## On Complementing Nondeterministic Büchi Automata (2003)

### Cached

### Download Links

- [www.cs.rice.edu]
- [sankarng.googlepages.com]
- DBLP

### Other Repositories/Bibliography

Citations: | 20 - 8 self |

### BibTeX

@MISC{Gurumurthy03oncomplementing,

author = {Sankar Gurumurthy and Orna Kupferman and Fabio Somenzi and Moshe Y. Vardi},

title = {On Complementing Nondeterministic Büchi Automata},

year = {2003}

}

### Years of Citing Articles

### OpenURL

### Abstract

Several optimal algorithms have been proposed for the complementation of nondeterministic B uchi word automata. Due to the intricacy of the problem and the exponential blow-up that complementation involves, these algorithms have never been used in practice, even though an effective complementation construction would be of significant practical value. Recently, Kupferman and Vardi described a complementation algorithm that goes through weak alternating automata and that seems simpler than previous algorithms. We combine their algorithm with known and new minimization techniques. Our approach is based on optimizations of both the intermediate weak alternating automaton and the final nondeterministic automaton, and involves techniques of rank and height reductions, as well as direct and fair simulation.

### Citations

767 | Design and Validation of Computer Protocols - Holzmann - 1991 |

585 |
An automata-theoretic approach to automatic program verification
- Vardi, Wolper
- 1986
(Show Context)
Citation Context ...to check the latter, we check that the intersection of the system with an automaton that accepts exactly all the computations that violate the specification is empty. For instance, LTL model checking =-=[15,25]-=- usually proceeds by translating the negation of an LTL formula into a Büchi automaton. When properties are specified by ω-regular automata, one needs to complement the property automaton. Due to the ... |

336 |
On a decision method in restricted second order arithmatic. In Proceeding of the 1960 international Congress on Logic, Methodology. and Philosophy of Science
- Büchi
- 1962
(Show Context)
Citation Context ...lation. 1 Introduction Efforts for developing simple complementation algorithms for nondeterministic Büchi automata started early in the 60s, motivated by decision problems of second order logics. In =-=[5]-=-, Büchi suggested a complementation construction that involved a complicated combinatorial argument and a doubly-exponential blow-up in the state space. Thus, complementing an automaton with n states ... |

235 |
Checking that finite state concurrent programs satisfy their linear specification
- Lichtenstein, Pnueli
- 1985
(Show Context)
Citation Context ...to check the latter, we check that the intersection of the system with an automaton that accepts exactly all the computations that violate the specification is empty. For instance, LTL model checking =-=[15,25]-=- usually proceeds by translating the negation of an LTL formula into a Büchi automaton. When properties are specified by ω-regular automata, one needs to complement the property automaton. Due to the ... |

198 |
Temporal logic can be more expressive
- Wolper
- 1983
(Show Context)
Citation Context ...y deterministic Büchi automata [14] (it is easy to complement a deterministic automaton), or to supply the automaton for the negation of the property [9]. Similarly, specification formalisms like ETL =-=[26]-=-, which have automata within the logic, involve complementation of automata, and the difficulty of complementing Büchi automata is an obstacle to practical use [3]. In fact, even when the properties a... |

179 | Recognizing safety and liveness
- Alpern, Schneider
- 1987
(Show Context)
Citation Context ... of rank i. We show that the hierarchy collapses at R3 (that is, all ω-regular languages have rank at most 3) and characterize its four levels. For a definition of safety and co-safety languages, see =-=[1,21]-=-. Theorem 5. R3 = ω-regular languages, R2 = DBW, R1 = co-safety languages, and R0 = safety languages. The hierarchy induced by ranks is closely related to a hierarchy induced by heights of AWW. Intuit... |

131 |
The complementation problem for Büchi automata with applications to temporal logic
- Sistla, Vardi, et al.
- 1987
(Show Context)
Citation Context ... that involved a complicated combinatorial argument and a doubly-exponential blow-up in the state space. Thus, complementing an automaton with n states resulted in an automaton with 22O(n) states. In =-=[22]-=-, Sistla, Vardi, and Wolper suggested an improved construction, with 2O(n2 ) states. Only in [20], however, Safra introduced an optimal determinization construction, which also enabled a 2O(n log n) c... |

