Abstract

In this paper we revisit Safra's determinization constructions. We show how to construct deterministic automata with fewer states and, most importantly, parity acceptance conditions. Specifically, starting from a nondeterministic Buchi automaton with n states our construction yields a deterministic parity automaton with n2n+2 states and index 2n (instead of a Rabin automaton with (12)nn2n states and n pairs). Starting from a nondeterministic Streett automaton with n states and k pairs our construction yields a deterministic parity automaton with nn(k+2)+2(k+1)2n(K+1) states and index 2n(k+1) (instead of a Rabin automaton with (12)n(k+1)n n(k+2)(k+1)2n(k+1) states and n(k+1) pairs). The parity condition is much simpler than the Rabin condition. In applications such as solving games and emptiness of tree automata handling the Rabin condition involves an additional multiplier of n2n!(or(n(k+1))2(n(k+1))! in the case of Streett) which is saved using our construction

Highlights

  • One of the fundamental questions in the theory of automata is determinism vs. nondeterminism

  • As before, when compared to Safra’s construction, we reduce the number of states and get a parity automaton

  • We reduce the number of states and more important construct directly a parity automaton

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Summary

Introduction

One of the fundamental questions in the theory of automata is determinism vs. nondeterminism. Automata reduces to reasoning about nondeterministic Rabin tree automata and reasoning about general games reduces to reasoning about Rabin games Some of these applications use co-determinization, the deterministic automaton for the complementary language. States and index 2n(k + 1) (instead of Rabin automaton with (12)n(k+1)nn(k+1)n(k+1)(n(k+1))n(k+1) states and n(k + 1) pairs) For both constructions, complementation is done by considering the same automaton with a dual parity condition. Applications like nondeterminization of alternating tree automata use co-determinization but require the result to be a Rabin or parity automaton. Kupferman and Vardi showed that they can check the emptiness of an alternating parity tree automaton without directly using Safra’s determinization [KV05] Their construction can be used for many game / tree automata applications that require determinization. Our improved construction implies that the complexity of their algorithm reduces from (12)n2 n4n2+2n(n!)2n to (2nnn!)2n

Nondeterministic Automata
Determinization of Buchi Automata
Determinization of Streett Automata
G J instead of
Conclusions and Future Work
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