# On the Complexity of Parity Word Automata

@inproceedings{King2001OnTC, title={On the Complexity of Parity Word Automata}, author={Valerie King and Orna Kupferman and Moshe Y. Vardi}, booktitle={FoSSaCS}, year={2001} }

Different types of nondeterministic automata on infinite words differ in their succinctness and in the complexity for their nonemptiness problem. A simple translation of a parity automaton to an equivalent Buchi automaton is quadratic: a parity automaton with n states, m transitions, and index k may result in a Buchi automaton of size O((n + m)k). The best known algorithm for the nonemptiness problem of parity automata goes through Buchi automata, leading to a complexity of O((n + m)k). In this…

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## References

SHOWING 1-10 OF 20 REFERENCES

### The complexity of tree automata and logics of programs

- Computer Science[Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science
- 1988

It is shown that for tree automata with m states and n pairs nonemptiness can be tested in time O((mn)/sup 3n/), even though the problem is in general NP-complete, and it follows that satisfiability for propositional dynamic logic with a repetition construct and for the propositional mu-calculus can be tests in deterministic single exponential time.

### Regular expressions for infinite trees and a standard form of automata

- Computer Science, MathematicsSymposium on Computation Theory
- 1984

For Rabin pair automata [R1] a standard form is defined /def. 2/ i.e. such that an ordered subset {s1,...,s2I-1} of states is distinguished in such a way that a path of a run is accepting /rejecting…

### An Algorithm for Strongly Connected Component Analysis in n log n Symbolic Steps

- Computer ScienceFormal Methods Syst. Des.
- 2000

The algorithm can be used to decide emptiness of Büchi automata with the same complexity bound, improving Emerson and Lei's quadratic bound, and emptiness of Streett automata, with a similar bound in terms of nodes.

### Tree automata, mu-calculus and determinacy

- Mathematics, Computer Science[1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science
- 1991

It is shown that the propositional mu-calculus is equivalent in expressive power to finite automata on infinite trees, which provides a radically simplified, alternative proof of M.O. Rabin's (1989) complementation lemma for tree automata, which is the heart of one of the deepest decidability results.

### Decidability of second-order theories and automata on infinite trees

- Mathematics
- 1968

Introduction. In this paper we solve the decision problem of a certain secondorder mathematical theory and apply it to obtain a large number of decidability results. The method of solution involves…

### Reasoning About Infinite Computations

- Computer ScienceInf. Comput.
- 1994

This work investigates extensions of temporal logic by connectives defined by finite automata on infinite words and shows that they do not increase the expressive power of the logic or the complexity of the decision problem.

### Automata for the mu-calculus and Related Results

- Computer Science
- 1995

It is shown that every formula is semantically equivalent to a disjunctive formula and this kind of formula gives rise to a new notion of finite automaton which characterizes the expressive power of the mu-calculus over all transition systems.

### Automata for the Modal mu-Calculus and related Results

- Computer ScienceMFCS
- 1995

It is shown that every formula is semantically equivalent to a disjunctive formula and this kind of formula gives rise to a new notion of finite automaton which characterizes the expressive power of the Μ-calculus over all transition systems.

### On a Decision Method in Restricted Second Order Arithmetic

- Mathematics
- 1990

Let SC be the interpreted formalism which makes use of individual variables t, x, y, z,... ranging over natural numbers, monadic predicate variables q( ), r( ), s( ), i( ),... ranging over arbitrary…