Structural complexity of ω-automata
In this paper we relate expressiveness of ω-automata to their complexity. Expressiveness is related to the different subclasses of the ω-regular languages that are accepted by automata that arise by restrictions on the acceptance conditions used. For example, different subclasses of the ω-regular languages arise from identifying the ω-languages with different classes and levels in the Borel hierarchy. Within the class of ω-regular languages, Wagner and Kaminski identified a strict hierarchy of languages induced by restricting the number of pairs allowed in a deterministic Rabin automaton (DRA).
Complexity relates to the smallest size automaton required to realize an ω-regular language. Safra shows that there are ω-regular languages for which deterministic Streett automata (DSA) are exponentially smaller than nondeterministic Buchi automata; in contrast, we show that for every DSA or DRA whose language is in class Gδ, there exists a DBA of size linear in the original automaton. We show in particular that the language of a DRA is in class Gδ if and only if the language can be realized as a DBA on the same transition structure as the DRA. We present a simple construction to transform a DRA with h pairs and n states to an equivalent DRA with O(n.hk) states and k pairs (i.e. DR(n,h)→ DR(n.hk, k)), where k is the Rabin Index (RI) of the language-the minimum number of pairs required to realize the language as a DRA. We also present a construction to translate a DSA into a minimum-pair DRA, achieving a transformation DS(n,h)→DR(n.hk, r), where k is the Streett index (SI), and r the RI of the language.
We prove that it is NP-hard to determine the RI (SI) of a language specified by a DRA (DSA). However, for a DRA (DSA) with h pairs, determining whether the RI (SI) is h, or any constant c, is in polynomial time.
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