We present a new abstract machine for Abadi and Cardelli's untyped calculus of objects. What is special about this semantic artifact (i.e., man-made construct) is that is mechanically corresponds to both the reduction semantics (i.e., small-step operational semantics) and the natural semantics (i.e., big-step operational semantics) specified in Abadi and Cardelli's monograph. This abstract machine therefore embodies the soundness of Abadi and Cardelli's reduction semantics and natural semantics relative to each other. To move closer to actual implementations, which use environments rather than actual substitutions, we then represent object methods as closures and in the same inter-derivational spirit, we present three new semantic artifacts: a reduction semantics for a version of Abadi and Cardelli's untyped calculus of objects with explicit substitutions, an environment-based abstract machine, and a natural semantics (i.e., an interpreter) with environments. These three new semantic artifacts mechanically correspond to each other, and furthermore, they are coherent with the previous ones since as we show, the two abstract machines are bisimilar. Overall, though, the significance of these artifacts lies in them not having been designed from scratch and then proved correct: instead, they were mechanically inter-derived.
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