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Prove the LTL formula (FFGa) ↔ (FGa) to be valid by applying these steps: Translate the negation of the formula to a symbolic ω-automaton.

In the solution's acceptance condition there are three GF's conjuncted. First one is GF[a -> q0], this one is clearly Ga's omega translation. The next GF[q1 -> q0] and GF[q2 -> q1] are Fa's translation.

My question is why did we only considered the first part FFGa although the problem asks to translate the negation of whole formula? (after negation of <-> it becomes xor)

https://es.cs.uni-kl.de/teaching/vrs/exams/2018.08.21.vrs/2018.08.21.vrs.solutions.pdf

Update:

Professor answered on https://q2a.cs.uni-kl.de/1835/august-2018-problem-7b-prove-validity-of-ffga-fga

When we make the abbreviations, we first get