Convergence to nearly minimal size grammars by vacillating learning machines

In Gold's influential language learning paradigm a learning machine converges in the limit to one correct grammar. In an attempt to improve Gold's paradigm, Case considered the question whether people might converge to vacillating between up to (some integer) n > 1 distinct, but equivalent, correct grammars. He showed that larger classes of languages can be algorithmically learned (in the limit) by converging to up to n +1 rather than up to n correct grammars. He also argued that, for “small” n > 1, it is plausible that people might sometimes converge to vacillating between up to n grammars. The insistence on small n was motivated by the consideration that, for “large” n , at least one of n grammars would be too large to fit in peoples' heads. This latter assumes, of course, that human brain storage is not magic, admitting of infinite regress, etc. Of course, even for Gold's n = 1 case, the single grammar converged to in the limit may be infeasibly large. An interesting complexity restriction to make, then, on the final grammar(s) converged to in the limit is that they all have small size. In this paper we study some of the tradeoffs in learning power involved in making a well-defined version of this restriction.

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