Of Hummingbirds and Helicopters: an Algebraic Framework for Interdisciplinary Studies of Imitation and Its Applications

Imitation is a powerful mechanism for eecient learning of novel behaviors that both supports and takes advantage of sociality. A fundamental problem for imitation is to create an appropriate (partial) mapping between the body of the system being imitated and the imitator. By considering for each of these two systems an associated automata (resp. transformation semigroup) structure, attempts at such mapping can be considered (partial) relational homomorphisms. This chapter shows how mathematical techniques can be applied to characterize how far a behavior is from a successful imitation and to evaluate attempts at imitation arising from a particular correspondence between the imitator's and model's transformation semigroups. For the imitator and the imitated, aaordances in the agent-environment structural coupling are likely to be diierent, all the more so in the case of dissimilar embodiment. We argue that the use of what is aaorded to the imitator to attain corresponding eeects or, as in dance, sequences of eeects, is necessary and suu-cient for successful imitation. Moreover, the judged degree of success or failure of an imitation depends on some externally imposed or | in the case of automonous agents | internally determined criteria on eeects of the attempted imitative behavior (including eeects attained successively as well as nal eeects). We discuss how identiication of states in the system-environment coupling which are equivalent for suchòbserver-dependent' purposes allows one to formally deene successful imitation with respect to such criteria. The resulting measures can be used to compare various candidate mappings (e.g. body plan or perception-action correspondences). Additionally, this may be applied in the automated construction of mappings to be used in imitation for artiicial, hardware and software systems.