Abstract In the last decade, a large body of work has emerged on robustness of neural networks, i.e., checking if the decision remains unchanged when the input is slightly perturbed. However, most of these approaches ignore the confidence of a neural network on its output. In this work, we aim to develop a generalized framework for formally reasoning about the confidence along with robustness in neural networks. We propose a simple yet expressive grammar that captures various confidence-based specifications. We develop a novel and unified technique to verify all instances of the grammar in a homogeneous way, viz., by adding a few additional layers to the neural network, which enables the use any state-of-the-art neural network verification tool. We perform an extensive experimental evaluation over a large suite of 8870 benchmarks, where the largest network has 138M parameters, and show that this outperforms ad-hoc encoding approaches by a significant margin.

Out-of-distribution (OOD) detection attempts to distinguish outlier samples to prevent models trained on the in-distribution (ID) dataset from producing unavailable outputs. Most OOD detection methods require many ID samples for training, which seriously limits their real-world applications. To this end, we target a challenging setting: few-shot OOD detection, where only a few labeled ID samples are available. Therefore, few-shot OOD detection is much more challenging than the traditional OOD detection setting. Previous few-shot OOD detection works ignore the distinct diversity between different classes. In this paper, we propose a novel network: Adaptive Multi-prompt Contrastive Network (AMCN), which adapts the ID-OOD separation boundary by learning inter- and intra-class distribution. To compensate for the absence of OOD and scarcity of ID image samples, we leverage CLIP, connecting text with images, engineering learnable ID and OOD textual prompts. Specifically, we first generate adaptive prompts (learnable ID prompts, label-fixed OOD prompts, and label-adaptive OOD prompts). Then, we generate an adaptive class boundary for each class by introducing a class-wise threshold. Finally, we propose a prompt-guided ID-OOD separation module to control the margin between ID and OOD prompts. Experimental results show that AMCN outperforms other state-of-the-art works.

Decision Trees (DTs) constitute one of the major highly non-linear AI models, valued, e.g., for their efficiency on tabular data. Learning accurate DTs is, however, complicated, especially for oblique DTs, and does take a significant training time. Further, DTs suffer from overfitting, e.g., they proverbially “do not generalize” in regression tasks. Recently, some works proposed ways to make (oblique) DTs differentiable. This enables highly efficient gradient-descent algorithms to be used to learn DTs. It also enables generalizing capabilities by learning regressors at the leaves simultaneously with the decisions in the tree. Prior approaches to making DTs differentiable rely either on probabilistic approximations at the tree’s internal nodes (soft DTs) or on approximations in gradient computation at the internal node (quantized gradient descent). In this work, we propose DTSemNet, a novel semantically equivalent and invertible encoding for (hard, oblique) DTs as Neural Networks (NNs), that uses standard vanilla gradient descent. Experiments across various classification and regression benchmarks show that oblique DTs learned using DTSemNet are more accurate than oblique DTs of similar size learned using state-of-the-art techniques. Further, DT training time is significantly reduced. We also experimentally demonstrate that DTSemNet can learn DT policies as efficiently as NN policies in the Reinforcement Learning (RL) setup with physical inputs (dimensions ≤32). The code is available at https://github.com/CPS-research-group/dtsemnet.

The Skolem problem is a long-standing open problem in linear dynamical systems: can a linear recurrence sequence (LRS) ever reach 0 from a given initial configuration? Similarly, the positivity problem asks whether the LRS stays positive from an initial configuration. Deciding Skolem (or positivity) has been open for half a century: the best known decidability results are for LRS with special properties (e.g., low order recurrences). But these problems are easier for "uninitialized" variants, where the initial configuration is not fixed but can vary arbitrarily: checking if there is an initial configuration from which the LRS stays positive can be decided in polynomial time (Tiwari in 2004, Braverman in 2006). In this paper, we consider problems that lie between the initialized and uninitialized variants. More precisely, we ask if 0 (resp. negative numbers) can be avoided from every initial configuration in a neighborhood of a given initial configuration. This can be considered as a robust variant of the Skolem (resp. positivity) problem. We show that these problems lie at the frontier of decidability: if the neighbourhood is given as part of the input, then robust Skolem and robust positivity are Diophantine hard, i.e., solving either would entail major breakthroughs in Diophantine approximations, as happens for (non-robust) positivity. However, if one asks whether such a neighbourhood exists, then the problems turn out to be decidable with PSPACE complexity. Our techniques also allow us to tackle robustness for ultimate positivity, which asks whether there is a bound on the number of steps after which the LRS remains positive. There are two variants depending on whether we ask for a "uniform" bound on this number of steps. For the non-uniform variant, when the neighbourhood is open, the problem turns out to be tractable, even when the neighbourhood is given as input.

