Rabin vs. Streett Automata

The Rabin and Streett acceptance conditions are dual. Accordingly, deterministic Rabin and Streett automata are dual. Yet, when adding nondeterminsim, the picture changes dramatically. In fact, the state blowup involved in translations between Rabin and Streett automata is a longstanding open problem, having an exponential gap between the known lower and upper bounds. We resolve the problem, showing that the translation of Streett to Rabin automata involves a state blowup in $Theta(n^2)$, whereas in the other direction, the translations of both deterministic and nondeterministic Rabin automata to nondeterministic Streett automata involve a state blowup in $2^{Theta(n)}$. Analyzing this substantial difference between the two directions, we get to the conclusion that when studying translations between automata, one should not only consider the state blowup, but also the emph{size} blowup, where the latter takes into account all of the automaton elements. More precisely, the size of an automaton is defined to be the maximum of the alphabet length, the number of states, the number of transitions, and the acceptance condition length (index). Indeed, size-wise, the results are opposite. That is, the translation of Rabin to Streett involves a size blowup in $Theta(n^2)$ and of Streett to Rabin in $2^{Theta(n)}$. The core difference between state blowup and size blowup stems from the tradeoff between the index and the number of states. (Recall that the index of Rabin and Streett automata might be exponential in the number of states.) We continue with resolving the open problem of translating deterministic Rabin and Streett automata to the weaker types of deterministic co-B"uchi and B"uchi automata, respectively. We show that the state blowup involved in these translations, when possible, is in $2^{Theta(n)}$, whereas the size blowup is in $Theta(n^2)$

On the (In)Succinctness of Muller Automata

There are several types of finite automata on infinite words, differing in their acceptance conditions. As each type has its own advantages, there is an extensive research on the size blowup involved in translating one automaton type to another. Of special interest is the Muller type, providing the most detailed acceptance condition. It turns out that there is inconsistency and incompleteness in the literature results regarding the translations to and from Muller automata. Considering the automaton size, some results take into account, in addition to the number of states, the alphabet length and the number of transitions while ignoring the length of the acceptance condition, whereas other results consider the length of the acceptance condition while ignoring the two other parameters. We establish a full picture of the translations to and from Muller automata, enhancing known results and adding new ones. Overall, Muller automata can be considered less succinct than parity, Rabin, and Streett automata: translating nondeterministic Muller automata to the other nondeterministic types involves a polynomial size blowup, while the other way round is exponential; translating between the deterministic versions is exponential in both directions; and translating nondeterministic automata of all types to deterministic Muller automata is doubly exponential, as opposed to a single exponent in the translations to the other deterministic types

Translating to Co-Büchi made tight, unified, and useful

We solve the longstanding open problems of the blow-up involved in the translations, when possible, of a nondeterministic Büchi word automaton (NBW) to a nondeterministic co-Büchi word automaton (NCW) and to a deterministic co-Büchi word automaton (DCW). For the NBW to NCW translation, the currently known upper bound is 2o(nlog n) and the lower bound is 1.5n. We improve the upper bound to n2n and describe a matching lower bound of 2ω(n). For the NBW to DCW translation, the currently known upper bound is 2o(nlog n). We improve it to 2 o(n), which is asymptotically tight. Both of our upper-bound constructions are based on a simple subset construction, do not involve intermediate automata with richer acceptance conditions, and can be implemented symbolically. We continue and solve the open problems of translating nondeterministic Streett, Rabin, Muller, and parity word automata to NCW and to DCW. Going via an intermediate NBW is not optimal and we describe direct, simple, and asymptotically tight constructions, involving a 2o(n) blow-up. The constructions are variants of the subset construction, providing a unified approach for translating all common classes of automata to NCW and DCW. Beyond the theoretical importance of the results, we point to numerous applications of the new constructions. In particular, they imply a simple subset-construction based translation, when possible, of LTL to deterministic Büchi word automata