Laws of Timed State Machines

State machines are widely used in industry and academia to capture behavioural models of control. They are included in popular notations, such as UML and its variants, and used (sometimes informally) to describe computational artefacts. In this paper, we present laws for state machines that we prove sound with respect to a process algebraic semantics for refinement, and complete, in that they are sufficient to reduce an arbitrary model to a normal form that isolates basic (action and control) elements. We consider two variants of UML-like state machines, both enriched with facilities to deal with time budgets, timeouts and deadlines over triggers and actions. In the first variant, machines are self-contained components, declaring all the variables, events and operations that they require or define. In contrast, in the second variant, machines are open, like in UML for instance. Laws for open state machines do not depend on a specific context of variables, events and operations, and normalization uses a novel operator for open-machine (de)composition. Our laws can be used in behaviour-preservation transformation techniques. Their applications are automated by a model-transformation engine.