How Many Queries Are Needed to Learn One Bit of Information?

In this paper we study the question how many queries are needed to halve a given version space. In other words: how many queries are needed to extract from the learning environment the one bit of information that rules out fifty percent of the concepts which are still candidates for the unknown target concept. We relate this problem to the classical exact learning problem. For instance, we show that lower bounds on the number of queries needed to halve a version space also apply to randomized learners (whereas the classical adversary arguments do not readily apply). Furthermore, we introduce two new combinatorial parameters, the halving dimension and the strong halving dimension, which determine the halving complexity (modulo a small constant factor) for two popular models of query learning: learning by a minimum adequate teacher (equivalence queries combined with membership queries) and learning by counterexamples (equivalence queries alone). These parameters are finally used to characterize the additional power provided by membership queries (compared to the power of equivalence queries alone). All investigations are purely information-theoretic and ignore computational issues.