On Aggregating Teams of Learning

1 The present paper studies the problem of when a team of learning machines can be aggregated into a single learning machine without any loss in learning power. The main results concern aggregation ratios for vacillatory identi cation of languages from texts. For a positive integer n, a machine is said to TxtFexn-identify a language L just in case the machine converges to up to n grammars for L on any text for L. For such identi cation criteria, the aggregation ratio is derived for the n = 2 case. It is shown that the collection of languages that can be TxtFex2 identi ed by teams with success ratio greater than 5=6 are the same as those collections of languages that can be TxtFex2identi ed by a single machine. It is also established that 5=6 is indeed the cut-o point by showing that there are collections of languages that can be TxtFex2-identi ed by a team employing 6 machines, at least 5 of which are required to be successful, but cannot be TxtFex2-identi ed by any single machine. Additionally, aggregation ratios are also derived for nite identi cation of languages from positive data and for numerous criteria involving language learning from both positive and negative data. 1A preliminary version of this paper was presented at the Fourth International Workshop on Algorithmic Learning Theory , Tokyo, November 1993.