Unlike Counterexample-Guided Abstraction Refinement (CEGAR), Three-Valued Abstraction Refinement (TVAR) is able to verify all properties of the \(\upmu \) -calculus. We present a novel algorithmic framework for TVAR that employs a simulator-like approach to build and refine the abstract state space with input-based splitting. This leads to a state space formalism that is much simpler than in previous TVAR frameworks, which use modal transitions. We implemented the framework in our open-source tool machine-check and verified properties of machine-code systems for the AVR architecture, showing the ability to verify systems and \(\upmu \) -calculus properties not verifiable by naïve model checking or CEGAR, respectively. This is the first practical use of TVAR for machine-code verification.