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An Overview of Schema Theory
The purpose of this paper is to give an introduction to the field of Schema
Theory written by a mathematician and for mathematicians. In particular, we
endeavor to to highlight areas of the field which might be of interest to a
mathematician, to point out some related open problems, and to suggest some
large-scale projects. Schema theory seeks to give a theoretical justification
for the efficacy of the field of genetic algorithms, so readers who have
studied genetic algorithms stand to gain the most from this paper. However,
nothing beyond basic probability theory is assumed of the reader, and for this
reason we write in a fairly informal style.
Because the mathematics behind the theorems in schema theory is relatively
elementary, we focus more on the motivation and philosophy. Many of these
results have been proven elsewhere, so this paper is designed to serve a
primarily expository role. We attempt to cast known results in a new light,
which makes the suggested future directions natural. This involves devoting a
substantial amount of time to the history of the field.
We hope that this exposition will entice some mathematicians to do research
in this area, that it will serve as a road map for researchers new to the
field, and that it will help explain how schema theory developed. Furthermore,
we hope that the results collected in this document will serve as a useful
reference. Finally, as far as the author knows, the questions raised in the
final section are new.Comment: 27 pages. Originally written in 2009 and hosted on my website, I've
decided to put it on the arXiv as a more permanent home. The paper is
primarily expository, so I don't really know where to submit it, but perhaps
one day I will find an appropriate journa
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