The computer and the Collatz conjecture seem to be in perfect harmony. One, Jeremy Avigad is a logician at Carnegie Mellon and a professor of philosophy. He notes that the idea of an iterative algorithms is the core of computer science. The Collatz sequences, which follow a deterministic rule, are one example of such an iterative algorithm. Computer science is also plagued by the problem of proving that a process has ended. Avigad states that computer scientists want to know when their algorithms end. Heule and his colleagues are using that technology to tackle the Collatz conjecture. It is really a termination problem.This automated system is very simple to use. You can simply turn on the computer and wait. Jeffrey LagariasHeules' expertise lies with a computational tool called a SAT Solveror a Satisfiability Solver, a computer program that determines if there is a solution to a problem or formula given a set constraints. However, it is crucial that a SAT solver must first understand the problem in order to solve a mathematical problem. Yolcu, a doctoral student at Heule, says that representation matters a lot.It's a long shot, but it is worth a try.Aaronson was skeptical when Heule mentioned that Collatz could be solved using a SAT solver. However, Heule was able to see subtle ways to make the problem more pliable. Hed had noticed that a group of computer scientists were using SAT solutions to find termination proofs for an abstract representation computation called a rewrite method. Aaronson suggested that transforming Collatz's conjecture into a rewrite program might be possible. (Aronson had helped to transform the Riemann hypothesis into a computing system by coding it in a small Turing Machine). Aaronson created the system that evening. He describes it as a homework assignment. It was also a fun exercise."In a very literal meaning, I was fighting a Terminatorat minimum a termination theorem profferr." Scott AaronsonAaronsons system solved the Collatz problem using 11 rules. The Collatz conjecture would be proved true if the researchers could prove that the analogous system can be terminated by applying the 11 rules in any order.Heule tried state-of the-art tools to prove that rewrite systems were terminated. It didn't work, but it was disappointing. These tools can solve problems in minutes, but Collatz will require a long time of computation. This motivated them to refine their approach and create their own tools to convert the rewrite problem to a SAT question.This is a representation of the Collatz conjecture's 11-rule rewrite. MARIJN HEULEAaronson thought it would be easier to solve the system without one of the 11 rulesleaving an Collatz-like system as a litmus test for the bigger goal. He challenged humans to compete against computers: Whoever solves all subsystems using 10 rules wins. Aaronson tried hand. Heule tried SAT solver. He encoded the system to satisfy the satisfaction problem with yet another layer of representation. He then translated the system into computers lingo, which can be either 0s or 1s. Then, he let his SAT solutionr run the cores, looking for evidence of termination.This system follows the Collatz sequence. The starting value of 2727 is at top left, while 1 is at bottom. The researchers used an equivalent Collatz algorithm, which is 71 steps rather than 111. If the number is even, then divide it by 2, otherwise multiply by 3 and add 1. Divide the result by 2. MARIJN HEULEBoth proved that the system ends with the 10 sets of rules. It was sometimes trivial for both the program and the human. The automated Heules approach took only 24 hours. Aaronsons method required considerable intellectual effort. It took him a few hours to complete 10 rules that he didn't prove. He believes he could have done it with more effort. Aaronson said that he was fighting a Terminator in a literal sense.Yolcu has since refined the SAT solver and calibrated it to better match the nature of Collatz's problem. These tricks were a game changer, speeding up termination proofs for subsystems of the 10 rules and reducing runtimes down to seconds.Aaronson says that the main question is: What about the complete set of 11? The Collatz problem is when the Collatz set runs the system.Heule believes that most automated reasoning research has turned a blinder on problems that require a lot of computation. Based on his past breakthroughs, he believes that these problems can be solved. While Collatz has been transformed into a rewrite tool, it is his strategy of using a highly tuned SAT solver at large scale and with incredible compute power that could help him gain momentum towards proving it.Heule has so far conducted the Collatz investigation with approximately 5,000 cores (the processing unit powering computers; consumer computer have four to eight cores). Heule is an Amazon Scholar and has access to virtually unlimited resources, including as many as one million cores. He isn't willing to use much more.He says he wants to know if this is a real attempt. Heule believes that he would be wasting trust and resources if he didn't. Although I don't require 100% certainty, I would love to see evidence that it is likely to succeed.Transform your business by supercharging itThis automated method has the advantage that you can simply turn on the computer and wait. Jeffrey Lagarias from the University of Michigan says this. After playing with Collatz for over fifty years, he became a keeper of the knowledge and compiles annotated bibliographies. He also edited a book about the subject called The Ultimate Challenge. John Horton Conway, a Princeton mathematician, suggested that Collatz might be one of a few elusive classes of problems that is true but not provably unsolvable. Conway pointed out that even if they were not provable, it could be that their assertion is not necessarily provable.Conway may be right, Lagarias claims, but there won't be any proof, whether automated or not. We will never know the truth.
