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A cyber–cyber digital twin is a simulation of a software system. By contrast, a cyber–physical digital twin is a simulation of a non-software (physical) system. Although cyber–physical digital twins have received a lot of recent attention, their cyber–cyber counterparts have been comparatively overlooked. In this paper we show how the unique properties of cyber–cyber digital twins open up exciting opportunities for research and development. Like all digital twins, the cyber–cyber digital twin is both informed by and informs the behaviour of the twin it simulates. It is therefore a software system that simulates another software system, making it conceptually truly a twin, blurring the distinction between the simulated and the simulator. Cyber–cyber digital twins can be twins of other cyber–cyber digital twins, leading to a hierarchy of twins. As we shall see, these apparently philosophical observations have practical …