Bounded exhaustive testing is a very effective technique for bug finding, which proposes to test a given program under all valid bounded inputs, for a bound provided by the developer. Existing bounded exhaustive testing techniques require the developer to provide a precise specification of the valid inputs. Such specifications are rarely present as part of the software under test, and writing them can be costly and challenging. To address this situation we propose BEAPI, a tool that given a Java class under test, generates a bounded exhaustive set of objects of the class solely employing the methods of the class, without the need for a specification. BEAPI creates sequences of calls to methods from the class' public API, and executes them to generate inputs. BEAPI implements very effective pruning techniques that allow it to generate inputs efficiently. We experimentally assessed BEAPI in several case studies from the literature, and showed that it performs comparably to the best existing specification-based bounded exhaustive generation tool (Korat), without requiring a specification of the valid inputs.