Canonical representation genetic programming

Search spaces sampled by the process of Genetic Programming often consist of programs which can represent a function in many different ways. Thus, when the space is examined it is highly likely that different programs may be tested which represent the same function, which is an undesirable waste of resources. It is argued that, if a search space can be constructed where only unique representations of a function are permitted, then this will be more successful than employing multiple representations. When the search space consists of canonical representations it is called a canonical search space, and when Genetic Programming is applied to this search space, it is called Canonical Representation Genetic Programming. The challenge lies in constructing these search spaces. With some function sets this is a trivial task, and with some function sets this is impossible to achieve. With other function sets it is not clear how the goal can be achieved. In this paper, we specically examine the search space dened by the function set f+;À; _; =g and the terminal set fx; 1g. Drawing inspiration from the fundamental theorem of arithmetic, and results regarding the fundamental theorem of algebra, we construct a representation where each function that can be constructed with this primitive set has a unique representation. Abbreviations: Genetic Programming (GP), Genetic Algorithm (GA) , The No Free Lunch Theorem (NFL) Keywords: canonical representation, standard form, evolutionary computation, genetic programming, no free lunch, bias, symmetric functions, inverse functions, complementary functions, isomorphic representations.