Abstract: Specifications play an important role in many software engineering activities. Despite their usefulness, formal specifications are often unavailable in practice. Specification mining techniques try to automatically recover specifications from existing programs. Unfortunately, mined specifications are often overly general, which hampers their applications in the downstream analysis and testing. Nowadays, programmers develop software systems by utilizing existing components that usually have some available specifications. However, benefits of these available specifications are not explored by current specification miners. In this paper, we propose an approach to leverage available specifications of subcomponents to improve the precision of specifications of the composite component mined by state-based mining techniques. We monitor subcomponents against their specifications during the mining process and use states that are reached to construct abstract states of the composite component. Our approach makes subcomponents' states encoded within their specifications visible to their composite component, and improves the precision of mined specifications by effectively increasing the number of their states. The empirical evaluation shows that our approach can significantly improve the precision of mined specifications by removing erroneous behavior without noticeable loss of recall.