Timed Circus: Timed CSP with the Miracle

Timed Circus is a compact extension to Circus, that is, it inherits only the CSP part of Circus while introducing time. Although it looks much like timed CSP from the viewpoint of syntax, its semantics is very different from that of timed CSP because it uses a complete lattice in the implication ordering instead of the complete partial order of the standard failures-divergences model of CSP. The complete lattice gives rise to a number of strange processes which violate some axioms of CSP, especially when the miracle (the top element) and SKIP meet time. In this paper, compared with timed CSP, we will extensively explore such strange processes which turn out to be very useful in specifying a distinct property that "something must occur". Finally, we use a simple example to demonstrate how our model can contribute to modelling temporal behaviours with multiple time scales in complex systems.