310 research outputs found
Complementing BĂĽchi automata with a subset-tuple construction
Complementation of Büchi automata is well known for being difficult. In the worst case, a state-space growth of (0:76n)n is unavoidable. Recent studies suggest that “simpler” algorithms perform better than more involved ones on practical cases. In this paper, we present a simple “direct” algorithm for complementing Büchi automata. It involves a structured subset construction (using tuples of subsets of states) that produces a deterministic automaton. This construction leads to a complementation procedure that resembles the straightforward complementation algorithm for deterministic Büchi automata, the latter algorithm actually being a special case of our construction
Parity and generalised BĂĽchi automata - determinisation and complementation
In this thesis, we study the problems of determinisation and complementation of finite automata on infinite words. We focus on two classes of automata that occur naturally: generalised Büchi automata and nondeterministic parity automata. Generalised Büchi and parity automata occur naturally in model-checking, realisability checking and synthesis procedures. We first review a tight determinisation procedure for Büchi automata, which uses a simplification of Safra trees called history trees. As Büchi automata are special types of both generalised Büchi and parity automata, we adjust the data structure to arrive at suitably tight determinisation constructions for both generalised Büchi and parity automata. As the parity condition describes combinations of Büchi and CoBüchi conditions, instead of immediately modifying the data structure to handle parity automata, we arrive at a suitable data structure by first looking at a special case, Rabin automata with one accepting pair. One pair Rabin automata correspond to parity automata with three priorities and serve as a starting point to modify the structures that result from Büchi determinisation: we then nest these structures to reflect the standard parity condition and describe a direct determinisation construction. The generalised Büchi condition is characterised by an accepting family with 'k' accepting sets. It is easy to extend classic determinisation constructions to handle generalised Büchi automata by incorporating the degeneralization algorithm in the determinisation construction. We extend the tight Büchi construction to do exactly this. Our determinisation constructions go to deterministic Rabin automata. It is known that one can determinise to the more convenient parity condition by incorporating the standard Latest Appearance Record construction in the determinisation procedure. We determinise to parity automata using this technique. We prove lower bounds on these constructions. In the case of determinisation to Rabin automata, our constructions are tight to the state. In the case of determinisation to parity, there is a constant factor ≤ 1.5 between upper and lower bounds reducing to optimal(to the state) in the case of Büchi and 1-pair Rabin. We also reconnect tight determinisation and complementation and provide constructions for complementing generalised Büchi and parity automata by starting withour data structure for determinisation. We introduce suitable data structures for the complementation procedures based on the data structure used for determinisation. We prove lower bounds for both constructions that are tight upto an O(n) factor where 'n' is the number of states of the nondeterministic automaton that is complemented
Partially Ordered Two-way B\"uchi Automata
We introduce partially ordered two-way B\"uchi automata and characterize
their expressive power in terms of fragments of first-order logic FO[<].
Partially ordered two-way B\"uchi automata are B\"uchi automata which can
change the direction in which the input is processed with the constraint that
whenever a state is left, it is never re-entered again. Nondeterministic
partially ordered two-way B\"uchi automata coincide with the first-order
fragment Sigma2. Our main contribution is that deterministic partially ordered
two-way B\"uchi automata are expressively complete for the first-order fragment
Delta2. As an intermediate step, we show that deterministic partially ordered
two-way B\"uchi automata are effectively closed under Boolean operations.
A small model property yields coNP-completeness of the emptiness problem and
the inclusion problem for deterministic partially ordered two-way B\"uchi
automata.Comment: The results of this paper were presented at CIAA 2010; University of
Stuttgart, Computer Scienc
Determinising Parity Automata
Parity word automata and their determinisation play an important role in
automata and game theory. We discuss a determinisation procedure for
nondeterministic parity automata through deterministic Rabin to deterministic
parity automata. We prove that the intermediate determinisation to Rabin
automata is optimal. We show that the resulting determinisation to parity
automata is optimal up to a small constant. Moreover, the lower bound refers to
the more liberal Streett acceptance. We thus show that determinisation to
Streett would not lead to better bounds than determinisation to parity. As a
side-result, this optimality extends to the determinisation of B\"uchi
automata
Finitary languages
The class of omega-regular languages provides a robust specification language
in verification. Every omega-regular condition can be decomposed into a safety
part and a liveness part. The liveness part ensures that something good happens
"eventually". Finitary liveness was proposed by Alur and Henzinger as a
stronger formulation of liveness. It requires that there exists an unknown,
fixed bound b such that something good happens within b transitions. In this
work we consider automata with finitary acceptance conditions defined by
finitary Buchi, parity and Streett languages. We study languages expressible by
such automata: we give their topological complexity and present a
regular-expression characterization. We compare the expressive power of
finitary automata and give optimal algorithms for classical decisions
questions. We show that the finitary languages are Sigma 2-complete; we present
a complete picture of the expressive power of various classes of automata with
finitary and infinitary acceptance conditions; we show that the languages
defined by finitary parity automata exactly characterize the star-free fragment
of omega B-regular languages; and we show that emptiness is NLOGSPACE-complete
and universality as well as language inclusion are PSPACE-complete for finitary
parity and Streett automata
Efficient Algorithms for Morphisms over Omega-Regular Languages
Morphisms to finite semigroups can be used for recognizing omega-regular
languages. The so-called strongly recognizing morphisms can be seen as a
deterministic computation model which provides minimal objects (known as the
syntactic morphism) and a trivial complementation procedure. We give a
quadratic-time algorithm for computing the syntactic morphism from any given
strongly recognizing morphism, thereby showing that minimization is easy as
well. In addition, we give algorithms for efficiently solving various decision
problems for weakly recognizing morphisms. Weakly recognizing morphism are
often smaller than their strongly recognizing counterparts. Finally, we
describe the language operations needed for converting formulas in monadic
second-order logic (MSO) into strongly recognizing morphisms, and we give some
experimental results.Comment: Full version of a paper accepted to FSTTCS 201
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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