Lower Bounds for Complementation of omega-Automata Via the Full Automata Technique

In this paper, we first introduce a lower bound technique for the state complexity of transformations of automata. Namely we suggest first considering the class of full automata in lower bound analysis, and later reducing the size of the large alphabet via alphabet substitutions. Then we apply such technique to the complementation of nondeterministic \omega-automata, and obtain several lower bound results. Particularly, we prove an \omega((0.76n)^n) lower bound for B\"uchi complementation, which also holds for almost every complementation or determinization transformation of nondeterministic omega-automata, and prove an optimal (\omega(nk))^n lower bound for the complementation of generalized B\"uchi automata, which holds for Streett automata as well

State of B\"uchi Complementation

Complementation of B\"uchi automata has been studied for over five decades since the formalism was introduced in 1960. Known complementation constructions can be classified into Ramsey-based, determinization-based, rank-based, and slice-based approaches. Regarding the performance of these approaches, there have been several complexity analyses but very few experimental results. What especially lacks is a comparative experiment on all of the four approaches to see how they perform in practice. In this paper, we review the four approaches, propose several optimization heuristics, and perform comparative experimentation on four representative constructions that are considered the most efficient in each approach. The experimental results show that (1) the determinization-based Safra-Piterman construction outperforms the other three in producing smaller complements and finishing more tasks in the allocated time and (2) the proposed heuristics substantially improve the Safra-Piterman and the slice-based constructions.Comment: 28 pages, 4 figures, a preliminary version of this paper appeared in the Proceedings of the 15th International Conference on Implementation and Application of Automata (CIAA

Determinising Parity Automata

Parity word automata and their determinisation play an important role in automata and game theory. We discuss a determinisation procedure for nondeterministic parity automata through deterministic Rabin to deterministic parity automata. We prove that the intermediate determinisation to Rabin automata is optimal. We show that the resulting determinisation to parity automata is optimal up to a small constant. Moreover, the lower bound refers to the more liberal Streett acceptance. We thus show that determinisation to Streett would not lead to better bounds than determinisation to parity. As a side-result, this optimality extends to the determinisation of B\"uchi automata

Lower Bounds for Complementation of omega-Automata Via the Full Automata Technique

In this paper, we first introduce a lower bound technique for the state complexity of transformations of automata. Namely we suggest first considering the class of full automata in lower bound analysis, and later reducing the size of the large alphabet via alphabet substitutions. Then we apply such technique to the complementation of nondeterministic \omega-automata, and obtain several lower bound results. Particularly, we prove an \omega((0.76n)^n) lower bound for B\"uchi complementation, which also holds for almost every complementation or determinization transformation of nondeterministic omega-automata, and prove an optimal (\omega(nk))^n lower bound for the complementation of generalized B\"uchi automata, which holds for Streett automata as well

Parity and generalised Büchi automata - determinisation and complementation

In this thesis, we study the problems of determinisation and complementation of finite automata on infinite words. We focus on two classes of automata that occur naturally: generalised Büchi automata and nondeterministic parity automata. Generalised Büchi and parity automata occur naturally in model-checking, realisability checking and synthesis procedures. We first review a tight determinisation procedure for Büchi automata, which uses a simplification of Safra trees called history trees. As Büchi automata are special types of both generalised Büchi and parity automata, we adjust the data structure to arrive at suitably tight determinisation constructions for both generalised Büchi and parity automata. As the parity condition describes combinations of Büchi and CoBüchi conditions, instead of immediately modifying the data structure to handle parity automata, we arrive at a suitable data structure by first looking at a special case, Rabin automata with one accepting pair. One pair Rabin automata correspond to parity automata with three priorities and serve as a starting point to modify the structures that result from Büchi determinisation: we then nest these structures to reflect the standard parity condition and describe a direct determinisation construction. The generalised Büchi condition is characterised by an accepting family with 'k' accepting sets. It is easy to extend classic determinisation constructions to handle generalised Büchi automata by incorporating the degeneralization algorithm in the determinisation construction. We extend the tight Büchi construction to do exactly this. Our determinisation constructions go to deterministic Rabin automata. It is known that one can determinise to the more convenient parity condition by incorporating the standard Latest Appearance Record construction in the determinisation procedure. We determinise to parity automata using this technique. We prove lower bounds on these constructions. In the case of determinisation to Rabin automata, our constructions are tight to the state. In the case of determinisation to parity, there is a constant factor ≤ 1.5 between upper and lower bounds reducing to optimal(to the state) in the case of Büchi and 1-pair Rabin. We also reconnect tight determinisation and complementation and provide constructions for complementing generalised Büchi and parity automata by starting withour data structure for determinisation. We introduce suitable data structures for the complementation procedures based on the data structure used for determinisation. We prove lower bounds for both constructions that are tight upto an O(n) factor where 'n' is the number of states of the nondeterministic automaton that is complemented