Automata Containing Evolutionary Algorithms: Behavior and Learning Under Law

Seth J. Chandler* and Christian Jacob**

* Associate Professor of Law
University of Houston Law Center, USA

** Assistant Prof., Dept. of Programming Languages
University of Erlangen, GERMANY

To date, the study of cellular automata and genetic programming have proceeded largely on parallel paths. This article studies the evolution of cellular automata in which the cells each contain data and genetic programs. The immediate application of this convergence is to study how different legal rules affect the evolution of learning and behavior in an economy.  The typical "Coase Theorem" scenario in which neighbors engage in activities that may detrimentally affect each other is used for this study [Chandler, 1997].

Using the Mathematica programming language, the authors create a ring of sites each of which contain genetic programs for determining the behavior and learning strategy of that site. The genetic programs are created using a flexible template mechanism that facilitates specification of a complex grammar for permissible programs (as was already demonstrated for the evolution of Lindenmayer systems and plant growth programs [Jacob, 1995]). The rich set of primitives of these programs include sensing operations such as determination of the behaviors of neighboring sites and mating operations that permit the programs of neighboring sites to be adopted.  These primitives thus permit the sites to learn from each other and, recursively, to learn how the other sites themselves learn. This is a novel approach of combining evolutionary algorithms and, in particular, genetic programming techniques [Jacob, 1997a, 1997b], with interacting agents modeled by cellular automata.

The ring of sites (the economy) evolves through a multi-stage iteration. First, the programs of the sites are evaluated to determine the behaviors at the various sites. The behaviors are in turn evaluated to determine certain global parameters (such as the prices of various commodities within the economy) and the consequences of the local interactions created as a result of the behaviors and the legal rules applicable thereto.  This process permits a computation of the fitness of each of the sites within the economy. The final state of each evolutionary iteration permits the programs at each site to mutate based in part on the fitness of the site.  Evolution is modelled through the repetition of these iterations.

The article then examines how different rules of law regarding the interaction of the sites alters both the evolution of learning and the development of the economy.