Determining concepts by group membership

Abstract

Abstract. One contribution of this paper is an efficient algorithm for deciding membership in a subgroup H of an Abelian group G when G and H are in a special form. Our approach is particularly fast because calculations are postponed until needed, and because some decisions can be made based on the existence of certain objects without actually calculating them. This mathematical problem arises naturally in machine learning, and is particularly relevant to concept modeling. We use genetic algorithms as a nontrivial example of how concept formation may correspond to subgroup formation and to illustrate that analysis may reveal abstract concepts determined by group membership which are not initially apparent. This example forms another contribution of this paper; genetic search has not previously been framed within an algebraic context.

Genetic set recombination and its application to neural network topology optimisation

“…The example in Fig. 13, which is due to Vose [36], suffices to show that directed edge formae are also non-separable. It should be noted, however, that this second example relies on the introduction of a cycle, which is only permitted if the cycle forms the entire tour.…”

Structural Search Spaces and Genetic Operators

“…Suppose, however, that you understand enough about the structure of the problem that you can identify certain inherent symmetries, and certain related properties that you consider significant (so that you would wish offspring to inherit them from parents). If you can characterise these symmetries by defining a group of permutations that acts transitively on the search space, and if you can identify your important properties with the subgroup structure of this group (they correspond, if you will, to the natural "folds" in the search space-see (Vose, 1991)), then our theory applies and will tell you something about the kinds of crossover and mutation operators that can be defined on your problem. That is the subject of our previous paper.…”