85 research outputs found
Recurrence and Transience for Probabilistic Automata
In a context of -regular specifications for infinite execution
sequences, the classical B"uchi condition, or repeated liveness
condition, asks that an accepting state is visited infinitely often. In
this paper, we show that in a probabilistic context it is relevant to
strengthen this infinitely often condition. An execution path is now
accepting if the emph{proportion} of time spent on an accepting state
does not go to zero as the length of the path goes to infinity. We
introduce associated notions of recurrence and transience for
non-homogeneous finite Markov chains and study the computational
complexity of the associated problems. As Probabilistic B"uchi Automata
(PBA) have been an attempt to generalize B"uchi automata to a
probabilistic context, we define a class of Constrained Probabilistic
Automata with our new accepting condition on runs. The accepted language
is defined by the requirement that the measure of the set of accepting
runs is positive (probable semantics) or equals 1 (almost-sure
semantics). In contrast to the PBA case, we prove that
the emptiness problem for the language of a constrained probabilistic
B"uchi automaton with the probable semantics is decidable
The Decidability Frontier for Probabilistic Automata on Infinite Words
We consider probabilistic automata on infinite words with acceptance defined
by safety, reachability, B\"uchi, coB\"uchi, and limit-average conditions. We
consider quantitative and qualitative decision problems. We present extensions
and adaptations of proofs for probabilistic finite automata and present a
complete characterization of the decidability and undecidability frontier of
the quantitative and qualitative decision problems for probabilistic automata
on infinite words
Introducing Divergence for Infinite Probabilistic Models
Computing the reachability probability in infinite state probabilistic models
has been the topic of numerous works. Here we introduce a new property called
\emph{divergence} that when satisfied allows to compute reachability
probabilities up to an arbitrary precision. One of the main interest of
divergence is that our algorithm does not require the reachability problem to
be decidable. Then we study the decidability of divergence for probabilistic
versions of pushdown automata and Petri nets where the weights associated with
transitions may also depend on the current state. This should be contrasted
with most of the existing works that assume weights independent of the state.
Such an extended framework is motivated by the modeling of real case studies.
Moreover, we exhibit some divergent subclasses of channel systems and pushdown
automata, particularly suited for specifying open distributed systems and
networks prone to performance collapsing in order to compute the probabilities
related to service requirements.Comment: 31 page
About Decisiveness of Dynamic Probabilistic Models
Decisiveness of infinite Markov chains with respect to some (finite or infinite) target set of states is a key property that allows to compute the reachability probability of this set up to an arbitrary precision. Most of the existing works assume constant weights for defining the probability of a transition in the considered models. However numerous probabilistic modelings require the (dynamic) weight to also depend on the current state. So we introduce a dynamic probabilistic version of counter machine (pCM). After establishing that decisiveness is undecidable for pCMs even with constant weights, we study the decidability of decisiveness for subclasses of pCM. We show that, without restrictions on dynamic weights, decisiveness is undecidable with a single state and single counter pCM. On the contrary with polynomial weights, decisiveness becomes decidable for single counter pCMs under mild conditions. Then we show that decisiveness of probabilistic Petri nets (pPNs) with polynomial weights is undecidable even when the target set is upward-closed unlike the case of constant weights. Finally we prove that the standard subclass of pPNs with a regular language is decisive with respect to a finite set whatever the kind of weights
Markov two-components processes
We propose Markov two-components processes (M2CP) as a probabilistic model of
asynchronous systems based on the trace semantics for concurrency. Considering
an asynchronous system distributed over two sites, we introduce concepts and
tools to manipulate random trajectories in an asynchronous framework: stopping
times, an Asynchronous Strong Markov property, recurrent and transient states
and irreducible components of asynchronous probabilistic processes. The
asynchrony assumption implies that there is no global totally ordered clock
ruling the system. Instead, time appears as partially ordered and random. We
construct and characterize M2CP through a finite family of transition matrices.
M2CP have a local independence property that guarantees that local components
are independent in the probabilistic sense, conditionally to their
synchronization constraints. A synchronization product of two Markov chains is
introduced, as a natural example of M2CP.Comment: 34 page
Extensive amenability and an application to interval exchanges
Extensive amenability is a property of group actions which has recently been
used as a tool to prove amenability of groups. We study this property and prove
that it is preserved under a very general construction of semidirect products.
As an application, we establish the amenability of all subgroups of the group
IET of interval exchange transformations that have angular components of
rational rank~.
In addition, we obtain a reformulation of extensive amenability in terms of
inverted orbits and use it to present a purely probabilistic proof that
recurrent actions are extensively amenable. Finally, we study the triviality of
the Poisson boundary for random walks on IET and show that there are subgroups
admitting no finitely supported measure with trivial boundary.Comment: 28 page
Stochastic Finite State Control of POMDPs with LTL Specifications
Partially observable Markov decision processes (POMDPs) provide a modeling framework for autonomous decision making under uncertainty and imperfect sensing, e.g. robot manipulation and self-driving cars. However, optimal control of POMDPs is notoriously intractable. This paper considers the quantitative problem of synthesizing sub-optimal stochastic finite state controllers (sFSCs) for POMDPs such that the probability of satisfying a set of high-level specifications in terms of linear temporal logic (LTL) formulae is maximized. We begin by casting the latter problem into an optimization and use relaxations based on the Poisson equation and McCormick envelopes. Then, we propose an stochastic bounded policy iteration algorithm, leading to a controlled growth in sFSC size and an any time algorithm, where the performance of the controller improves with successive iterations, but can be stopped by the user based on time or memory considerations. We illustrate the proposed method by a robot navigation case study
Recommended from our members
Strongly Correlated Random Interacting Processes
The focus of the workshop was to discuss the recent developments and future research directions in the area of large scale random interacting processes, with main emphasis in models where local microscopic interactions either produce strong correlations at macroscopic levels, or generate non-equilibrium dynamics. This report contains extended abstracts of the presentations, which featured research in several directions including selfinteracting random walks, spatially growing processes, strongly dependent percolation, spin systems with long-range order, and random permutations
- …