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The Büchi Complementation Saga
, 2007
"... The complementation problem for nondeterministic word automata has numerous applications in formal verification. In particular, the languagecontainment problem, to which many verification problems are reduced, involves complementation. For automata on finite words, which correspond to safety prop ..."
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Cited by 16 (3 self)
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The complementation problem for nondeterministic word automata has numerous applications in formal verification. In particular, the languagecontainment problem, to which many verification problems are reduced, involves complementation. For automata on finite words, which correspond to safety properties, complementation involves determinization. The 2n blowup that is caused by the subset construction is justified by a tight lower bound. For Büchi automata on infinite words, which are required for the modeling of liveness properties, optimal complementation constructions are quite complicated, as the subset construction is not sufficient. We review here progress on this problem, which dates back to its introduction in Büchi's seminal 1962 paper.
Avoiding determinization
 In Proc. 21st IEEE Symp. on Logic in Computer Science
, 2006
"... Automata on infinite objects are extensively used in system specification, verification, and synthesis. While some applications of the automatatheoretic approach have been well accepted by the industry, some have not yet been reduced to practice. Applications that involve determinization of automat ..."
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Cited by 7 (2 self)
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Automata on infinite objects are extensively used in system specification, verification, and synthesis. While some applications of the automatatheoretic approach have been well accepted by the industry, some have not yet been reduced to practice. Applications that involve determinization of automata on infinite words have been doomed to belong to the second category. This has to do with the intricacy of Safra’s optimal determinization construction, the fact that the state space that results from determinization is awfully complex and is not amenable to optimizations and a symbolic implementation, and the fact that determinization requires the introduction of acceptance conditions that are more complex than the Büchi acceptance condition. Examples of applications that involve determinization and belong to the unfortunate second category include model checking of ωregular properties, decidability of branching temporal logics, and synthesis and control of open systems. We offer an alternative to the standard automatatheoretic approach. The crux of our approach is avoiding determinization. Our approach goes instead via universal coBüchi automata. Like nondeterministic automata, universal automata may have several runs on every input. Here, however, an input is accepted if all of the runs are accepting. We show how the use of universal automata simplifies significantly known complementation constructions for automata on infinite words, known decision procedures for branching temporal logics, known synthesis algorithms, and other applications that are now based on determinization. Our algorithms are less difficult to implement and have practical advantages like being amenable to optimizations and a symbolic implementation.
Lower bounds for complementation of ωautomata via the full automata technique
 In Proc. 33rd ICALP, LNCS 4052
, 2006
"... Abstract. In this paper, we first introduce a new lower bound technique for the state complexity of transformations of automata. Namely we suggest considering the class of full automata in lower bound analysis. Then we apply such technique to the complementation of nondeterministic ωautomata and obt ..."
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Abstract. In this paper, we first introduce a new lower bound technique for the state complexity of transformations of automata. Namely we suggest considering the class of full automata in lower bound analysis. Then we apply such technique to the complementation of nondeterministic ωautomata and obtain several lower bound results. Particularly, we prove an Ω((0.76n) n) lower bound for Büchi complementation, which also holds for almost every complementation and determinization transformation of nondeterministic ωautomata, and prove an optimal (Ω(nk)) n lower bound for the complementation of generalized Büchi automata, which holds for Streett automata as well. 1
A Generalization of Cobham’s Theorem to Automata over Real Numbers
, 2008
"... This article studies the expressive power of finitestate automata recognizing sets of real numbers encoded positionally. It is known that the sets that are definable in the firstorder additive theory of real and integer variables 〈R, Z, +, < 〉 can all be recognized by weak deterministic Büchi auto ..."
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Cited by 4 (1 self)
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This article studies the expressive power of finitestate automata recognizing sets of real numbers encoded positionally. It is known that the sets that are definable in the firstorder additive theory of real and integer variables 〈R, Z, +, < 〉 can all be recognized by weak deterministic Büchi automata, regardless of the encoding base r> 1. In this article, we prove the reciprocal property, i.e., that a subset of R that is recognizable by weak deterministic automata in every base r> 1 is necessarily definable in 〈R, Z, +, <〉. This result generalizes to real numbers the wellknown Cobham’s theorem on the finitestate recognizability of sets of integers. Our proof gives interesting insight into the internal structure of automata recognizing sets of real numbers, which may lead to efficient data structures for handling these sets.
An Improved Lower Bound for the Complementation of Rabin Automata
"... Automata on infinite words (ωautomata) have wide applications in formal language theory as well as in modeling and verifying reactive systems. Complementation ofωautomata is a crucial instrument in many these applications, and hence there have been great interests in determining the state complexit ..."
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Cited by 4 (3 self)
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Automata on infinite words (ωautomata) have wide applications in formal language theory as well as in modeling and verifying reactive systems. Complementation ofωautomata 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 stateoftheart lower bound 2Ω(N lg N) and upper bound 2Ω(kN lg N) , 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
Alternation Removal in Büchi Automata
"... Abstract. Alternating automata play a key role in the automatatheoretic approach to specification, verification, and synthesis of reactive systems. Many algorithms on alternating automata, and in particular, their nonemptiness test, involve removal of alternation: a translation of the alternating a ..."
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Cited by 4 (3 self)
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Abstract. Alternating automata play a key role in the automatatheoretic approach to specification, verification, and synthesis of reactive systems. Many algorithms on alternating automata, and in particular, their nonemptiness test, involve removal of alternation: a translation of the alternating automaton to an equivalent nondeterministic one. For alternating Büchi automata, the best known translation uses the “breakpoint construction ” and involves an O(3 n) state blowup. The translation was described by Miyano and Hayashi in 1984, and is widely used since, in both theory and practice. Yet, the best known lower bound is only 2 n. In this paper we develop and present a complete picture of the problem of alternation removal in alternating Büchi automata. In the lower bound front, we show that the breakpoint construction captures the accurate essence of alternation removal, and provide a matching Ω(3 n) lower bound. Our lower bound holds already for universal (rather than alternating) automata with an alphabet of a constant size. In the upperbound front, we point to a class of alternating Büchi automata for which the breakpoint construction can be replaced by a simpler n2 n construction. Our class, of ordered alternating Büchi automata, strictly contains the class of veryweak alternating automata, for which an n2 n construction is known. 1
A Tight Lower Bound for Streett
"... Finite automata on infinite words (ωautomata) proved to be a powerful weapon for modeling and reasoning infinite behaviors of reactive systems. Complementation of ωautomata is crucial in many of these applications. But the problem is nontrivial; even after extensive study during the past two deca ..."
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Finite automata on infinite words (ωautomata) proved to be a powerful weapon for modeling and reasoning infinite behaviors of reactive systems. Complementation of ωautomata is crucial in many of these applications. But the problem is nontrivial; even after extensive study during the past two decades, we still have an important type of ωautomata, namely Streett automata, for which the gap between the current best lower bound 2 Ω(n lg nk) and upper bound 2 Ω(nk lg nk) is substantial, for the Streett index size k can be exponential in the number of states n. In [4] we showed a construction for complementing Streett automata with the upper bound 2 O(n lg n+nk lg k) for k = O(n) and 2 O(n2 lg n) for k = ω(n). In this paper we establish a matching lower bound 2 Ω(n lg n+nk lg k) for k = O(n) and 2 Ω(n2 lg n) for k = ω(n), and therefore showing that the construction is asymptotically optimal with respect to the 2 Θ(·) notation.