The Skolem problem is a long-standing open problem in linear dynamical systems: can a linear recurrence sequence (LRS) ever reach 0 from a given initial configuration? Similarly, the positivity problem asks whether the LRS stays positive from an initial configuration. Deciding Skolem (or positivity) has been open for half a century: The best known decidability results are for LRS with special properties (e.g., low order recurrences). On the other hand, these problems are much easier for "uninitialized" variants, where the initial configuration is not fixed but can vary arbitrarily: checking if there is an initial configuration from which the LRS stays positive can be decided by polynomial time algorithms (Tiwari in 2004, Braverman in 2006). In this paper, we consider problems that lie between the initialized and uninitialized variant. More precisely, we ask if 0 (resp. negative numbers) can be avoided from every initial configuration in a neighborhood of a given initial configuration. This can be considered as a robust variant of the Skolem (resp. positivity) problem. We show that these problems lie at the frontier of decidability: if the neighborhood is given as part of the input, then robust Skolem and robust positivity are Diophantine-hard, i.e., solving either would entail major breakthrough in Diophantine approximations, as happens for (non-robust) positivity. Interestingly, this is the first Diophantine-hardness result on a variant of the Skolem problem, to the best of our knowledge. On the other hand, if one asks whether such a neighborhood exists, then the problems turn out to be decidable in their full generality, with PSPACE complexity. Our analysis is based on the set of initial configurations such that positivity holds, which leads to new insights into these difficult problems, and interesting geometrical interpretations.

Erroneous behaviour in safety critical real-time systems may inflict serious consequences. In this paper, we show how to synthesize timed shields from timed safety properties given as timed automata. A timed shield enforces the safety of a running system while interfering with the system as little as possible. We present timed post-shields and timed pre-shields. A timed pre-shield is placed before the system and provides a set of safe outputs. This set restricts the choices of the system. A timed post-shield is implemented after the system. It monitors the system and corrects the system's output only if necessary. We further extend the timed post-shield construction to provide a guarantee on the recovery phase, i.e., the time between a specification violation and the point at which full control can be handed back to the system. In our experimental results, we use timed post-shields to ensure the safety in a reinforcement learning setting for controlling a platoon of cars, during the learning and execution phase, and study the effect.

Learning models from observations of a system is a powerful tool with many applications. In this paper, we consider learning Discrete Time Markov Chains (DTMC), with different methods such as frequency estimation or Laplace smoothing . While models learnt with such methods converge asymptotically towards the exact system, a more practical question in the realm of trusted machine learning is how accurate a model learnt with a limited time budget is. Existing approaches provide bounds on how close the model is to the original system, in terms of bounds on local (transition) probabilities, which has unclear implication on the global behavior. In this work, we provide global bounds on the error made by such a learning process, in terms of global behaviors formalized using temporal logic . More precisely, we propose a learning process ensuring a bound on the error in the probabilities of these properties. While such learning process cannot exist for the full LTL logic, we provide one ensuring a bound that is uniform over all the formulas of CTL. Further, given one time-to-failure property, we provide an improved learning algorithm. Interestingly, frequency estimation is sufficient for the latter, while Laplace smoothing is needed to ensure non-trivial uniform bounds for the full CTL logic.

Abstract Negotiations were introduced in [6] as a model for concurrent systems with multiparty decisions. What is very appealing with negotiations is that it is one of the very few non-trivial concurrent models where several interesting problems, such as soundness, i.e. absence of deadlocks, can be solved in PTIME [3]. In this paper, we introduce the model of timed negotiations and consider the problem of computing the minimum and the maximum execution times of a negotiation. The latter can be solved using the algorithm of [10] computing costs in negotiations, but surprisingly minimum execution time cannot. This paper proposes new algorithms to compute both minimum and maximum execution time, that work in much more general classes of negotiations than [10], that only considered sound and deterministic negotiations. Further, we uncover the precise complexities of these questions, ranging from PTIME to $$\varDelta _2^P$$ Δ2P -complete. In particular, we show that computing the minimum execution time is more complex than computing the maximum execution time in most classes of negotiations we consider.

