Exact Exploration

Recent analysis of classical algorithms resulted in their axiomatization as transition systems satisfying simple postulates, and in the Abstract State Machine Theorem, stating that any such transition system can be programmed in a most general model of computation, called “abstract state machines”. In this way, classical abstract state machines answer the question “What is an algorithm?” for the discrete case. The extension of this work to analog and hybrid computing, however, is hard. The very first step is to deal with real-number data, which leads to the issues pursued here. Whereas the original analysis assumes equality is total, and neglects details of intra-step computation, here we must take those details into account, since an algorithm employing reals, for example, may hang on account of an unnecessary test x=0 or division 1/x. We modify the original axiomatization appropriately, and show that there is an abstract state machine that not only has the same state transitions as does a given algorithm but also performs the exact same tests on states when determining how to proceed to the next state.