# LTL to GBA versus LTL to BA

Let's assume that I have an LTL formula and I want to convert it to a Buchi automaton. For which fragment of LTL, GBA is more succinct and for which fragment BA has the same size as GBA.

2. It is easy to construct a language for which smaller GBA exist. Example would be (in LTL) $\mathsf{GF}(a \wedge b) \wedge \mathsf{GF}(a \wedge \neg b) \wedge \mathsf{GF}(\neg a \wedge b) \wedge \mathsf{GF}(\neg a \wedge \neg b)$ - a four-state GBA does the trick, but the BA needs more states to encode waiting for the next event (out of $a \wedge b$, $\ldots$, $\neg a \wedge \neg b$) to happen. So whatever fragment of LTL you build, this formula will not be contained therein. But if you now take a disjunction of this property with $\mathsf{GF}(a)$ and then with $\mathsf{GF}(\neg a)$, then the smallest BA and GBA have the same size (one state). So you cannot give a precise syntactic representation of such a fragment (without making a containment check at least PSPACE-hard).