Angluin et al. proved that population protocols compute exactly the predicates definable in Presburger arithmetic (PA), the first-order theory of addition. As part of this result, they presented a procedure that translates any formula $φ$ of quantifier-free PA with remainder predicates (which has the same expressive power as full PA) into a population protocol with $2^{O(\text{poly}(|φ|))}$ states that computes $φ$. More precisely, the number of states of the protocol is exponential in both the bit length of the largest coefficient in the formula, and the number of nodes of its syntax tree. In this paper, we prove that every formula $φ$ of quantifier-free PA with remainder predicates is computable by a leaderless population protocol with $O(\text{poly}(|φ|))$ states. Our proof is based on several new constructions, which may be of independent interest. Given a formula $φ$ of quantifier-free PA with remainder predicates, a first construction produces a succinct protocol (with $O(|φ|^3)$ leaders) that computes $φ$; this completes the work initiated in [STACS'18], where we constructed such protocols for a fragment of PA. For large enough inputs, we can get rid of these leaders. If the input is not large enough, then it is small, and we design another construction producing a succinct protocol with one leader that computes $φ$. Our last construction gets rid of this leader for small inputs.

We are interested in studying the evolution of large homogeneous populations of cells, where each cell is assumed to be composed of a group of biological players (species) whose dynamics is governed by a complex biological pathway, identical for all cells. Modeling the inherent variability of the species concentrations in different cells is crucial to understand the dynamics of the population. In this work, we focus on handling this variability by modeling each species by a random variable that evolves over time. This appealing approach runs into the curse of dimensionality since exactly representing a joint probability distribution involving a large set of random variables quickly becomes intractable as the number of variables grows. To make this approach amenable to biopathways, we explore different techniques to (i) approximate the exact joint distribution at a given time point, and (ii) to track its evolution as time elapses. We start with the problem of approximating the probability distribution of biological species in a population of cells at some given time point. Data come from different fine-grained models of biological pathways of increasing complexities, such as (perturbed) Ordinary Differential Equations (ODEs). Classical approximations rely on the strong and unrealistic assumption that variables/species are independent, or that they can be grouped into small independent clusters. We propose instead to use the Chow-Liu tree representation, based on overlapping clusters of two variables, which better captures correlations between variables. Our experiments show that the proposed approximation scheme is more accurate than existing ones to model probability distributions deriving from biopathways. Then we address the problem of tracking the dynamics of a population of cells, that is computing from an initial distribution the evolution of the (approximate) joint distribution of species over time, called the inference problem. We evaluate several approximate inference algorithms (e.g., [14] , [17] ) for coarse-grained abstractions [12], [16] of biological pathways. Using the Chow-Liu tree approximation, we develop a new inference algorithm which is very accurate according to the experiments we report, for a minimal computation overhead. Our implementation is available at https://codeocean.com/capsule/6491669/tree.

An important task in AI is one of classifying an observation as belonging to one class among several (e.g. image classification). We revisit this problem in a verification context: given k partially observable systems modeled as Hidden Markov Models (also called labeled Markov chains), and an execution of one of them, can we eventually classify which system performed this execution, just by looking at its observations? Interestingly, this problem generalizes several problems in verification and control, such as fault diagnosis and opacity. Also, classification has strong connections with different notions of distances between stochastic models. In this paper, we study a general and practical notion of classifiers, namely limit-sure classifiers, which allow misclassification, i.e. errors in classification, as long as the probability of misclassification tends to 0 as the length of the observation grows. To study the complexity of several notions of classification, we develop techniques based on a simple but powerful notion of stationary distributions for HMMs. We prove that one cannot classify among HMMs iff there is a finite separating word from their stationary distributions. This provides a direct proof that classifiability can be checked in PTIME, as an alternative to existing proofs using separating events (i.e. sets of infinite separating words) for the total variation distance. Our approach also allows us to introduce and tackle new notions of classifiability which are applicable in a security context.

We introduce a new setting where a population of agents, each modelled by a finite-state system, are controlled uniformly: the controller applies the same action to every agent. The framework is largely inspired by the control of a biological system, namely a population of yeasts, where the controller may only change the environment common to all cells. We study a synchronisation problem for such populations: no matter how individual agents react to the actions of the controller, the controller aims at driving all agents synchronously to a target state. The agents are naturally represented by a non-deterministic finite state automaton (NFA), the same for every agent, and the whole system is encoded as a 2-player game. The first player (Controller) chooses actions, and the second player (Agents) resolves non-determinism for each agent. The game with m agents is called the m -population game. This gives rise to a parameterized control problem (where control refers to 2 player games), namely the population control problem: can Controller control the m-population game for all m in N whatever Agents does? Comment: This is a journal version of the extended abstract arXiv:1707.02058 which appeared in Concur 2017, together with proofs

