon. This is a movie that needs to be seen on a big screen – and w a full return to IMAX screens for 1 week only starting this weekend, make… https://t.co/suK8NYpgWM — Christopher Miller (@chrizmillr) April 16, 2026 “It won’t be on streaming any time soon,” Miller’s tweet reads in part. “This is a movie that needs to be seen on a big screen […] Bring friends and loved ones. It’s an experience to share with others.” Project Hail Mary launched on March 20, so it’s not too surprising that it’s not headed home just yet—it’s just shy of a month into its theatrical window, which has now been extended by at least another week with the return to IMAX. But as studios begin to try realigning towards more theatrical releases with longer exclusivity windows again (one of the lingering aftereffects of covid’s impact on movie theaters), we should probably expect some of the biggest films of the year and beyond to try and hold off of hitting streaming for as long as they can.
At least in Project Hail Mary‘s case, you can still go and see it somewhere, even if it’s not at home. Good things come to those who wait, but for now, you can head to a movie theater to get your fix again. Want more io9 news? Check out when to expect the latest Marvel, Star Wars, and Star Trek releases, what’s next for the DC Universe on film and TV, and everything you need to know about the future of Doctor Who. #Project #Hail #Mary #Wont #Coming #Streaming #TimeAmazon MGM,Project Hail Mary,Streaming](https://gizmodo.com/app/uploads/2026/04/project-hail-mary-ryan-gosling-1280x853.jpg)
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