103 |
On the complexity of ω-automata
- Safra
- 1988
(Show Context)
Citation Context ... space. Thus, complementing an automaton with n states resulted in an automaton with 22O(n) states. In [22], Sistla, Vardi, and Wolper suggested an improved construction, with 2O(n2 ) states. Only in =-=[20]-=-, however, Safra introduced an optimal determinization construction, which also enabled a 2O(n log n) complementation construction, matching the known lower bound [18]. Another 2O(n log n) constructio... |

103 | Efficient Büchi automata from LTL formulae
- Somenzi, Bloem
- 2000
(Show Context)
Citation Context ... to practical use [3]. In fact, even when the properties are specified in LTL, complementation is useful: the translators from LTL into automata have reached a remarkable level of sophistication (cf. =-=[23,8]-=-). Even though complementation of the automata is not explicitly required, the translations are so involved that it is useful to checks their correctness, which involves complementation 1 . Complement... |

78 | The ForSpec temporal logic: A new temporal property-specification language - Armoni, Fix, et al. - 2002 |

77 | Weak alternating automata are not that weak
- Kupferman, Vardi
- 2001
(Show Context)
Citation Context ...timization techniques that are based on language containment rather than simulation. Thus, an effective algorithm for the complementation of Büchi automata would be of significant practical value. In =-=[12]-=-, Kupferman and Vardi describe a complementation procedure that is simpler than those in [20,10]. The key idea of [12] is to go via weak alternating automata.Inan alternating automaton [6], both exist... |

42 |
liveness, and fairness in temporal logic
- Safety
- 1994
(Show Context)
Citation Context ... of rank i. We show that the hierarchy collapses at R3 (that is, all ω-regular languages have rank at most 3) and characterize its four levels. For a definition of safety and co-safety languages, see =-=[1,21]-=-. Theorem 5. R3 = ω-regular languages, R2 = DBW, R1 = co-safety languages, and R0 = safety languages. The hierarchy induced by ranks is closely related to a hierarchy induced by heights of AWW. Intuit... |

40 | Efficient decision procedures for model checking of linear time logic properties
- Bloem, Ravi, et al.
- 1999
(Show Context)
Citation Context ...his special case avoids the potentially expensive simplification of W and makes complementation of NWB efficient. This is practically relevant because many natural specifications induce weak automata =-=[11,4]-=-. (In [17] it is shown that the intersection of ACTL and LTL is UCW[1], which is included in UWW.) If C is not weak, first its rank is determined, and W is built accordingly, simplifying transitions a... |

39 |
Alternating finite automata on ω-words
- Miyano, Hayashi
- 1984
(Show Context)
Citation Context ... complementation trivial— one only has to dualize the transition function and the acceptance condition. Removing alternation from Büchi automata involves a simple extension of the subset construction =-=[19]-=-. Unfortunately, by dualizing the given nondeterministic Büchi automaton, one gets a universal co-Büchi automaton, creating a gap in the construction. This gap is closed in [12], whose complementation... |

32 |
Complementation is more difficult with automata on infinite words
- Michel
- 1988
(Show Context)
Citation Context ...on, with 2O(n2 ) states. Only in [20], however, Safra introduced an optimal determinization construction, which also enabled a 2O(n log n) complementation construction, matching the known lower bound =-=[18]-=-. Another 2O(n log n) construction was suggested by Klarlund in [10], which circumvented the need for determinization. While being the heart of many complexity results in verification, the constructio... |

19 | Relating linear and branching model checking
- Kupferman, Vardi
- 1998
(Show Context)
Citation Context ...his special case avoids the potentially expensive simplification of W and makes complementation of NWB efficient. This is practically relevant because many natural specifications induce weak automata =-=[11,4]-=-. (In [17] it is shown that the intersection of ACTL and LTL is UCW[1], which is included in UWW.) If C is not weak, first its rank is determined, and W is built accordingly, simplifying transitions a... |