We study quantitative properties of the response time in stochastic models. For instance, we are interested in quantifying bounds such that a high percentage of the runs answers a query within these bounds. To study such problems, computing probabilities on a state-space blown-up by a factor depending on the bound could be used, but this solution is not satisfactory when the bound is large. In this paper, we propose a new symbolic method to quantify bounds on the response time, using the moments of the distribution of simple stochastic systems. We prove that the distribution (and hence the bounds) is uniquely defined given its moments. We provide optimal bounds for the response time over all distributions having a pair of these moments. We explain how to symbolically compute in polynomial time any moment of the distribution of response times using adequately-defined semirings. This allows us to compute optimal bounds in parametric models and to reduce complexity for computing optimal bounds in hierarchical models. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We consider distribution-based objectives for Markov Decision Processes (MDP). This class of objectives gives rise to an interesting trade-off between full and partial information. As in full observation, the strategy in the MDP can depend on the state of the system, but similar to partial information, the strategy needs to account for all the states at the same time.

MOTIVATION: Quantitative models are increasingly used in systems biology. Usually, these quantitative models involve many molecular species and their associated reactions. When simulating a tissue with thousands of cells, using these large models becomes computationally and time limiting. RESULTS: In this paper, we propose to construct abstractions using information theory notions. Entropy is used to discretize the state space and mutual information is used to select a subset of all original variables and their mutual dependencies. We apply our method to an hybrid model of TRAIL-induced apoptosis in HeLa cell. Our abstraction, represented as a Dynamic Bayesian Network (DBN), reduces the number of variables from 92 to 10, and accelerates numerical simulation by an order of magnitude, yet preserving essential features of cell death time distributions. AVAILABILITY AND IMPLEMENTATION: This approach is implemented in the tool DBNizer, freely available at http://perso.crans.org/genest/DBNizer . CONTACT: [email protected] or [email protected]. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

We consider two-person zero-sum stochastic games with signals, a standard model of stochastic games with imperfect information. The only source of information for the players consists of the signals they receive; they cannot directly observe the state of the game, nor the actions played by their opponent, nor their own actions. We are interested in the existence of almost-surely winning or positively winning strategies, under reachability, safety, Büchi, or co-Büchi winning objectives, and the computation of these strategies when the game has finitely many states and actions. We prove two qualitative determinacy results. First, in a reachability game, either player 1 can achieve almost surely the reachability objective, or player 2 can achieve surely the dual safety objective, or both players have positively winning strategies. Second, in a Büchi game, if player 1 cannot achieve almost surely the Büchi objective, then player 2 can ensure positively the dual co-Büchi objective. We prove that players only need strategies with finite memory . The number of memory states needed to win with finite-memory strategies ranges from one (corresponding to memoryless strategies) to doubly exponential, with matching upper and lower bounds. Together with the qualitative determinacy results, we also provide fix-point algorithms for deciding which player has an almost-surely winning or a positively winning strategy and for computing an associated finite-memory strategy. Complexity ranges from EXPTIME to 2EXPTIME, with matching lower bounds. Our fix-point algorithms also enjoy a better complexity in the cases where one of the players is better informed than their opponent. Our results hold even when players do not necessarily observe their own actions. The adequate class of strategies, in this case, is mixed or general strategies (they are equivalent). Behavioral strategies are too restrictive to guarantee determinacy: it may happen that one of the players has a winning general strategy but none of them has a winning behavioral strategy. On the other hand, if a player can observe their actions, then general, mixed, and behavioral strategies are equivalent. Finite-memory strategies are sufficient for determinacy to hold, provided that randomized memory updates are allowed.

