Process mining is a technique that can discover and enhance business processes by extracting knowledge from the event logs generated in information systems. A mixed multiple-concurrency short-loop structure is a frequently appearing structure in business processes. It cannot be mined accurately (or well) from incomplete logs by existing methods to the best of our knowledge. In this paper, an AlphaMining algorithm is proposed to discover mixed multiple-concurrency short-loop structures via Petri nets. First, the activities are matched with a triangular two-degree loop or a quadrilateral two-degree loop. Then, two kinds of short-loop structures are identified from the incomplete logs. Algorithms are proposed to correctly construct models with multiple-concurrency short-loop structures. Finally, the proposed method is integrated as a plug-in into an open-source process mining tool named ProM. The correctness and effectiveness of the proposed method are verified by experiments.