Marijn Heule, an Amazon-Damn and Professor of Computer Science at Carnegie Mellon University, along with his colleague Manfred Scheucher of Technische University Berlin, have solved a geometry problem that was posed almost 100 years ago by the Hungarian Australian mathematical esters.
Paul Erdős, the legendary Hungarian mathematician who gave his name to the ERDőS number, called it the “happy-ending problem” because it led to it led to Esther, f.
The problem asks the minimum number of points in a plane, of which no three is tray required to guarantee it n of the points that make up a convex polygon that contains no one of the other points. (“Convex” means that a line segment that connects two points within Polygon itself is enthusiastically within the polygon.)
Esther Szekeres sent the case of n = 4 in the 1930s. It was almost 50 years before Heiko Harth decides that 10 points are needed to guarantee an empty pentagon. Around the same time, Joseph Horton showed that the problem is insoluble for polygons with seven or more pages: No number of points will guarantee that a convex 7-gon can be found that contains no other points in the collection.
But the remaining case – the empty hexagon – was still unique. That’s the problem that Heule and Scheucher solved. They showed that 30 points are sufficient to guarantee a convex hexagon that contains none of the other points.
To prove this result, Heule and Scheucher used one Set SolverAn automated-reasonable tool that determines how long chains of logical limitations may be Sat.Isfied. SAT -SOLVER generates one proof The special task of values for variable is prohibited by the limitations. Confirmation of correctness correct Proof checker.
Evidence, Evever, can be dogeds of terabytes in size, and just controlling input-output (I/O) and data collection during the proof control process can be extremely time-consuming. “The cost of control can be, for example, 100% to 200% of the original solution time,” says Heule.
Helu, a member of the Amazon Web Services’ (AWS’s) automated reasoning group, worked with its AWS colleagues to develop the infrastructure of a new streaming approach to awareness, where a dedicated server core controls the evidence when generated. This reduces proof control over the head from 100% to 200% to a place about 10%.
This innovation, in turn, will be of benefit to the automated reasoning group in its future work on, says, software security, provublic correct software and hardwar validation. Of race, these applications still require developers to create strict formal models of the systems they validate. But in the proof control phase, “If we can do things to say 10% overhead instaad of 150%, it’s a clear win,” says Heule.
Geometric restrictions
SAT PROBLEMS is NP-COMPLETE, which means that the SAT problems can be reassured, which would be insoluble by all computers in the world in the life of the universe.
But that does not mean that all the SAT problem, or even SAT-problem with a large number of variables, is insoluble, and part of the automated researchers’ art is the formulation of problems in such a way that a SAT solver can solve them.
“Marijn is best in the world when mapping complex problems with solver,” says Robert Jones, a senior main application science in AWS Automed Reasoning Group.
The setup of the happy ending problem can be described using binary (Boolian) variables, each describing the orientations of three points. The variables all have the same general form: given three points in general position (ie not hill), Hair,,,,,,,, Band C,,,,,,,, C is over the line through Hair and B. (If the variable is false, C are requirements below the line.) Watch enough of these together and you can specify the 30 points in the 6-gon case (or 29 points or any other number).
Within this framework, the difficulty is to describe the condition that there is at least one hexagon without meaning inside it. Scheucher’s group had struck that problem around for years without arriving at a formulation that a Sat solver could handle. This is where Heule came in.
People who map problems for SAT expressions often focus on conclusion, heule explains; The more briefly the expression they justify, the fewer options the solver must consider. It may be true in general, says Heule, but in his experience long chains of simple limitations are often easier to resonate, albeit short chains with more complex limitations.
Simplifies the problem
The natural way of approaching the empty-hexagon problem is to break hexagons into triangles and reason about where each triangle has a point in its inner. Previous attempts to map this problem to a SAT expression had taken a general approach, specifically a set of logical limitations that could be used for any triangle in the collection and all hexagons that include this triangle. The resulting expression, says Heule, was easy to formulate, but difficult to resonate.
Heule suggests that he and Scheucher take the opposite tack, explicitly marking of any possible configuration of each hexagon that specifies the individual triangles using these labels, and controls each of the named triangles pointing into its interior.
“In this case, you really need to blow it up to get much less later,” explains Heule. “I did 10 times greater and then realized that the new expression could be compressed significantly. This compression step is also possible with existing automated rattle tools.
One of the ways that sat Solvers reduces the complexity of the problems they tackle is by looking for logical redundancies and removing them. In his initial special of the Tom-Hexagon problem, Heule shared each hexagon at the point set in an oven triangle and controlled each triangle for a point in its interior.
However, he noted that SAT solver reduced this step to control only one triangle per day. Hexagon. After thinking it through, Heule and Scheucher realized that in each hexagon there was a single triangle – calling it internal Triangle – that shared all sides with Hexagon’s other three triangles – calling them bite triangles. If the inner triangle was empty, it was possible to derive the existence of an empty hexagon from the points in the point set.
Suppose one of the whole triangles contains pointing. Then it is possible to draw a new triangle that contains this point and shares a side with the inner triangle. Repeating this process as needed is GuaranRald to y vestx hexagon without points in its interior.
Heule and Scheucher extracted this reasoning from the SAT solver itself. “I have often seen that Solver delivers useful feedback, even if it is feedback to an expert,” says Heule. “I think it’s really important that this feedback becomes available to non -experts. For example, you implement something and the solver says, ‘Okay, you are trying to do this, but the expression is not necessary.’ This feedback can be used to reformulate the expression in such a way that it is much easier to solve.
When Heule and Scheucher understood what the Solver told them, they were able to motivate a more practical special of the SAT problem. The Solver was able to resonate through all the possibility of a 30-point point set and pro, which within this set must exist at least one hexagon whose inner triangle did not continue other points.
It was still an extremely long proof, but a Heule and his AWS colleagues’ new proof control mechanism were able to confirm its validity relatively quickly.
“One of the problems here is that many uses of these tools not now how to make the most of them,” says Heule. “And it’s not just for this specific problem for many other issues as well. Within Amazon, there are a lot of applications where SAT -SOLDER could verify developers’ work or find better solutions. I can help by writing an effective coding, but ideally everything would be done automatically. I would love to see myylf be taken out of the equation.