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We consider the task of optimally sensing a two-state Markovian channel with an observation cost and without any prior information regarding the channel's transition probabilities. This task is of interest in the field of cognitive radio as a model for opportunistic access to a communication network by a secondary user. The optimal sensing problem may be cast into the framework of model-based reinforcement learning in a specific class of partially observable Markov decision processes (POMDPs). We propose the Tiling Algorithm, an original method aimed at reaching an optimal tradeoff between the exploration (or estimation) and exploitation requirements. It is shown that this algorithm achieves finite horizon regret bounds that are as good as those recently obtained for multi-armed bandits and finite-state Markov decision processes (MDPs).