An Improved Lower Bound for the Complementation of Rabin Automata

Automata on infinite words (ω-automata) have wide applications in formal language theory as well as in modeling and verifying reactive systems. Complementation of ω-automata is a crucial instrument in many these applications, and hence there have been great interests in determining the state complexity of the complementation problem. However, obtaining nontrivial lower bounds has been difficult. For the complementation of Rabin automata, a significant gap exists between the state-of-the-art lower bound 2^Ω(NlgN) and upper bound 2^O(kNlgN), where k, the number of Rabin pairs, can be as large as 2N. In this paper we introduce multidimensional rankings to the full automata technique. Using the improved technique we establish an almost tight lower bound for the complementation of Rabin automata. We also show that the same lower bound holds for the determinization of Rabin automata.

[1]  Dana S. Scott,et al.  Finite Automata and Their Decision Problems , 1959, IBM J. Res. Dev..

[2]  Juraj Hromkovic,et al.  Lower Bounds on the Size of Sweeping Automata , 2009, J. Autom. Lang. Comb..

[3]  R. P. Kurshan,et al.  Automata-theoretic verification of coordinating processes , 1994 .

[4]  Pierre Wolper,et al.  Reasoning About Infinite Computations , 1994, Inf. Comput..

[5]  Christof Löding.  Optimal Bounds for Transformations of omega-Automata , 1999, FSTTCS.

[6]  William J. Sakoda,et al.  Nondeterminism and the size of two way finite automata , 1978, STOC.

[7]  M. Nivat.  Theoretical Computer Science Volume 213-214 , 1999 .

[8]  Nils Klarlund,et al.  Progress measures for complementation omega -automata with applications to temporal logic , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[9]  S. Safra,et al.  On the complexity of omega -automata , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[10]  Nissim Francez,et al.  Generalized fair termination , 1984, POPL '84.

[11]  Orna Kupferman,et al.  Weak alternating automata are not that weak , 1997, Proceedings of the Fifth Israeli Symposium on Theory of Computing and Systems.

[12]  Orna Kupferman,et al.  Complementation Constructions for Nondeterministic Automata on Infinite Words , 2005, TACAS.

[13]  Lutz Priese,et al.  Fairness , 1988, Bull. EATCS.

[14]  J. R. Büchi.  On a Decision Method in Restricted Second Order Arithmetic , 1990 .

[15]  R. J. Joenk,et al.  IBM journal of research and development: information for authors , 1978 .

[16]  Orna Kupferman,et al.  Büchi Complementation Made Tighter , 2006, Int. J. Found. Comput. Sci..

[17]  Qiqi Yan.  Lower Bounds for Complementation of ω-Automata Via the Full Automata Technique , 2006 .

[18]  H. Robbins.  A Remark on Stirling’s Formula , 1955 .

[19]  Sven Schewe,et al.  Büchi complementation made tight , 2009, STACS.

[20]  Moshe Y. Vardi.  The Büchi Complementation Saga , 2007, STACS.

[21]  Shmuel Safra,et al.  Exponential determinization for ω-automata with strong-fairness acceptance condition (extended abstract) , 1992, STOC '92.

[22]  Pierre Wolper,et al.  The Complementation Problem for Büchi Automata with Appplications to Temporal Logic , 1987, Theor. Comput. Sci..

[23]  Nils Klarlund,et al.  Rabin measures and their applications to fairness and automata theory , 1991, [1991] Proceedings Sixth Annual IEEE Symposium on Logic in Computer Science.

[24]  S. Sieber.  On a decision method in restricted second-order arithmetic , 1960 .