On the Translation of Automata to Linear Temporal Logic

While the complexity of translating future linear temporal logic (LTL) into automata on infinite words is well-understood, size increase involved in turning back to LTL not. In particular, there no known elementary bound deterministic $\omega$-regular LTL. Our first contribution consists tight bounds for over a unary alphabet: alternating, nondeterministic and can be exactly exponentially, quadratically linearly more succinct, respectively, than any equivalent formula. main translation general counter-free formulas double exponential temporal-nesting depth triple length, using an intermediate Krohn-Rhodes cascade decomposition automaton. To our knowledge, this translation. Furthermore, preserves acceptance condition automaton sense that it turns looping, weak, B\"uchi, coB\"uchi or Muller formula belongs matching class syntactic hierarchy. used translate recognising safety language belonging fragment (over both finite words).