Peter, the ‘GPWS’ situation is more complex if you consider EGPWS which uses a database and navigation position in its computation. You may have meant this but your response related to GPWS; as you realise, it is important to distinguish between the two. This is particularly so for the crew as the later system (EGPWS/TAWS) is more capable and reliable than previous systems, but then opens greater opportunity for error due to the range of alerting and warning modes and the terrain display, e.g. during an approach a pilot may pull up in response to a warning and incorrectly level at MDA, but due to a (gross) error in range, MDA is not safe.
Also note that some versions of EGPWS use GPS altitude as a gross error check of altitude, mainly incorrect pressure setting. GPS (geometric) altitude does not replace the rad alt (there are exceptions) as its accuracy is less than that of normal altimetry, and therefore at this time it would not be a candidate for ACAS altitude as suggested earlier in the thread.
As for the solution to the three aircraft ACAS problem, this is done reasonably successfully by combat pilots in 1 vs 2 situations. My experiences and existence today is relevant, but based on the same experiences I would be less confident for situations involving 4 vs 4, due to the human limitations in continuously computing a 4D (space – time) solution, that’s assuming that you can see everyone!
Your ACAS problem relates to a similar 4D situation and thus might be modelled in the form the 4th power.
Considering three aircraft avoidance, x, y and z, then a solution could be in the form of x^4 = y^4 + z^4.
This form of equation for powers greater than 2 has been proven to have no solution (Fermat’s conjecture). With appropriate deductive logic, ‘no solution’ relates to a collision condition, thus I claim my PhD !