Wolfram Computation Meets Knowledge

Wolfram Summer School


Ed Hopkins

Summer School

Class of 2004


After doing his undergraduate work at Dartmouth College in 1992, Ed Hopkins spent 12 years in software development. Recently, the state of the economy forced his return to school. In 2004, he finished his master’s at Thayer School of Engineering, Dartmouth College (B.E., M.S., 2004), where he designed an inertial measurement unit for his B.E. project and his thesis was titled “Nonlinear System Identification for High Dimensional Systems.”

Currently, he is working for SignalQuest of Lebanon, NH. In the fall, he will be starting his Ph.D. program at Thayer in lasers and fiber optics.

Project: System Identification for Continuous Cellular Automata

The fundamental problem in system identification is to model reality. Traditionally, analytical equations are used to represent various properties and characteristics in systems. For example, if a baseball is thrown, dynamics teaches us to use projectile motion to describe the trajectory of the baseball. The initial time, height, velocity, and acceleration are specified, and the accepted value for the earth’s gravity. These values are fed into the projectile motion equation, and the coordinates of the ball at t seconds from time of relase can be computed with reasonable accuracy.

Many problems can be addressed with sets of differential equations, and if these problems are not solvable analytically they are frequently solvable numerically. In numerical solvers, the problem is typically phrased in a continuous context, but then is discretized as part of the numerical solution procedure.

However, many problems in nature cannot be addressed with sets of differential equations. Some “simple” examples include the growth of plants and trees, and shorelines such as the fjords of Norway and Sweden. Fractals have been developed that adequately mimic this behavior but do not provide a mathematical foundation, and they fail to address significantly complex biological modeling problems such as the response of the human immune system to hantaviruses.

With cellular automata, it is not as straightforward to fit a rule to a situation like the baseball’s motion. We’d like to explore methods for using continuous cellular automata (CAs)for modeling. In particular, given an image we’d like to find parameters a, b, and c so that Fractional_Part[a + (b x) + (c x^2)] gives close approximation.

Finally, a useful and practical implementation of this approach would be to scan an image (generating a 2D bitmap) and conduct a modeling fit to a continuous cellular automaton.

Favorite Two-Color, Radius-2 Rule

Rule chosen: 4

My favorite two-color two-step range rule is rule 4 because it translates into the math equation y = x. From a system identification perspective one could potentially back out the system physics from raw data converted into NKS format. Although these sorts of binary discrete automata are generally considered “one dimensional” in NKS terminology, I would point out that truly they are two dimensional as vertically they change over successive iterations (time steps). Therefore one can represent two-dimensional systems with one-dimensional automata, three-dimensional systems with two-dimensional automata, and so on.