88 research outputs found
Can Nondeterminism Help Complementation?
Complementation and determinization are two fundamental notions in automata
theory. The close relationship between the two has been well observed in the
literature. In the case of nondeterministic finite automata on finite words
(NFA), complementation and determinization have the same state complexity,
namely Theta(2^n) where n is the state size. The same similarity between
determinization and complementation was found for Buchi automata, where both
operations were shown to have 2^\Theta(n lg n) state complexity. An intriguing
question is whether there exists a type of omega-automata whose determinization
is considerably harder than its complementation. In this paper, we show that
for all common types of omega-automata, the determinization problem has the
same state complexity as the corresponding complementation problem at the
granularity of 2^\Theta(.).Comment: In Proceedings GandALF 2012, arXiv:1210.202
Tight Upper Bounds for Streett and Parity Complementation
Complementation of finite automata on infinite words is not only a
fundamental problem in automata theory, but also serves as a cornerstone for
solving numerous decision problems in mathematical logic, model-checking,
program analysis and verification. For Streett complementation, a significant
gap exists between the current lower bound and upper
bound , where is the state size, is the number of
Streett pairs, and can be as large as . Determining the complexity
of Streett complementation has been an open question since the late '80s. In
this paper show a complementation construction with upper bound for and for ,
which matches well the lower bound obtained in \cite{CZ11a}. We also obtain a
tight upper bound for parity complementation.Comment: Corrected typos. 23 pages, 3 figures. To appear in the 20th
Conference on Computer Science Logic (CSL 2011
Lower Bounds for Complementation of omega-Automata Via the Full Automata Technique
In this paper, we first introduce a lower bound technique for the state
complexity of transformations of automata. Namely we suggest first considering
the class of full automata in lower bound analysis, and later reducing the size
of the large alphabet via alphabet substitutions. Then we apply such technique
to the complementation of nondeterministic \omega-automata, and obtain several
lower bound results. Particularly, we prove an \omega((0.76n)^n) lower bound
for B\"uchi complementation, which also holds for almost every complementation
or determinization transformation of nondeterministic omega-automata, and prove
an optimal (\omega(nk))^n lower bound for the complementation of generalized
B\"uchi automata, which holds for Streett automata as well
State of B\"uchi Complementation
Complementation of B\"uchi automata has been studied for over five decades
since the formalism was introduced in 1960. Known complementation constructions
can be classified into Ramsey-based, determinization-based, rank-based, and
slice-based approaches. Regarding the performance of these approaches, there
have been several complexity analyses but very few experimental results. What
especially lacks is a comparative experiment on all of the four approaches to
see how they perform in practice. In this paper, we review the four approaches,
propose several optimization heuristics, and perform comparative
experimentation on four representative constructions that are considered the
most efficient in each approach. The experimental results show that (1) the
determinization-based Safra-Piterman construction outperforms the other three
in producing smaller complements and finishing more tasks in the allocated time
and (2) the proposed heuristics substantially improve the Safra-Piterman and
the slice-based constructions.Comment: 28 pages, 4 figures, a preliminary version of this paper appeared in
the Proceedings of the 15th International Conference on Implementation and
Application of Automata (CIAA
A Tight Lower Bound for Streett Complementation
Finite automata on infinite words (-automata) proved to be a powerful
weapon for modeling and reasoning infinite behaviors of reactive systems.
Complementation of -automata is crucial in many of these applications.
But the problem is non-trivial; even after extensive study during the past four
decades, we still have an important type of -automata, namely Streett
automata, for which the gap between the current best lower bound and upper bound is substantial, for the
Streett index size can be exponential in the number of states . In
arXiv:1102.2960 we showed a construction for complementing Streett automata
with the upper bound for and for . In this paper we establish a matching lower bound
for and for
, and therefore showing that the construction is asymptotically
optimal with respect to the notation.Comment: Typo correction and section reorganization. To appear in the
proceeding of the 31st Foundations of Software Technology and Theoretical
Computer Science conference (FSTTCS 2011
Rabin vs. Streett Automata
The Rabin and Streett acceptance conditions are dual. Accordingly, deterministic Rabin and Streett automata are dual. Yet, when adding nondeterminsim, the picture changes dramatically. In fact, the state blowup involved in translations between Rabin and Streett automata is a longstanding open problem, having an exponential gap between the known lower and upper bounds.
We resolve the problem, showing that the translation of Streett to Rabin automata involves a state blowup in , whereas in the other direction, the translations of both deterministic and nondeterministic Rabin automata to nondeterministic Streett automata involve a state blowup in .
Analyzing this substantial difference between the two directions, we get to the conclusion that when studying translations between automata, one should not only consider the state blowup, but also the emph{size} blowup, where the latter takes into account all of the automaton elements. More precisely, the size of an automaton is defined to be the maximum of the alphabet length, the number of states, the number of transitions, and the acceptance condition length (index).
Indeed, size-wise, the results are opposite. That is, the translation of Rabin to Streett involves a size blowup in and of Streett to Rabin in . The core difference between state blowup and size blowup stems from the tradeoff between the index and the number of states. (Recall that the index of Rabin and Streett automata might be exponential in the number of states.)
We continue with resolving the open problem of translating deterministic Rabin and Streett automata to the weaker types of deterministic co-B"uchi and B"uchi automata, respectively. We show that the state blowup involved in these translations, when possible, is in , whereas the size blowup is in
How Deterministic are Good-For-Games Automata?
In GFG automata, it is possible to resolve nondeterminism in a way that only
depends on the past and still accepts all the words in the language. The
motivation for GFG automata comes from their adequacy for games and synthesis,
wherein general nondeterminism is inappropriate. We continue the ongoing effort
of studying the power of nondeterminism in GFG automata. Initial indications
have hinted that every GFG automaton embodies a deterministic one. Today we
know that this is not the case, and in fact GFG automata may be exponentially
more succinct than deterministic ones.
We focus on the typeness question, namely the question of whether a GFG
automaton with a certain acceptance condition has an equivalent GFG automaton
with a weaker acceptance condition on the same structure. Beyond the
theoretical interest in studying typeness, its existence implies efficient
translations among different acceptance conditions. This practical issue is of
special interest in the context of games, where the Buchi and co-Buchi
conditions admit memoryless strategies for both players. Typeness is known to
hold for deterministic automata and not to hold for general nondeterministic
automata.
We show that GFG automata enjoy the benefits of typeness, similarly to the
case of deterministic automata. In particular, when Rabin or Streett GFG
automata have equivalent Buchi or co-Buchi GFG automata, respectively, then
such equivalent automata can be defined on a substructure of the original
automata. Using our typeness results, we further study the place of GFG
automata in between deterministic and nondeterministic ones. Specifically,
considering automata complementation, we show that GFG automata lean toward
nondeterministic ones, admitting an exponential state blow-up in the
complementation of a Streett automaton into a Rabin automaton, as opposed to
the constant blow-up in the deterministic case
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