If your computer has a pressure sensor and it is with you (you don't have to wear it) it may not know your current tissue saturation level unless it stores the last levels (which most if not all computers do). It also needs to constantly calculate the tissue pressures and store the results at least until saturation. Once the current calculations for all the tissues agree with what has been saved over so many iterations or time periods, the computer "knows" the tissues are in saturation.
Just because your display is turned off that doesn't mean the computer doesn't calculate tissue pressures. The Shearwater Perdix, once on the surface after the dive has ended, calculates every 16 seconds rather than 1 second or (maybe) faster intervals. So, the computer doesn't have to assume anything in terms of what the tissue levels are. Once at saturation the computer may stop calculating tissue pressures and ceilings until a large pressure change is detected, either a reduction in pressure (going up in altitude) or, an increase in pressure (the start of a dive). The Perdix stores the surface pressure level so this is how it detects a change in pressure. Then, when it detects a dive (an increase of pressure) it saves the last good known reading of surface pressure. It then uses that pressure to compare to the pressure when surfacing to end the dive.
