Animal Pattern-Learning Experiments: Some Mathematical Background∗

A number of recent studies on the learning of invented patterns in symbol sequences by animals such as cottontop tamarins (Hauser, Chomsky & Fitch 2002:1578; Fitch & Hauser 2004; Hauser 2005; O’Donnell, Hauser & Fitch 2005), songbirds (Gentner 2005), and undergraduates (Fitch & Hauser 2004; Perruchet & Rey, in press) have shown interest in the contrast between two classes of sets of strings: the finite-state and the context-free. The claim of Fitch & Hauser (2004) is that tamarins can learn characteristic finitestate string patterns but not those that are context-free, while humans can learn both. Perruchet & Rey dispute the latter claim about humans; Gentner suggests that starlings show signs of being able to learn distinctively context-free patterns of recursion (self-embedding); and so on. Showing that a mechanism can learn some stringset in a certain class does not, of course, show that it can learn every stringset in that class (unless that stringset is complete for that class in the technical sense of computational complexity theory). But it is possible to provide evidence that the learnability boundary for some learning mechanism (the line between the stringsets it can and cannot learn) may fall between two hierarchically related stringset classes. One could do this by utilizing a pair of stringsets: one in the stronger (larger) class which cannot be learned (showing that not all stringsets in the stronger class are learnable) and one in the weaker (smaller) class which can be learned (showing that at least one stringset in the weaker class is learnable). But one would want to find a pair of stringsets that were relatively close to each other with respect to the hierarchy. One would not, for instance, choose a finite stringset and a strictly context-sensitive stringset if the aim was to find evidence that some mechanism could learn context-free but not context-sensitive stringsets.