TY - JOUR
T1 - Unifying büchi complementation constructions
AU - Fogarty, Seth
AU - Kupferman, Orna
AU - Wilke, Thomas
AU - Vardi, Moshe Y.
PY - 2013/3/26
Y1 - 2013/3/26
N2 - Complementation of B̈uchi automata, required for checking automata containment, is of major theoretical and practical interest in formal verification. We consider two recent approaches to complementation. The first is the rank-based approach of Kupferman and Vardi, which operates over a dag that embodies all runs of the automaton. This approach is based on the observation that the vertices of this dag can be ranked in a certain way, termed an odd ranking, iff all runs are rejecting. The second is the slice-based approach of K̈ahler and Wilke. This approach tracks levels of "split trees" - run trees in which only essential information about the history of each run is maintained. While the slice-based construction is conceptually simple, the complementing automata it generates are exponentially larger than those of the recent rank-based construction of Schewe, and it suffers from the difficulty of symbolically encoding levels of split trees. In this work we reformulate the slice-based approach in terms of run dags and preorders over states. In doing so, we begin to draw parallels between the rank-based and slice-based approaches. Through deeper analysis of the slice-based approach, we strongly restrict the nondeterminism it generates. We are then able to employ the slice-based approach to provide a new odd ranking, called a retrospective ranking, that is different from the one provided by Kupferman and Vardi. This new ranking allows us to construct a deterministic-in-the-limit rank-based automaton with a highly restricted transition function. Further, by phrasing the slice-based approach in terms of ranks, our approach affords a simple symbolic encoding and achieves the tight bound of Schewe's construction.
AB - Complementation of B̈uchi automata, required for checking automata containment, is of major theoretical and practical interest in formal verification. We consider two recent approaches to complementation. The first is the rank-based approach of Kupferman and Vardi, which operates over a dag that embodies all runs of the automaton. This approach is based on the observation that the vertices of this dag can be ranked in a certain way, termed an odd ranking, iff all runs are rejecting. The second is the slice-based approach of K̈ahler and Wilke. This approach tracks levels of "split trees" - run trees in which only essential information about the history of each run is maintained. While the slice-based construction is conceptually simple, the complementing automata it generates are exponentially larger than those of the recent rank-based construction of Schewe, and it suffers from the difficulty of symbolically encoding levels of split trees. In this work we reformulate the slice-based approach in terms of run dags and preorders over states. In doing so, we begin to draw parallels between the rank-based and slice-based approaches. Through deeper analysis of the slice-based approach, we strongly restrict the nondeterminism it generates. We are then able to employ the slice-based approach to provide a new odd ranking, called a retrospective ranking, that is different from the one provided by Kupferman and Vardi. This new ranking allows us to construct a deterministic-in-the-limit rank-based automaton with a highly restricted transition function. Further, by phrasing the slice-based approach in terms of ranks, our approach affords a simple symbolic encoding and achieves the tight bound of Schewe's construction.
KW - Automata theory
KW - Büchi automata
KW - Büchi complementation
KW - Model checking
KW - Omega automata
UR - http://www.scopus.com/inward/record.url?scp=84875749309&partnerID=8YFLogxK
U2 - 10.2168/LMCS-9(1:13)2013
DO - 10.2168/LMCS-9(1:13)2013
M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???
AN - SCOPUS:84875749309
SN - 1860-5974
VL - 9
JO - Logical Methods in Computer Science
JF - Logical Methods in Computer Science
IS - 1
ER -