| [](ENABLED <<A>>_v => <><<A>>_v) is weaker than []<><<A>>_v. In particular, the former is true of a behavior in which <<A>>_v is never enabled. Neither of the two notions is "feasible" [1] (or machine closed) in the sense that any finite computation can be extended to a computation satisfying the fairness condition. Clearly, []<><<A>>_v cannot be made true if <<A>>_v is never enabled, and the same is true for
XF_v(A) == [](ENABLED <<A>>_v => <><<A>>_v)
Suppose that you have two actions A and B that can be simultaneously enabled but where each one disables the other, then it may be impossible to ensure XF_v(A) /\ XF_v(B) for an execution.
In contrast, countable conjunctions of weak and strong fairness conditions are machine closed.
Stephan
On 13 Nov 2025, at 17:59, Andrew Helwer <andrew.helwer@xxxxxxxxx> wrote:
Thanks, Stephan! Another interesting question that occurs to me is whether the first formula is equivalent to unconditional fairness, since presumably a step must be enabled before it can be taken. Thus the ENABLED A precondition seems redundant.
Andrew As Andrew suspects, the second formula is equivalent to weak fairness:
THEOREM ASSUME NEW ACTION A, NEW STATE v PROVE WF_v(A) <=> []( <>[]ENABLED <<A>>_v => <><<A>>_v) BY PTL
Stephan [](ENABLED <<A>>_v => <><<A>>_v) is equivalent to []((ENABLED A) ~> <><<A>>_v) which means if an A step is enabled (like at all), eventually it will be taken; moreover, it will also be taken after every time an A step is enabled in the future. This is like an even stronger fairness assumption, but looking through the texts I have it does not seem to be a standard one. The usual next strongest fairness assumption in temporal logic is unconditional fairness, which is just []<><<A>>_v. There are a lot of fairness definitions out there though so I'm sure someone has covered & named it.
For the second one, since P => <>P, it seems to be implied by weak fairness. So it is at most as strong as weak fairness. Is it any weaker than weak fairness? I can't think of how it could be, but then we should be able to rewrite it into weak fairness. And there I am somewhat at a loss.
Andrew
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