Concepts of Automata Construction from LTL

We present an algorithm for the conversion of very weak alternating Buchi automata into nondeterministic Buchi automata (NBA), and we introduce a local optimization criterion for deleting superfluous transitions in these NBA. We show how to use this algorithm in the translation of LTL formulas into NBA, matching the worst-case upper bounds of other LTL-to-NBA translations. We compare the NBA resulting from our translation to the results of two popular algorithms for the translation of LTL to generalized Buchi automata: the translation of Gerth et al. of 1995 (resulting in the GPVW-automaton), and the translation of Daniele et al. of 1999 (resulting in the DGV-automaton), which improves on the GPVW algorithm. We show that the redundancy check by syntactical implication used in the construction of the DGV-automaton is covered by our local optimization, that is, all transitions removed by the redundancy check will also be removed according to our local optimization criterion. Moreover, for a fixed input formula in next normal form, our locally optimized NBA from LTL and the locally optimized GPVW- and DGV-automaton are all essentially the same. Both these results give a “structural” explanation for the syntactic approaches by Gerth et al. and Daniele et al. We show that a bottom-up variant of our algorithm allows to pass simplifications of NBA for subformulas on to the NBA for the entire LTL formula.

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