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I was wondering when we construct an automata starting from a language, is there a reason when to use the past operator and when not?  In the solution is mostly used the language with the past operators.

For example 29.08.2017 problem 6 e) is using the third variant of property 1, but would it be also possible to use the second variant for the construction?

Exam 28.07.2015 problem 6 c) is using both variants.

in * TF "Emb. Sys. and Rob." by (150 points)

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You can use either future or past operators. The advantage of using past operators is that a translation to an equivalent ω-automaton will give us directly a deterministic one, and deterministic automata are easier to deal with when it comes to boolean combinations, in particular, negations.
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