Adding real time information to Petri net models often leads to undecidability of classical verification problems such as reachability and boundedness. For instance, models such as Timed-Transition Petri nets (TPNs) [ 22 ] are intractable except in a bounded setting. On the other hand, the model of Timed-Arc Petri nets [ 26 ] enjoys decidability results for boundedness and control-state reachability problems at the cost of disallowing urgency (the ability to enforce actions within a time delay). Our goal is to investigate decidable classes of Petri nets with time that capture some urgency and still allow unbounded behaviors, which go beyond finite state systems. We present, up to our knowledge, the first decidability results on reachability and boundedness for Petri net variants that combine unbounded places, time, and urgency. For this, we introduce the class of Timed-Arc Petri nets with restricted Urgency, where urgency can be used only on transitions consuming tokens from bounded places. We show that control-state reachability and boundedness are decidable for this new class, by extending results from Timed-Arc Petri nets (without urgency) [ 2 ]. Our main result concerns (marking) reachability, which is undecidable for both TPNs (because of unrestricted urgency) [ 20 ] and Timed-Arc Petri Nets (because of infinite number of “clocks”) [ 25 ]. We obtain decidability of reachability for unbounded TPNs with restricted urgency under a new, yet natural, timed-arc semantics presenting them as Timed-Arc Petri Nets with restricted urgency. Decidability of reachability under the intermediate marking semantics is also obtained for a restricted subclass. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The quantitative verification of Probabilistic Automata (PA) is undecidable in general. Unary PA are a simpler model where the choice of action is fixed. Still, the quantitative verification problem is open and known to be as hard as Skolem's problem, a problem on linear recurrence sequences, whose decidability is open for at least 40 years. In this paper, we approach this problem by studying the languages generated by unary PAs (as defined below), whose regularity would entail the decidability of quantitative verification. Given an initial distribution, we represent the trajectory of a unary PA over time as an infinite word over a finite alphabet, where the n-th letter represents a probability range after n steps. We extend this to a language of trajectories (a set of words), one trajectory for each initial distribution from a (possibly infinite) set. We show that if the eigenvalues of the transition matrix associated with the unary PA are all distinct positive real numbers, then the language is effectively regular. Further, we show that this result is at the boundary of regularity, as non-regular languages can be generated when the restrictions are even slightly relaxed. The regular representation of the language allows us to reason about more general properties, e.g., robustness of a regular property in a neighbourhood around a given distribution.

A finite-state Markov chain M can be regarded as a linear transform operating on the set of probability distributions over its node set. The iterative applications of M to an initial probability distribution μ 0 will generate a trajectory of probability distributions. Thus, a set of initial distributions will induce a set of trajectories. It is an interesting and useful task to analyze the dynamics of M as defined by this set of trajectories. The novel idea here is to carry out this task in a symbolic framework. Specifically, we discretize the probability value space [0,1] into a finite set of intervals I = { I 1 , I 2 ,..., I m }. A concrete probability distribution μ over the node set {1, 2,..., n } of M is then symbolically represented as D , a tuple of intervals drawn from I where the i th component of D will be the interval in which μ( i ) falls. The set of discretized distributions D is a finite alphabet. Hence, the trajectory, generated by repeated applications of M to an initial distribution, will induce an infinite string over this alphabet. Given a set of initial distributions, the symbolic dynamics of M will then consist of a language of infinite strings L over the alphabet D . Our main goal is to verify whether L meets a specification given as a linear-time temporal logic formula φ. In our logic, an atomic proposition will assert that the current probability of a node falls in the interval I from I . If L is an ω-regular language, one can hope to solve our model-checking problem (whether L ⊧ φ?) using standard techniques. However, we show that, in general, this is not the case. Consequently, we develop the notion of an ϵ-approximation, based on the transient and long-term behaviors of the Markov chain M . Briefly, the symbolic trajectory ξ' is an ϵ-approximation of the symbolic trajectory ξ iff (1) ξ' agrees with ξ during its transient phase; and (2) both ξ and ξ' are within an ϵ-neighborhood at all times after the transient phase. Our main results are that one can effectively check whether (i) for each infinite word in L , at least one of its ϵ-approximations satisfies the given specification; (ii) for each infinite word in L , all its ϵ-approximations satisfy the specification. These verification results are strong in that they apply to all finite state Markov chains.

Zielonka's theorem, established 25 years ago, states that any regular language closed under commutation is the language of an asynchronous automaton (a tuple of automata, one per process, exchanging information when performing common actions). Since then, constructing asynchronous automata has been simplified and improved ([Cori/Métivier/Zielonka,1993],[Klarlund/Mukund/Sohoni,1994], [Diekert/Rozenberg,1995], [Genest/Muscholl,2006], [Genest/Gimbert/Muscholl/Walukiewicz,2010], [Baudru/Morin, 2006], [Baudru,2009], [Pighizzini,1993], [Stefanescu/Esparza/Muscholl,2003]). We first survey these constructions and conclude that the synthesized systems are not realistic in the following sense: existing constructions are either plagued by deadends, non deterministic guesses, or the acceptance condition or choice of actions are not distributed. We tackle this problem by giving (effectively testable) necessary and sufficient conditions which ensure that deadends can be avoided, acceptance condition and choices of action can be distributed, and determinism can be maintained. Finally, we implement our constructions, giving promising results when compared with the few other existing prototypes synthesizing asynchronous automata.

