We introduce the problem for a robot to localize itself, and, simultaneously, actively infer the existence and properties of phenomena present in its surrounding environment: the SLAPI problem. A phenomenon is a representation of an entity "as the robot experiences it" through interaction. The SLAPI problem relates to the SLAM (simultaneous localization and mapping) problem but differs in that it does not aim at constructing a precise map of the environment, and it can apply to robots with coarse sensors. We demonstrate a SLAPI algorithm to control a robot equipped with omni-directional wheels, an echo-localization sensor, photosensitive sensors, and an inertial measurement unit, but no precise sensors like camera, lidar, or odometry. As the robot circles around an object, it constructs the phenomenon corresponding to this object under the form of the set of the spatially-localized control loops of interaction that the object affords to the robot. SLAPI algorithms could help design companion robots that mimic intrinsic motivation such as curiosity and playfulness. Further studies of the SLAPI problem could improve the scientific understanding of how cognitive beings construct knowledge about objects from sensorimotor experience of interaction.