19 | Alternating automata and logics over infinite words
- Löding, Thomas
- 2000
(Show Context)
Citation Context ...ges, namely the minimal rank of a universal co-Büchi automaton that recognizes their complement. We show that, surprisingly, the rank of all ω-regular languages is 3 (a nice corollary, also proved in =-=[16]-=-, is that all ω-regular languages can be recognized by an alternating weak automaton of height 3). Reducing the rank to 3, however, has a flavor of determinization, and involves an exponential blow-up... |

15 | State space reductions for alternating Büchi automata
- Fritz, Wilke
- 2002
(Show Context)
Citation Context ...(A). In general, simulations between states of an ABW can be used to merge states (in case of simulation equivalence), remove transitions, or simplify transitions. 3 The last 3 This is in contrast to =-=[7]-=-, which only considers simulation equivalence quotients. Besides, its model of alternating automata with existential and universal states makes even direct simulation unsafe for minimization.sOn Compl... |

14 |
Automata-TheoreticVerification of Coordinating Processes
- Kurshan
- 1993
(Show Context)
Citation Context ...omata, one needs to complement the property automaton. Due to the lack of a simple complementation construction, the user is typically required to specify the property by deterministic Büchi automata =-=[14]-=- (it is easy to complement a deterministic automaton), or to supply the automaton for the negation of the property [9]. Similarly, specification formalisms like ETL [26], which have automata within th... |

13 |
Progress measures for complementation of ω-automata with applications to temporal logic
- Klarlund
- 1991
(Show Context)
Citation Context ...optimal determinization construction, which also enabled a 2O(n log n) complementation construction, matching the known lower bound [18]. Another 2O(n log n) construction was suggested by Klarlund in =-=[10]-=-, which circumvented the need for determinization. While being the heart of many complexity results in verification, the constructions in [20,10] are complicated and difficult to program. We know of n... |

13 |
Language containment using non-deterministic omega-automata
- Tasiran, Hojati, et al.
- 1995
(Show Context)
Citation Context ...exity results in verification, the constructions in [20,10] are complicated and difficult to program. We know of no implementation of Klarlund’s algorithm, and the implementation of Safra’s algorithm =-=[24]-=- has to cope with the involved structure of the states in the complementary automaton. The lack of a simple implementation is not due to a lack of need. In the automatatheoretic approach to verificati... |

12 | simulation minimization
- Gurumurthy, Bloem, et al.
- 2002
(Show Context)
Citation Context ... to practical use [3]. In fact, even when the properties are specified in LTL, complementation is useful: the translators from LTL into automata have reached a remarkable level of sophistication (cf. =-=[23,8]-=-). Even though complementation of the automata is not explicitly required, the translations are so involved that it is useful to checks their correctness, which involves complementation 1 . Complement... |

11 |
Complementing Deterministic Büchi Automata
- Kurshan
- 1987
(Show Context)
Citation Context ...e computing the NBW equivalent to W. If B is a DBW that is not weak, the resulting AWW is an NWW, and the subset construction does not change it. In such a case, our algorithm behaves like the one of =-=[13]-=-. In some cases, simplification of an AWW also produces an NWW, making the subset construction redundant.sOn Complementing Nondeterministic Büchi Automata 107 6 Simplification of Nondeterministic Büch... |

1 |
Alternating refinement relations. Concurrency Theory
- Alur, Henzinger, et al.
- 1998
(Show Context)
Citation Context ...ification is based on simulation minimization we apply to W, and which often reduces the state space and the transitions even more. Our simulation relation is similar to the alternating simulation of =-=[2]-=-, extended to automata with acceptance conditions on the states (direct simulation) as well as an extension of it in which acceptance conditions are moved to the arcs. Finally, we reduce the height of... |

1 |
The common fragment of
- Maidl
- 2000
(Show Context)
Citation Context ...case avoids the potentially expensive simplification of W and makes complementation of NWB efficient. This is practically relevant because many natural specifications induce weak automata [11,4]. (In =-=[17]-=- it is shown that the intersection of ACTL and LTL is UCW[1], which is included in UWW.) If C is not weak, first its rank is determined, and W is built accordingly, simplifying transitions as discusse... |