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Containment and equivalence of weighted automata: Probabilistic and max-plus cases
This paper surveys some results regarding decision problems for probabilistic and max-plus automata, such as containment and equivalence. Probabilistic and max-plus automata are part of the general family of weighted automata, whose semantics are maps from words to real values. Given two weighted automata, the equivalence problem asks whether their semantics are the same, and the containment problem whether one is point-wise smaller than the other one. These problems have been studied intensively and this paper will review some techniques used to show (un)decidability and state a list of open questions that still remain
A Generalised Twinning Property for Minimisation of Cost Register Automata
Weighted automata (WA) extend finite-state automata by associating with transitions weights from a semiring S, defining functions from words to S. Recently, cost register automata (CRA) have been introduced as an alternative model to describe any function realised by a WA by means of a deterministic machine. Unambiguous WA over a monoid (M, ⊗) can equivalently be described by cost register automata whose registers take their values in M, and are updated by operations of the form x: = y ⊗ c, with c ∈ M. This class is denoted by CRA⊗c(M).
We introduce a twinning property and a bounded variation property parametrised by an integer k, such that the corresponding notions introduced originally by Choffrut for finite-state transducers are obtained for k = 1. Given an unambiguous weighted automaton W over an infinitary group (G, ⊗) realizing some function f, we prove that the three following properties are equivalent: i) W satisfies the twinning property of order k, ii) f satisfies the k-bounded variation property, and iii) f can be described by a CRA⊗c(G) with at most k registers.
In the spirit of tranducers, we actually prove this result in a more general setting by considering machines over the semiring of finite sets of elements from (G, ⊗): the three properties are still equivalent for such finite-valued weighted automata, that is the ones associating with words subsets of G of cardinality at most ℓ, for some natural ℓ. Moreover, we show that if the operation ⊗ of G is commutative and computable, then one can decide whether a WA satisfies the twinning property of order k. As a corollary, this allows to decide the register minimisation problem for the class CRA⊗c(G).
Last, we prove that a similar result holds for finite-valued finite-state transducers, and that the register minimisation problem for the class CRA.c (B*) is Pspace-complete
When is Containment Decidable for Probabilistic Automata?
The containment problem for quantitative automata is the natural quantitative generalisation of the classical language inclusion problem for Boolean automata. We study it for probabilistic automata, where it is known to be undecidable in general. We restrict our study to the class of probabilistic automata with bounded ambiguity. There, we show decidability (subject to Schanuel's conjecture) when one of the automata is assumed to be unambiguous while the other one is allowed to be finitely ambiguous. Furthermore, we show that this is close to the most general decidable fragment of this problem by proving that it is already undecidable if one of the automata is allowed to be linearly ambiguous
Unambiguous Separators for Tropical Tree Automata
In this paper we show that given a max-plus automaton (over trees, and with real weights) computing a function f and a min-plus automaton (similar) computing a function g such that f ? g, there exists effectively an unambiguous tropical automaton computing h such that f ? h ? g.
This generalizes a result of Lombardy and Mairesse of 2006 stating that series which are both max-plus and min-plus rational are unambiguous. This generalization goes in two directions: trees are considered instead of words, and separation is established instead of characterization (separation implies characterization). The techniques in the two proofs are very different
Restricted density classification in one dimension
The density classification task is to determine which of the symbols
appearing in an array has the majority. A cellular automaton solving this task
is required to converge to a uniform configuration with the majority symbol at
each site. It is not known whether a one-dimensional cellular automaton with
binary alphabet can classify all Bernoulli random configurations almost surely
according to their densities. We show that any cellular automaton that washes
out finite islands in linear time classifies all Bernoulli random
configurations with parameters close to 0 or 1 almost surely correctly. The
proof is a direct application of a "percolation" argument which goes back to
Gacs (1986).Comment: 13 pages, 5 figure
One-Counter Stochastic Games
We study the computational complexity of basic decision problems for
one-counter simple stochastic games (OC-SSGs), under various objectives.
OC-SSGs are 2-player turn-based stochastic games played on the transition graph
of classic one-counter automata. We study primarily the termination objective,
where the goal of one player is to maximize the probability of reaching counter
value 0, while the other player wishes to avoid this. Partly motivated by the
goal of understanding termination objectives, we also study certain "limit" and
"long run average" reward objectives that are closely related to some
well-studied objectives for stochastic games with rewards. Examples of problems
we address include: does player 1 have a strategy to ensure that the counter
eventually hits 0, i.e., terminates, almost surely, regardless of what player 2
does? Or that the liminf (or limsup) counter value equals infinity with a
desired probability? Or that the long run average reward is >0 with desired
probability? We show that the qualitative termination problem for OC-SSGs is in
NP intersection coNP, and is in P-time for 1-player OC-SSGs, or equivalently
for one-counter Markov Decision Processes (OC-MDPs). Moreover, we show that
quantitative limit problems for OC-SSGs are in NP intersection coNP, and are in
P-time for 1-player OC-MDPs. Both qualitative limit problems and qualitative
termination problems for OC-SSGs are already at least as hard as Condon's
quantitative decision problem for finite-state SSGs.Comment: 20 pages, 1 figure. This is a full version of a paper accepted for
publication in proceedings of FSTTCS 201
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
Pumping Lemmas for Weighted Automata
We present three pumping lemmas for three classes of functions definable by fragments of weighted automata over the min-plus semiring and the semiring of natural numbers. As a corollary we show that the hierarchy of functions definable by unambiguous, finitely-ambiguous, polynomially-ambiguous weighted automata, and the full class of weighted automata is strict for the min-plus semiring
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