# Meaning of conjunction or disjunction of fairness

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### Jones Martins

Dec 21, 2021, 6:13:14 PM12/21/21
to tlaplus
Hello, everyone

I was wondering what does each option say about our system (or if it's incorrect).

\A v \in Values:
\/ WF_Vars(AtoB(v))
\/ WF_Vars(BtoC(v))

\E v \in Values:
\/ WF_Vars(AtoB(v))
\/ WF_Vars(BtoC(v))

\A v \in Values:
/\ WF_Vars(AtoB(v))
/\ WF_Vars(BtoC(v))

\E v \in Values:
\/ WF_Vars(AtoB(v))
\/ WF_Vars(BtoC(v))

Regards,

Jones

### Stephan Merz

Dec 22, 2021, 4:28:41 AM12/22/21
Hello,

universal quantification is stronger than existential quantification (assuming Values is non-empty) and conjunction is stronger than disjunction. For example,

\A v \in Values:
/\ WF_Vars(AtoB(v))
/\ WF_Vars(BtoC(v))

says that both transitions AtoB and BtoC must eventually occur when they remain enabled, for every possible value. This is typically what you want : any value that is at "A" will eventually move to "B" and then to "C", assuming that there are no conflicting transitions.

In contrast,

\E v \in Values:
\/ WF_Vars(AtoB(v))
\/ WF_Vars(BtoC(v))

is a very weak condition. Assuming that at least one value v is initially at "A", a behavior that stutters forever satisfies this fairness condition: v is never at "B", thus the transition BtoC is always disabled for value v, and therefore the fairness condition WF_Vars(BtoC(v)) holds, hence the entire formula.

Figuring out the meaning of the other conditions is a nice exercise.

Regards,

Stephan

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### Jones Martins

Dec 22, 2021, 10:33:03 AM12/22/21
Thank you, Stephan

That's all I needed! I'm worried about writing specifications with realistic fairness. For example, I can't simply apply Strong Fairness to all actions.

Jones

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