37 research outputs found
On finitely ambiguous B\"uchi automata
Unambiguous B\"uchi automata, i.e. B\"uchi automata allowing only one
accepting run per word, are a useful restriction of B\"uchi automata that is
well-suited for probabilistic model-checking. In this paper we propose a more
permissive variant, namely finitely ambiguous B\"uchi automata, a
generalisation where each word has at most accepting runs, for some fixed
. We adapt existing notions and results concerning finite and bounded
ambiguity of finite automata to the setting of -languages and present a
translation from arbitrary nondeterministic B\"uchi automata with states to
finitely ambiguous automata with at most states and at most accepting
runs per word
Ambiguity, Weakness, and Regularity in Probabilistic B\"uchi Automata
Probabilistic B\"uchi automata are a natural generalization of PFA to
infinite words, but have been studied in-depth only rather recently and many
interesting questions are still open. PBA are known to accept, in general, a
class of languages that goes beyond the regular languages. In this work we
extend the known classes of restricted PBA which are still regular, strongly
relying on notions concerning ambiguity in classical omega-automata.
Furthermore, we investigate the expressivity of the not yet considered but
natural class of weak PBA, and we also show that the regularity problem for
weak PBA is undecidable
Degrees of Ambiguity for Parity Tree Automata
An automaton is unambiguous if for every input it has at most one accepting computation. An automaton is finitely (respectively, countably) ambiguous if for every input it has at most finitely (respectively, countably) many accepting computations. An automaton is boundedly ambiguous if there is k ? ?, such that for every input it has at most k accepting computations. We consider Parity Tree Automata (PTA) and prove that the problem whether a PTA is not unambiguous (respectively, is not boundedly ambiguous, not finitely ambiguous) is co-NP complete, and the problem whether a PTA is not countably ambiguous is co-NP hard
Index problems for game automata
For a given regular language of infinite trees, one can ask about the minimal
number of priorities needed to recognize this language with a
non-deterministic, alternating, or weak alternating parity automaton. These
questions are known as, respectively, the non-deterministic, alternating, and
weak Rabin-Mostowski index problems. Whether they can be answered effectively
is a long-standing open problem, solved so far only for languages recognizable
by deterministic automata (the alternating variant trivializes).
We investigate a wider class of regular languages, recognizable by so-called
game automata, which can be seen as the closure of deterministic ones under
complementation and composition. Game automata are known to recognize languages
arbitrarily high in the alternating Rabin-Mostowski index hierarchy; that is,
the alternating index problem does not trivialize any more.
Our main contribution is that all three index problems are decidable for
languages recognizable by game automata. Additionally, we show that it is
decidable whether a given regular language can be recognized by a game
automaton
Partially ordered two-way Büchi automata
We introduce partially ordered two-way Büchi automata over infinite words. As for finite words, the nondeterministic variant recognizes the fragment Sigma2 of first-order logic FO[<] and the deterministic version yields the Delta2-definable omega-languages. As a byproduct of our results, we show that deterministic partially ordered two-way Büchi automata are effectively closed under Boolean operations.
In addition, we have coNP-completeness results for the emptiness problem and the inclusion problem over deterministic partially ordered two-way Büchi automata
Bidimensional Linear Recursive Sequences and Universality of Unambiguous Register Automata
We study the universality and inclusion problems for register automata over
equality data. We show that the universality and the inclusion problems can be
solved with 2-EXPTIME complexity when the input automata are without guessing
and unambiguous, improving on the currently best-known 2-EXPSPACE upper bound
by Mottet and Quaas. When the number of registers of both automata is fixed, we
obtain a lower EXPTIME complexity, also improving the EXPSPACE upper bound from
Mottet and Quaas for fixed number of registers. We reduce inclusion to
universality, and then we reduce universality to the problem of counting the
number of orbits of runs of the automaton. We show that the orbit-counting
function satisfies a system of bidimensional linear recursive equations with
polynomial coefficients (linrec), which generalises analogous recurrences for
the Stirling numbers of the second kind, and then we show that universality
reduces to the zeroness problem for linrec sequences. While such a counting
approach is classical and has successfully been applied to unambiguous finite
automata and grammars over finite alphabets, its application to register
automata over infinite alphabets is novel. We provide two algorithms to decide
the zeroness problem for bidimensional linear recursive sequences arising from
orbit-counting functions. Both algorithms rely on techniques from linear
non-commutative algebra. The first algorithm performs variable elimination and
has elementary complexity. The second algorithm is a refined version of the
first one and it relies on the computation of the Hermite normal form of
matrices over a skew polynomial field. The second algorithm yields an EXPTIME
decision procedure for the zeroness problem of linrec sequences, which in turn
yields the claimed bounds for the universality and inclusion problems of
register automata.Comment: full version of the homonymous paper to appear in the proceedings of
STACS'2
History-deterministic Vector Addition Systems
We consider history-determinism, a restricted form of non-determinism, for
Vector Addition Systems with States (VASS) when used as acceptors to recognise
languages of finite words. History-determinism requires that the
non-deterministic choices can be resolved on-the-fly; based on the past and
without jeopardising acceptance of any possible continuation of the input word.
Our results show that the history-deterministic (HD) VASS sit strictly
between deterministic and non-deterministic VASS regardless of the number of
counters. We compare the relative expressiveness of HD systems, and
closure-properties of the induced language classes, with coverability and
reachability semantics, and with and without -labelled
transitions.
Whereas in dimension 1, inclusion and regularity remain decidable, from
dimension two onwards, HD-VASS with suitable resolver strategies, are
essentially able to simulate 2-counter Minsky machines, leading to several
undecidability results: It is undecidable whether a VASS is
history-deterministic, or if a language equivalent history-deterministic VASS
exists. Checking language inclusion between history-deterministic 2-VASS is
also undecidable.Comment: This is the full version of a paper published in CONCUR 202
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