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We present a reinforcement learning approach to explore and optimize a safety-constrained Markov Decision Process(MDP). In this setting, the agent must maximize discounted cumulative reward while constraining the probability of entering unsafe states, defined using a safety function being within some tolerance. The safety values of all states are not known a priori, and we probabilistically model them via aGaussian Process (GP) prior. As such, properly behaving in such an environment requires balancing a three-way trade-off of exploring the safety function, exploring the reward function, and exploiting acquired knowledge to maximize reward. We propose a novel approach to balance this trade-off. Specifically, our approach explores unvisited states selectively; that is, it prioritizes the exploration of a state if visiting that state significantly improves the knowledge on the achievable cumulative reward. Our approach relies on a novel information gain criterion based on Gaussian Process representations of the reward and safety functions. We demonstrate the effectiveness of our approach on a range of experiments, including a simulation using the real Martian terrain data.