Imitation is a powerful mechanism for efficient 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 automaton (respectively, transformation semigroup) structure, attempts at such mapping can be considered (partial) relational homomorphisms. This article shows how mathematical techniques can be applied to characterize how far a behavior is from a successful imitation and how to evaluate attempts at imitation arising from a particular correspondence between the imitator and model. For the imitator and the imitated, affordances in the agent-environment structural coupling are likely to be different, all the more so in the case of dissimilar embodiment. We argue that the use of what is afforded to the imitator to attain corresponding effects or, as in dance, sequences of effects, is necessary and sufficient for successful imitation. However, the judged degree of success or failure of an attempted behavioral match depends on some externally imposed or in the case ofautonomous agents internally determined criteria on effects of the attempted imitative behavior (including effects attained successively as well as final effects). These criteria correspond to metrics measures of difference which can guide the evaluation of a correspondence, the learning of a correspondence, or learning how to apply one. Metrics on states and sequences of action events in the system-environment coupling allow judgment of similarity for observer-dependent' purposes. This allows one to formally define 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 and learning of mappings to be used in imitation for artificial, hardware, and software systems.