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

Recurrence and Transience for Probabilistic Automata

In a context of $omega$-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

IST Austria Technical Report

We consider probabilistic automata on infinite words with acceptance defined by parity conditions. We consider three qualitative decision problems: (i) the positive decision problem asks whether there is a word that is accepted with positive probability; (ii) the almost decision problem asks whether there is a word that is accepted with probability 1; and (iii) the limit decision problem asks whether for every ε > 0 there is a word that is accepted with probability at least 1 − ε. We unify and generalize several decidability results for probabilistic automata over infinite words, and identify a robust (closed under union and intersection) subclass of probabilistic automata for which all the qualitative decision problems are decidable for parity conditions. We also show that if the input words are restricted to lasso shape words, then the positive and almost problems are decidable for all probabilistic automata with parity conditions

IST Austria Technical Report

We consider partially observable Markov decision processes (POMDPs) with ω-regular conditions specified as parity objectives. The class of ω-regular languages extends regular languages to infinite strings and provides a robust specification language to express all properties used in verification, and parity objectives are canonical forms to express ω-regular conditions. The qualitative analysis problem given a POMDP and a parity objective asks whether there is a strategy to ensure that the objective is satis- fied with probability 1 (resp. positive probability). While the qualitative analysis problems are known to be undecidable even for very special cases of parity objectives, we establish decidability (with optimal complexity) of the qualitative analysis problems for POMDPs with all parity objectives under finite- memory strategies. We establish asymptotically optimal (exponential) memory bounds and EXPTIME- completeness of the qualitative analysis problems under finite-memory strategies for POMDPs with parity objectives

Interannual memory effects for spring NDVI in semi-arid South Africa

Almost 20 years of Normalized Difference Vegetative Index (NDVI) and precipitation (PPT) data are analysed to better understand the interannual memory effects on vegetation dynamics observed at regional scales in Southern Africa (SA). The study focuses on a semi-arid region (25°S–31°S; 21°E–26°E) during the austral early summer (September–December). The memory effects are examined using simple statistical approaches (linear correlations and regressions) which require the definition of an early summer vegetation predictand (December NDVI minus September NDVI) and a consistent set of potential predictors (rainfall amount, number of rainy days, rainfall intensity, NDVI and Rain-Use-Efficiency) considered with 4 to 15-month time-lag. An analysis over six SA sub-regions, corresponding to the six major land-cover types of the area reveals two distinct memory effects. A “negative” memory effect (with both rainfall and vegetation) is detected at 7 to 10-month time-lag while a “positive” memory effect (with vegetation only) is observed at 12 to 14-month time-lag. These results suggest that interannual memory effects in early summer vegetation dynamics of semi-arid South Africa may preferably be driven by biological rather than hydrological mechanisms

Computing Distances between Probabilistic Automata

We present relaxed notions of simulation and bisimulation on Probabilistic Automata (PA), that allow some error epsilon. When epsilon is zero we retrieve the usual notions of bisimulation and simulation on PAs. We give logical characterisations of these notions by choosing suitable logics which differ from the elementary ones, L with negation and L without negation, by the modal operator. Using flow networks, we show how to compute the relations in PTIME. This allows the definition of an efficiently computable non-discounted distance between the states of a PA. A natural modification of this distance is introduced, to obtain a discounted distance, which weakens the influence of long term transitions. We compare our notions of distance to others previously defined and illustrate our approach on various examples. We also show that our distance is not expansive with respect to process algebra operators. Although L without negation is a suitable logic to characterise epsilon-(bi)simulation on deterministic PAs, it is not for general PAs; interestingly, we prove that it does characterise weaker notions, called a priori epsilon-(bi)simulation, which we prove to be NP-difficult to decide.Comment: In Proceedings QAPL 2011, arXiv:1107.074

Penser la crise

Le Congrès d’Épinay en juin 1971, au-delà du mythe, inaugure une décennie exceptionnelle pour le socialisme français. Face à une droite qui lui nie toute légitimité, et à un frère ennemi communiste toujours dominateur, le Parti socialiste de François Mitterrand prône l’union de la gauche, se met en ordre de bataille... et conquiert le pouvoir en dix ans. Le rassemblement des siens et son ouverture, aussi bien aux Chrétiens qu’aux héritiers de Mai 1968, comme les contacts renoués avec le monde culturel et intellectuel, peuvent expliquer son succès qui, pourtant, n’était pas écrit. La double victoire présidentielle et législative de mai-juin 1981 a masqué les rivalités internes entre les courants et leurs animateurs (Chevènement, Rocard, Poperen...). Elle invite donc l’historien à s’interroger sur le potentiel réel de cette décennie. Les auteurs analysent la façon dont le PS s’est préparé à exercer le pouvoir à différentes échelles, du national à l’international. Et ils évaluent ses programmes, du Changer la vie (1972) au Projet socialiste (1980), au regard des nouveaux enjeux sociétaux et de la crise

What is decidable about partially observable Markov decision processes with ω-regular objectives

We consider partially observable Markov decision processes (POMDPs) with ω-regular conditions specified as parity objectives. The class of ω-regular languages provides a robust specification language to express properties in verification, and parity objectives are canonical forms to express them. The qualitative analysis problem given a POMDP and a parity objective asks whether there is a strategy to ensure that the objective is satisfied with probability 1 (resp. positive probability). While the qualitative analysis problems are undecidable even for special cases of parity objectives, we establish decidability (with optimal complexity) for POMDPs with all parity objectives under finite-memory strategies. We establish optimal (exponential) memory bounds and EXPTIME-completeness of the qualitative analysis problems under finite-memory strategies for POMDPs with parity objectives. We also present a practical approach, where we design heuristics to deal with the exponential complexity, and have applied our implementation on a number of POMDP examples

What is Decidable about Partially Observable Markov Decision Processes with omega-Regular Objectives

We consider partially observable Markov decision processes (POMDPs) with ω-regular conditions specified as parity objectives. The qualitative analysis problem given a POMDP and a parity objective asks whether there is a strategy to ensure that the objective is satisfied with probability 1 (resp. positive probability). While the qualitative analysis problems are known to be undecidable even for very special cases of parity objectives, we establish decidability (with optimal EXPTIMEcomplete complexity) of the qualitative analysis problems for POMDPs with all parity objectives under finite-memory strategies. We also establish optimal (exponential) memory bounds