LTL-CTL clarification

Question

For translation logic, i want to know if any one of the below formulae are incorrect:

1) Fa & Fb & Fc = F(a & b & c)

2) F(a & b & c) = Fa & Fb & Fc

3) AFb=Fb

4)Fb=AFb

I think #2 and 4 are wrong, #1 and# 3 are right but I am not sure. Could you please clarify?

Answer 1

They are all wrong, also #1 and #2 are the same just written the other way around, and also #3 and #4 are the same. #3 and #4 compare state and path formulas, which can never be the same.

Comments

If 3 & 4 are wrong, then why it is solved that way in 2021.08.27 paper?(First questions) AG((AFa) & (AFb) & (AFc)) =>AG((Fa) & (Fb) & (Fc))?

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which problem is that in the 2021.08.27 paper?

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do you mean the 2019.08.27 paper?

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It seems the paper is of 2021.02, however the question paper is wrongly dated as 2021.08, I have attached the question screenshot in the edited question, plaese have a look.

Translation Logic - Question 8-a 2021.02

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Okay, I found it now. The point is that we have AGphi = AGAphi, so that we can translate AG(Fa&Fb) to AGA(Fa&Fb) and then to AG((AFa)&(AFb)). Still that does not mean that Fa is equivalent to AFa. Both are just the same under the context of AG, but not in general.

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This is what my second guess when arriving to this solution when you said AFa!=Fa as I was aware about AGphi=AGAphi. It is now clear to me how this is solved.
Thank you very much for clarifying.