Linear Temporal Logic (LTL) questions - logic

Question

[] = always
O = next
! = negation
<> = eventually
Wondering is it []<> is that equivalent to just []?
Also having a hard time understanding how to distribute temporal logic.
[][] (a OR !b)
!<>(!a AND b)
[]([] a ==> <> b)

Answer 1

I'll use the following notations:
F = eventually
G = always
X = next
U = until
In my model-checking course, we defined LTL the following way:
LTL: p | φ ∩ ψ | ¬φ | Xφ | φ U ψ
With F being a syntactic sugar for :
F (future)
Fφ = True U φ
and G:
G (global)
Gφ = ¬F¬φ
With that, your question is :
Is it true that : Gφ ?= GFφ
GFφ <=> G (True U φ)
Knowing that :
P ⊧ φ U ψ <=> exists i >= 0: P_(>= i) ⊧ ψ AND forall 0 <= j < i : P_(<= j) ⊧ φ
From that, we can clearly see that GFφ indicates that it must always be true that φ will be always be verified after some time i, and before that (j before i) True must be verified (trivial).
But Gφ indicates that φ must always be true, "from now to forever" and not "from i to forever".

Answer 2

G p indicates that at all times p holds. GF p indidcates that at all times, eventually p will hold. So while the infinite trace pppppp... satisfies both of the specifications, an infinite trace of the form p(!p)(!p!)p(!p)p... satisfies only GF p but not G p.
To be clear, both these example traces need to contain infinitely many locations, where p holds. But in the case of GF p, and only in this case, it is acceptable that there be locations in between, where p does not hold.
So the short answer to the above question by counterexample is: no, those two specifications aren't the same.