Lower Bounds on Witnesses for Nonemptiness of Universal Co-Büchi Automata

Abstract

The nonemptiness problem for nondeterministic automata on infinite words can be reduced to a sequence of reachability queries. The length of a shortest witness to the nonemptiness is then polynomial in the automaton. Nonemptiness algorithms for alternating automata translate them to nondeterministic automata. The exponential blow-up that the translation involves is justified by lower bounds for the nonemptiness problem, which is exponentially harder for alternating automata. The translation to nondeterministic automata also entails a blow-up in the length of the shortest witness. A matching lower bound here is known for cases where the translation involves a 2O(n) blow up, as is the case for finite words or Büchi automata.Alternating co-Büchi automata and witnesses to their nonemptiness have applications in model checking (complementing a nondeterministic Büchi word automaton results in a universal co-Büchi automaton) and synthesis (an LTL specification can be translated to a universal co-Büchi tree automaton accepting exactly all the transducers that realize it). Emptiness algorithms for alternating co-Büchi automata proceed by a translation to nondeterministic Büchi automata. The blow up here is 2O(n logn), and it follows from the fact that, on top of the subset construction, the nondeterministic automaton maintains ranks to the states of the alternating automaton. It has been conjectured that this super-exponential blow-up need not apply to the length of the shortest witness. Intuitively, since co-Büchi automata are memoryless, it looks like a shortest witness need not visit a state associated with the same set of states more than once. A similar conjecture has been made for the width of a transducer generating a tree accepted by an alternating co-Büchi tree automaton. We show that, unfortunately, this is not the case, and that the super-exponential lower bound on the witness applies already for universal co-Büchi word and tree automata.