Credit: Paderborn University, Besim Mazhiqi
The previous number in the Dedekind sequence, the 8th Dedekind number, was discovered in 1991 using a Cray 2, the most effective supercomputer at the time. “It, for that reason, appeared possible to us that it should be possible by now to determine the 9th number on a big supercomputer,” says Van Hirtum, describing the inspiration for the enthusiastic task, which he at first carried out jointly with the managers of his masters thesis at KU Leuven.
Grains of sand, chess and supercomputers
The main topic of Dedekind numbers are so-called monotone Boolean functions. Van Hirtum discusses, “Basically, you can consider a monotone Boolean function in two, three, and infinite measurements as a video game with an n-dimensional cube. You balance the cube on one corner and then color each of the remaining corners either white or red. There is just one rule: you should never ever place a white corner above a red one. This creates a sort of vertical red-white crossway. The item of the game is to count how numerous different cuts there are. Their number is what is defined as the Dedekind number. Even if it does not appear like it, the numbers quickly become massive while doing so: the 8th Dedekind number currently has 23 digits.”
he figure shows all possible cuts for measurements 0, 1, 2, and 3. The variety of these colored 2D, 3D,– N-dimensional cuts that can be formed is what is specified as the Dedekind number. Credit: Paderborn University
Comparably big– however incomparably much easier to determine– numbers are understood from a legend concerning the innovation of the game of chess. “According to this legend, the innovator of the chess video game asked the king for only a few grains of rice on each square of the chess board as a reward: one grain on the very first square, two grains on the 2nd, four on the third, and twice as numerous on each of the following squares. The king rapidly recognized that this demand was impossible to meet, due to the fact that so much rice does not exist in the entire world. The number of grains of rice on the complete board would have 20 digits– an unimaginable amount, but still less than D( 8 ). When you recognize these orders of magnitude, it is obvious that both an effective computational approach and a very fast computer would be needed to find D( 9 ),” Van Hirtum said.
Milestone: Years end up being months
To calculate D( 9 ), the scientists used a technique established by masters thesis consultant Patrick De Causmaecker referred to as the P-coefficient formula. It supplies a way to compute Dedekind numbers not by counting, but by a large sum. This allows D( 8) to be translated in just 8 minutes on a typical laptop. “What takes 8 minutes for D( 8) ends up being hundreds of thousands of years for D( 9 ). Even if you utilized a large supercomputer solely for this job, it would still take many years to complete the estimation,” Van Hirtum mentions. The primary issue is that the number of terms in this formula grows exceptionally fast. “In our case, by making use of proportions in the formula, we had the ability to lower the number of terms to just 5.5 * 10 ^ 18– a huge amount. By contrast, the number of grains of sand in the world is about 7.5 * 10 ^ 18, which is absolutely nothing to sneeze at, but for a modern-day supercomputer, 5.5 * 10 ^ 18 operations are quite manageable,” the computer scientist stated. The issue: The estimation of these terms on normal processors is slow and also an usage of GPUs as presently the fastest hardware accelerator innovation for numerous AI applications is not effective for this algorithm.
The service: application-specific hardware utilizing extremely specialized and parallel arithmetic systems– so-called FPGAs (field programmable gate ranges). Van Hirtum established an initial model for the hardware accelerator and began looking for a supercomputer that had the essential FPGA cards. In the process, he became aware of the Noctua 2 computer system at the “Paderborn Center for Parallel Computing (PC2)” at the University of Paderborn, which has one of the worlds most powerful FPGA systems.
Prof. Dr. Christian Plessl, head of PC2, explains: “When Lennart Van Hirtum and Patrick De Causmaeker called us, it was instantly clear to us that we desired to support this moonshot job. Fixing tough combinatorial issues with FPGAs is an appealing field of application and Noctua 2 is among the couple of supercomputers worldwide with which the experiment is feasible at all. The extreme reliability and stability requirements also posture a challenge and test for our infrastructure. The FPGA professional seeking advice from team worked carefully with Lennart to optimize the application and adapt for our environment.”
After numerous years of development, the program worked on the supercomputer for about five months. And then the time had come: on March 8, the scientists found the 9th Dedekind number: 286386577668298411128469151667598498812366.
Today, three years after the start of the Dedekind project, Van Hirtum is working as a fellow of the NHR Graduate School at the Paderborn Center for Parallel Computing to develop the next generation of hardware tools in his PhD. The NHR (National High Performance Computing) Graduate School is the joint graduate school of the NHR. He will report on his remarkable success together with Patrick De Causmaecker on June 27 at 2 p.m. in Lecture Hall O2 of the University of Paderborn. The interested public is cordially invited.
Making history with 42 digits: Scientists at Paderborn University and KU Leuven have actually unlocked a decades-old mystery of mathematics with the so-called ninth Dedekind number. The researchers join a renowned group with their work: Earlier numbers in the series were discovered by mathematician Richard Dedekind himself when he defined the issue in 1897, and later by greats of early computer science such as Randolph Church and Morgan Ward. Their number is what is specified as the Dedekind number. Even if it doesnt seem like it, the numbers quickly end up being massive in the procedure: the 8th Dedekind number currently has 23 digits.”
The number of these colored 2D, 3D,– N-dimensional cuts that can be formed is what is specified as the Dedekind number.
Mathematicians at Paderborn University and KU Leuven, utilizing the Noctua supercomputer and specialized hardware accelerators, have actually solved a decades-old issue by calculating the ninth Dedekind number, a mathematical sequence of massive complexity. The exact number, formerly thought uncomputable due to its size, is 286386577668298411128469151667598498812366.
Scientists from the Universities of Paderborn and Leuven fix long-known issue in mathematics.
Making history with 42 digits: Scientists at Paderborn University and KU Leuven have opened a decades-old secret of mathematics with the so-called ninth Dedekind number. The Paderborn researchers arrived at the precise sequence of numbers with the assistance of the Noctua supercomputer located there.
What started as a masters thesis job by Lennart Van Hirtum, then a computer technology student at KU Leuven and now a research partner at the University of Paderborn, has actually ended up being a huge success. The scientists sign up with a remarkable group with their work: Earlier numbers in the series were discovered by mathematician Richard Dedekind himself when he specified the problem in 1897, and later by greats of early computer science such as Randolph Church and Morgan Ward. “For 32 years, the estimation of D( 9) was an open difficulty, and it was doubtful whether it would ever be possible to determine this number at all,” Van Hirtum says.