# Re: [tlaplus] "As long as, eventually"

Right. A ~> B == [](A => <>B).

What you're looking for is, I believe, []([]A => <>B) (where A == "a" \in Objects B == "b" \in Objects)

This would be true, as that it pretty much follows from the definition of weak fairness,

WF_v(C) == []([]ENABLED <<C>>_v => <><<C>>_v)

and A => ENABLED Bar_1 and <<Bar_1>>_Objects => B'

On Wednesday, January 16, 2019 at 2:50:06 AM UTC, domin...@xxxxxxxxx wrote:

Thanks for your response. I'm truly not an expert in temporal formulae, but I believe this not to be correct:

This statements specifies A will lead to B eventually, so [](A => <> B). But a correct behavior of the specification could be

{ } - Foo_1 -> {"a"} - Foo_2->{ } Stuttering

So the presence of "a" does not necessarily lead to the presence of "b"

Best, Dominik

On Tuesday, January 15, 2019 at 12:11:12 PM UTC-8, Karolis Petrauskas wrote:
> I think it should be:
>
>
>     ("a" \in Objects) ~> ("b" \in Objects).
>
>
> Karolis
>
>
>
> On Tue, Jan 15, 2019 at 8:46 PM <domin...@xxxxxxxxx> wrote:
>
>
> Hello Stephan,
>
>
>
> Thanks for your prompt and helpful response. Also, you are correct, Bar_2 was meant to remove "b" if "a" is no longer present, sorry for the copy paste error.
>
>
>
> Bar_2 ==
>
>   /\ ~ "a" \in Objects
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>   /\ Objects' = Objects \ {"b"}
>
>
>
> In your opinion is the Invariant helpful in describing the system? I want to communicate the informal property or intuition that "if there is an "a" long enough there will eventually be a "b"" but I'm not certain that formula captures that property. Is there an alternative that I am not thinking of?
>
>
>
> Thanks again, Dominik
>
>
>
> On Tuesday, January 15, 2019 at 9:39:14 AM UTC-8, Stephan Merz wrote:
>
> > Hello Dominik,
>
> >
>
> > your temporal formula corresponds to your informal explanation (modulo the fact that the fairness part of the explanation is part of the specification, not the property). It is quite easy to see that the property is true of your specification:
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> >
>
> > 1. You assume "a" \in Objects to be persistently true in an execution.
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> > 2. This condition enables action <<Bar_1>>_Objects as long as "b" is not in Objects (this is actually a somewhat subtle point: Bar_1 is a stuttering action if both "a" and "b" are contained in Objects).
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> > 3. Using weak fairness for that action, we know that it must eventually execute and therefore establish "b" \in Objects.
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> > 4. No action ever removes "b" from Objects, and therefore it remains there persistently.
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> >
>
> > However, I suspect that item (4) actually does not correspond to your intuition and that you meant Bar_2 to remove "b" from set Objects rather than being a copy of Bar_1. Your property will not hold of the modified specification.
>
> >
>
> > Regards,
>
> > Stephan
>
> >
>
> >
>
> > > On 15 Jan 2019, at 18:17, domin...@xxxxxxxxx wrote:
>
> > >
>
> > >
>
> > > Hello,
>
> > >
>
> > > I have a question regarding a temporal formula. Let's assume I have the following Spec:
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> > >
>
> > > VARIABLE Objects
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> > >
>
> > > Foo_1 ==
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> > >  Objects' = Objects \cup {"a"}
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> > >
>
> > > Foo_2 ==
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> > >  Objects' = Objects \ {"a"}
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> > >
>
> > > Bar_1 ==
>
> > >  /\ "a" \in Objects
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> > >  /\ Objects' = Objects \cup {"b"}
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> > >
>
> > > Bar_2 ==
>
> > >  /\ "a" \in Objects
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> > >  /\ Objects' = Objects \cup {"b"}
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> > >
>
> > > Init ==
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> > >  Objects = {}
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> > >
>
> > > Next ==
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> > >  \/ Foo_1
>
> > >  \/ Foo_2
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> > >  \/ Bar_1
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> > >  \/ Bar_2
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> > >  \/ UNCHANGED Objects
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> > >
>
> > > Spec == /\ Init /\ [][Next]_<< Objects >> /\ WF_<< Objects >>(Bar_1)
>
> > >
>
> > > - Foo_1 adds "a" to Objects
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> > > - Foo_2 removes "a" from Objects
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> > > - Bar_1 adds "b" to Objects if "a" is in Objects
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> > > - Bar_2 removes "b" from Objects if "a" is not in Objects
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> > >
>
> > > For each behavior Objects could be {}, {"a"}, {"a", "b"}, {"b"}
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> > >
>
> > > I would like to use a temporal formula to document the behavior:
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> > >
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> > > <>[]("a" \in Objects) => <>[]("b" \in Objects)
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> > >
>
> > > In words:
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> > >
>
> > > "Given that Bar_1 is weakly fair, If eventually "a" is continuously in Objects then eventually "b" is continuously in Objects"
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> > >
>
> > > My Questions:
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> > >
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> > > - Is this temporal formula actually expressing what I put in words
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> > > - Does that make sense at all? Or is it more of a tautology (in logic and rhetoric)?
>
> > >
>
> > > Thanks, Dominik
>
> > >
>
> > > --
>
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