17 research outputs found
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
Sensor Synthesis for POMDPs with Reachability Objectives
Partially observable Markov decision processes (POMDPs) are widely used in
probabilistic planning problems in which an agent interacts with an environment
using noisy and imprecise sensors. We study a setting in which the sensors are
only partially defined and the goal is to synthesize "weakest" additional
sensors, such that in the resulting POMDP, there is a small-memory policy for
the agent that almost-surely (with probability~1) satisfies a reachability
objective. We show that the problem is NP-complete, and present a symbolic
algorithm by encoding the problem into SAT instances. We illustrate trade-offs
between the amount of memory of the policy and the number of additional sensors
on a simple example. We have implemented our approach and consider three
classical POMDP examples from the literature, and show that in all the examples
the number of sensors can be significantly decreased (as compared to the
existing solutions in the literature) without increasing the complexity of the
policies.Comment: arXiv admin note: text overlap with arXiv:1511.0845
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
POMDPs under Probabilistic Semantics
We consider partially observable Markov decision processes (POMDPs) with
limit-average payoff, where a reward value in the interval [0,1] is associated
to every transition, and the payoff of an infinite path is the long-run average
of the rewards. We consider two types of path constraints: (i) quantitative
constraint defines the set of paths where the payoff is at least a given
threshold lambda_1 in (0,1]; and (ii) qualitative constraint which is a special
case of quantitative constraint with lambda_1=1. We consider the computation of
the almost-sure winning set, where the controller needs to ensure that the path
constraint is satisfied with probability 1. Our main results for qualitative
path constraint are as follows: (i) the problem of deciding the existence of a
finite-memory controller is EXPTIME-complete; and (ii) the problem of deciding
the existence of an infinite-memory controller is undecidable. For quantitative
path constraint we show that the problem of deciding the existence of a
finite-memory controller is undecidable.Comment: Appears in Proceedings of the Twenty-Ninth Conference on Uncertainty
in Artificial Intelligence (UAI2013
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
Universality and Forall-Exactness of Cost Register Automata with Few Registers
The universality problem asks whether a given finite state automaton accepts all the input words. For quantitative models of automata, where input words are mapped to real values, this is naturally extended to ask whether all the words are mapped to values above (or below) a given threshold. This is known to be undecidable for commonly studied examples such as weighted automata over the positive rational (plus-times) or the integer tropical (min-plus) semirings, or equivalently cost register automata (CRAs) over these semirings. In this paper, we prove that when restricted to CRAs with only three registers, the universality problem is still undecidable, even with additional restrictions for the CRAs to be copyless linear with resets.
In contrast, we show that, assuming the unary encoding of updates, the ?-exact problem (does the CRA output zero on all the words?) for integer min-plus linear CRAs can be decided in polynomial time if the number of registers is constant. Without the restriction on the number of registers this problem is known to be PSPACE-complete
IST Austria Technical Report
POMDPs are standard models for probabilistic planning problems, where an agent interacts with an uncertain environment. We study the problem of almost-sure reachability, where given a set of target states, the question is to decide whether there is a policy to ensure that the target set is reached with probability 1 (almost-surely). While in general the problem is EXPTIME-complete, in many practical cases policies with a small amount of memory suffice. Moreover, the existing solution to the problem is explicit, which first requires to construct explicitly an exponential reduction to a belief-support MDP. In this work, we first study the existence of observation-stationary strategies, which is NP-complete, and then small-memory strategies. We present a symbolic algorithm by an efficient encoding to SAT and using a SAT solver for the problem. We report experimental results demonstrating the scalability of our symbolic (SAT-based) approach
Composition of Tree Series Transformations
Tree series transformations computed by bottom-up and top-down tree series transducers are called bottom-up and top-down tree series transformations, respectively. (Functional) compositions of such transformations are investigated. It turns out that the class of bottomup tree series transformations over a commutative and complete semiring is closed under left-composition with linear bottom-up tree series transformations and right-composition with boolean deterministic bottom-up tree series transformations. Moreover, it is shown that the class of top-down tree series transformations over a commutative and complete semiring is closed under right-composition with linear, nondeleting top-down tree series transformations. Finally, the composition of a boolean, deterministic, total top-down tree series transformation with a linear top-down tree series transformation is shown to be a top-down tree series transformation