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The Nonlinear Library allows you to easily listen to top EA and rationalist content on your podcast player. We use text-to-speech software to create an automatically updating repository of audio content from the EA Forum, Alignment Forum, LessWrong, and other EA blogs. To find out more, please visit us at nonlinear.org

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18 hours ago

LW - You Can’t Predict a Game of Pinball by AI Impacts

Link to original article Welcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is: You Can’t Predict a Game of Pinball, published by AI Impacts on March 30, 2023 on LessWrong. Jeffrey Heninger, 29 March 2023 Introduction When thinking about a new idea, it helps to have a particular example to use to gain intuition and to clarify your thoughts. Games are particularly helpful for this, because they have well defined rules and goals. Many of the most impressive abilities of current AI systems can be found in games.1 To demonstrate how chaos theory imposes some limits on the skill of an arbitrary intelligence, I will also look at a game: pinball. In this page, I will show that the uncertainty in the location of the pinball grows by a factor of about 5 every time the ball collides with one of the disks. After 12 bounces, an initial uncertainty in position the size of an atom grows to be as large as the disks themselves. Since you cannot launch a pinball with more than atom-scale precision, or even measure its position that precisely, you cannot make the ball bounce between the disks for more than 12 bounces. The challenge is not that we have not figured out the rules that determine the ball’s motion. The rules are simple; the ball’s trajectory is determined by simple geometry. The challenge is that the chaotic motion of the ball amplifies microscopic uncertainties. This is not a problem that is solvable by applying more cognitive effort. The Game Let’s consider a physicist’s game of pinball. Forget about most of the board and focus on the three disks at the top. Each disk is a perfect circle of radius R. The disks are arranged in an equilateral triangle. The minimum distance between any two disks is L. See Figure 1 for a picture of this setup. Figure 1: An idealization of the three disks near the top of a pinball table. Drawn by Jeffrey Heninger. The board is frictionless and flat, not sloped like in a real pinball machine. Collisions between the pinball and the disks are perfectly elastic, with no pop bumpers that come out of the disk and hit the ball. The pinball moves at a constant speed all of the time and only changes direction when it collides with a disk. The goal of the game is to get the pinball to bounce between the disks for as long as possible. As long as it is between the disks, it will not be able to get past your flippers and leave the board. A real game of pinball is more complicated than this – and correspondingly, harder to predict. If we can establish that the physicist’s game of pinball is impossible to predict, then a real game of pinball will be impossible to predict too. Collisions When the ball approaches a disk, it will not typically be aimed directly at the center of the disk. How far off center it is can be described by the impact parameter, b, which is the distance between the trajectory of the ball and a parallel line which passes through the center of the disk. Figure 2 shows the trajectory of the ball as it collides with a disk. The surface of the disk is at an angle relative to the ball’s trajectory. Call this angle θ. This is also the angle of the position of the collision on the disk relative to the line through the center parallel to the ball’s initial trajectory. This can be seen in Figure 2 because they are corresponding angles on a transversal. Figure 2: A single bounce of the pinball off of one of the disks. Drawn by Jeffrey Heninger. At the collision, the angle of incidence equals the angle of reflection. The total change in the direction of the ball’s motion is 2θ. We cannot aim the ball with perfect precision. We can calculate the effect of this imperfect precision by following two slightly different trajectories through the collision instead of one. The two trajectories have slightly different initial locations. The second trajectory has impact parameter b + db, with db ≪ b. Call db the uncertainty in the impac...