26 research outputs found
Deciding the value 1 problem for probabilistic leaktight automata
The value 1 problem is a decision problem for probabilistic automata over
finite words: given a probabilistic automaton, are there words accepted with
probability arbitrarily close to 1? This problem was proved undecidable
recently; to overcome this, several classes of probabilistic automata of
different nature were proposed, for which the value 1 problem has been shown
decidable. In this paper, we introduce yet another class of probabilistic
automata, called leaktight automata, which strictly subsumes all classes of
probabilistic automata whose value 1 problem is known to be decidable. We prove
that for leaktight automata, the value 1 problem is decidable (in fact,
PSPACE-complete) by constructing a saturation algorithm based on the
computation of a monoid abstracting the behaviours of the automaton. We rely on
algebraic techniques developed by Simon to prove that this abstraction is
complete. Furthermore, we adapt this saturation algorithm to decide whether an
automaton is leaktight. Finally, we show a reduction allowing to extend our
decidability results from finite words to infinite ones, implying that the
value 1 problem for probabilistic leaktight parity automata is decidable
What is known about the Value 1 Problem for Probabilistic Automata?
The value 1 problem is a decision problem for probabilistic automata over
finite words: are there words accepted by the automaton with arbitrarily high
probability? Although undecidable, this problem attracted a lot of attention
over the last few years. The aim of this paper is to review and relate the
results pertaining to the value 1 problem. In particular, several algorithms
have been proposed to partially solve this problem. We show the relations
between them, leading to the following conclusion: the Markov Monoid Algorithm
is the most correct algorithm known to (partially) solve the value 1 problem
Profinite Techniques for Probabilistic Automata and the Markov Monoid Algorithm
We consider the value 1 problem for probabilistic automata over finite words:
it asks whether a given probabilistic automaton accepts words with probability
arbitrarily close to 1. This problem is known to be undecidable. However,
different algorithms have been proposed to partially solve it; it has been
recently shown that the Markov Monoid algorithm, based on algebra, is the most
correct algorithm so far. The first contribution of this paper is to give a
characterisation of the Markov Monoid algorithm. The second contribution is to
develop a profinite theory for probabilistic automata, called the prostochastic
theory. This new framework gives a topological account of the value 1 problem,
which in this context is cast as an emptiness problem. The above
characterisation is reformulated using the prostochastic theory, allowing us to
give a simple and modular proof.Comment: Conference version: STACS'2016, Symposium on Theoretical Aspects of
Computer Science Journal version: TCS'2017, Theoretical Computer Scienc
Stamina : stabilisation nonoids in automata theory
We present Stamina, a tool solving three algorithmic problems in automata theory. First, compute the star height of a regular language, i.e. the minimal number of nested Kleene stars needed for expressing the language with a complement-free regular expression. Second, decide limitedness for regular cost functions. Third, decide whether a probabilistic leaktight automaton has value 1, i.e. whether a probabilistic leaktight automaton accepts words with probability arbitrarily close to 1.
All three problems reduce to the computation of the stabilisation monoid associated with an automaton, which is computationally challenging because the monoid is exponentially larger than the automaton. The compact data structures used in Stamina, together with optimisations and heuristics, allow us to handle automata with several hundreds of states.
This radically improves upon the performances of ACME, a similar tool solving a subset of these problems
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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
Characterisation of an Algebraic Algorithm for Probabilistic Automata
We consider the value 1 problem for probabilistic automata over finite words: it asks whether a given probabilistic automaton accepts words with probability arbitrarily close to 1. This problem is known to be undecidable. However, different algorithms have been proposed to partially solve it; it has been recently shown that the Markov Monoid algorithm, based on algebra, is the most correct algorithm so far. The first contribution of this paper is to give a characterisation of the Markov Monoid algorithm.
The second contribution is to develop a profinite theory for probabilistic automata, called the prostochastic theory. This new framework gives a topological account of the value 1 problem, which in this context is cast as an emptiness problem. The above characterisation is reformulated using the prostochastic theory, allowing to give a modular proof
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
Probabilistic Automata of Bounded Ambiguity
Probabilistic automata are a computational model introduced by Michael Rabin, extending nondeterministic finite automata with probabilistic transitions. Despite its simplicity, this model is very expressive and many of the associated algorithmic questions are undecidable. In this work we focus on the emptiness problem, which asks whether a given probabilistic automaton accepts some word with probability higher than a given threshold. We consider a natural and well-studied structural restriction on automata, namely the degree of ambiguity, which is defined as the maximum number of accepting runs over all words. We observe that undecidability of the emptiness problem requires infinite ambiguity and so we focus on the case of finitely ambiguous probabilistic automata.
Our main results are to construct efficient algorithms for analysing finitely ambiguous probabilistic automata through a reduction to a multi-objective optimisation problem, called the stochastic path problem. We obtain a polynomial time algorithm for approximating the value of finitely ambiguous probabilistic automata and a quasi-polynomial time algorithm for the emptiness problem for 2-ambiguous probabilistic automata
The Complexity of POMDPs with Long-run Average Objectives
We study the problem of approximation of optimal values in
partially-observable Markov decision processes (POMDPs) with long-run average
objectives. POMDPs are a standard model for dynamic systems with probabilistic
and nondeterministic behavior in uncertain environments. In long-run average
objectives rewards are associated with every transition of the POMDP and the
payoff is the long-run average of the rewards along the executions of the
POMDP. We establish strategy complexity and computational complexity results.
Our main result shows that finite-memory strategies suffice for approximation
of optimal values, and the related decision problem is recursively enumerable
complete
Deciding the Value 1 Problem for #-acyclic Partially Observable Markov Decision Processes
The value 1 problem is a natural decision problem in algorithmic game theory. For partially observable Markov decision processes with reachability objective, this problem is defined as follows: are there strategies that achieve the reachability objective with probability arbitrarily close to 1? This problem was shown undecidable recently. Our contribution is to introduce a class of partially observable Markov decision processes, namely #-acyclic partially observable Markov decision processes, for which the value 1 problem is decidable. Our algorithm is based on the construction of a two-player perfect information game, called the knowledge game, abstracting the behaviour of a #-acyclic partially observable Markov decision process M such that the first player has a winning strategy in the knowledge game if and only if the value of M is 1