119 research outputs found
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
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
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
Supervisory Controller Synthesis for Non-terminating Processes is an Obliging Game
We present a new algorithm to solve the supervisory control problem over non-terminating processes modeled as -regular automata. A solution to this problem was obtained by Thistle in 1995 which uses complex manipulations of automata. We show a new solution to the problem through a reduction to obliging games, which, in turn, can be reduced to -regular reactive synthesis. Therefore, our reduction results in a symbolic algorithm based on manipulating sets of states using tools from reactive synthesis
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
Optimal transformations of Muller conditions
In this paper, we are interested in automata over infinite words and infinite
duration games, that we view as general transition systems. We study
transformations of systems using a Muller condition into ones using a parity
condition, extending Zielonka's construction. We introduce the alternating
cycle decomposition transformation, and we prove a strong optimality result:
for any given deterministic Muller automaton, the obtained parity automaton is
minimal both in size and number of priorities among those automata admitting a
morphism into the original Muller automaton.
We give two applications. The first is an improvement in the process of
determinisation of B\"uchi automata into parity automata by Piterman and
Schewe. The second is to present characterisations on the possibility of
relabelling automata with different acceptance conditions
Alternative Automata-based Approaches to Probabilistic Model Checking
In this thesis we focus on new methods for probabilistic model checking (PMC) with linear temporal logic (LTL). The standard approach translates an LTL formula into a deterministic ω-automaton with a double-exponential blow up.
There are approaches for Markov chain analysis against LTL with exponential runtime, which motivates the search for non-deterministic automata with restricted forms of non-determinism that make them suitable for PMC. For MDPs, the approach via deterministic automata matches the double-exponential lower bound, but a practical application might benefit from approaches via non-deterministic automata.
We first investigate good-for-games (GFG) automata. In GFG automata one can resolve the non-determinism for a finite prefix without knowing the infinite suffix and still obtain an accepting run for an accepted word. We explain that GFG automata are well-suited for MDP analysis on a theoretic level, but our experiments show that GFG automata cannot compete with deterministic automata.
We have also researched another form of pseudo-determinism, namely unambiguity, where for every accepted word there is exactly one accepting run. We present a polynomial-time approach for PMC of Markov chains against specifications given by an unambiguous Büchi automaton (UBA). Its two key elements are the identification whether the induced probability is positive, and if so, the identification of a state set inducing probability 1.
Additionally, we examine the new symbolic Muller acceptance described in the Hanoi Omega Automata Format, which we call Emerson-Lei acceptance. It is a positive Boolean formula over unconditional fairness constraints. We present a construction of small deterministic automata using Emerson-Lei acceptance. Deciding, whether an MDP has a positive maximal probability to satisfy an Emerson-Lei acceptance, is NP-complete. This fact has triggered a DPLL-based algorithm for deciding positiveness
Streett Automata Model Checking of Higher-Order Recursion Schemes
We propose a practical algorithm for Streett automata model checking of higher-order recursion schemes (HORS), which checks whether the tree generated by a given HORS is accepted by a given Streett automaton. The Streett automata model checking of HORS is useful in the context of liveness verification of higher-order functional programs. The previous approach to Streett automata model checking converted Streett automata to parity automata and then invoked a parity tree automata model checker. We show through experiments that our direct approach outperforms the previous approach. Besides being able to directly deal with Streett automata, our algorithm is the first practical Streett or parity automata model checking algorithm that runs in time polynomial in the size of HORS, assuming that the other parameters are fixed. Previous practical fixed-parameter polynomial time algorithms for HORS could only deal with the class of trivial tree automata. We have confirmed through experiments that (a parity automata version of) our model checker outperforms previous parity automata model checkers for HORS
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