A finite state Markov chain M is often viewed as a probabilistic transition system. An alternative view - which we follow here - is to regard M as a linear transform operating on the space of probability distributions over its set of nodes. The novel idea here is to discretize the probability value space [0,1] into a finite set of intervals. A concrete probability distribution over the nodes is then symbolically represented as a tuple D of such intervals. The i-th component of the discretized distribution D will be the interval in which the probability of node i falls. The set of discretized distributions is a finite set and each trajectory, generated by repeated applications of M to an initial distribution, will induce a unique infinite string over this finite set of letters. Hence, given a set of initial distributions, the symbolic dynamics of M will consist of an infinite language L over the finite alphabet of discretized distributions. We investigate whether L meets a specification given as a linear time temporal logic formula whose atomic propositions will assert that the current probability of a node falls in an interval. Unfortunately, even for restricted Markov chains (for instance, irreducible and aperiodic chains), we do not know at present if and when L is an (omega)-regular language. To get around this we develop the notion of an epsilon-approximation, based on the transient and long term behaviors of M. Our main results are that, one can effectively check whether (i) for each infinite word in L, at least one of its epsilon-approximations satisfies the specification; (ii) for each infinite word in L all its epsilon approximations satisfy the specification. These verification results are strong in that they apply to all finite state Markov chains. Further, the study of the symbolic dynamics of Markov chains initiated here is of independent interest and can lead to other applications.

Dynamic Bayesian Networks (DBNs) can serve as succinct probabilistic dynamic models of biochemical networks. To analyze these models, one must compute the probability distribution over system states at a given time point. Doing this exactly is infeasible for large models; hence one must use approximate algorithms. The Factored Frontier algorithm (FF) is one such algorithm. However FF as well as the earlier Boyen-Koller (BK) algorithm can incur large errors. To address this, we present a new approximate algorithm called the Hybrid Factored Frontier (HFF) algorithm. At each time slice, in addition to maintaining probability distributions over local states-as FF does-HFF explicitly maintains the probabilities of a number of global states called spikes. When the number of spikes is 0, we get FF and with all global states as spikes, we get the exact inference algorithm. We show that by increasing the number of spikes one can reduce errors while the additional computational effort required is only quadratic in the number of spikes. We validated the performance of HFF on large DBN models of biopathways. Each pathway has more than 30 species and the corresponding DBN has more than 3,000 nodes. Comparisons with FF and BK show that HFF is a useful and powerful approximate inferencing algorithm for DBNs.

For security and efficiency reasons, most systems do not give the users a full access to their information. One key specification formalism for these systems are the so called Partially Observable Markov Decision Processes (POMDP for short), which have been extensively studied in several research communities, among which AI and model-checking. In this paper we tackle the problem of the minimal information a user needs at runtime to achieve a simple goal, modeled as reaching an objective with probability one. More precisely, to achieve her goal, the user can at each step either choose to use the partial information, or pay a fixed cost and receive the full information. The natural question is then to minimize the cost the user needs to fulfill her objective. This optimization question gives rise to two different problems, whether we consider to minimize the worst case cost, or the average cost. On the one hand, concerning the worst case cost, we show that efficient techniques from the model checking community can be adapted to compute the optimal worst case cost and give optimal strategies for the users. On the other hand, we show that the optimal average price (a question typically considered in the AI community) cannot be computed in general, nor can it be approximated in polynomial time even up to a large approximation factor.

We consider the standard model of finite two person zero sum stochastic games with signals. We are interested in the existence of almost surely winning or positively winning strategies, under reachability, safety, Buchi or co-Buchi winning objectives. We prove two qualitative determinacy results. First, in a reachability game either player 1 can achieve almost-surely the reachability objective, or player 2 can ensure surely the complementary safety objective, or both players have positively winning strategies. Second, in a Buchi game if player 1 cannot achieve almost-surely the Buchi objective, then player 2 can ensure positively the complementary co-Buchi objective. We prove that players only need strategies with finite memory, whose sizes range from no memory at all to doubly-exponential number of states, with matching lower bounds. Together with the qualitative determinacy results, we also provide fix point algorithms for deciding which player has an almost surely winning or a positively winning strategy and for computing the finite memory strategy. Complexity ranges from EXPTIME to 2-EXPTIME with matching lower bounds, and better complexity can be achieved for some special cases where one of the players is better informed than her opponent.