This paper studies the synthesis of control policies for an agent that has to satisfy a temporal logic specification in a partially observable environment, in the presence of an adversary. The interaction of the agent (defender) with the adversary is modeled as a partially observable stochastic game. The search for policies is limited to over the space of finite state controllers, which leads to a tractable approach to determine policies. The goal is to generate a defender policy to maximize satisfaction of a given temporal logic specification under any adversary policy. We relate the satisfaction of the specification in terms of reaching (a subset of) recurrent states of a Markov chain. We then present a procedure to determine a set of defender and adversary finite state controllers of given sizes that will satisfy the temporal logic specification. We illustrate our approach with an example.
Synthesis of memoryless strategies for POMDPs in order to satisfy a specification was shown to be NP-hard and in PSPACE in @cite_43 . In @cite_26 , a discretization of the belief space is carried out , resulting in a fully observable MDP. However, this approach might not be practical if the state space is large @cite_16 . The complexity of determining a winning strategy to solve the problem of determining the probability of satisfaction of parity objectives was shown to be undecidable in @cite_0 . However, determining finite-memory strategies for the qualitative problem of parity objective satisfaction was shown to be EXPTIME-complete in @cite_18 .
@cite_18 @cite_26 @cite_0 @cite_43 @cite_16
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Standard value function approaches to finding policies for Partially Observable Markov Decision Processes (POMDPs) are generally considered to be intractable for large models. The intractability of these algorithms is to a large extent a consequence of computing an exact, optimal policy over the entire belief space. However, in real-world POMDP problems, computing the optimal policy for the full belief space is often unnecessary for good control even for problems with complicated policy classes. The beliefs experienced by the controller often lie near a structured, low-dimensional subspace embedded in the high-dimensional belief space. Finding a good approximation to the optimal value function for only this subspace can be much easier than computing the full value function. We introduce a new method for solving large-scale POMDPs by reducing the dimensionality of the belief space. We use Exponential family Principal Components Analysis (Collins, Dasgupta, & Schapire, 2002) to represent sparse, high-dimensional belief spaces using small sets of learned features of the belief state. We then plan only in terms of the low-dimensional belief features. By planning in this low-dimensional space, we can find policies for POMDP models that are orders of magnitude larger than models that can be handled by conventional techniques. We demonstrate the use of this algorithm on a synthetic problem and on mobile robot navigation tasks. 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. We consider Markov decision processes (MDPs) with multiple discounted reward objectives. Such MDPs occur in design problems where one wishes to simultaneously optimize several criteria, for example, latency and power. The possible trade-offs between the different objectives are characterized by the Pareto curve. We show that every Pareto-optimal point can be achieved by a memoryless strategy; however, unlike in the single-objective case, the memoryless strategy may require randomization. Moreover, we show that the Pareto curve can be approximated in polynomial time in the size of the MDP. Additionally, we study the problem if a given value vector is realizable by any strategy, and show that it can be decided in polynomial time; but the question whether it is realizable by a deterministic memoryless strategy is NP-complete. These results provide efficient algorithms for design exploration in MDP models with multiple objectives. We consider partially observable Markov decision processes (POMDPs), that are a standard framework for robotics applications to model uncertainties present in the real world, with temporal logic specifications. All temporal logic specifications in linear-time temporal logic (LTL) can be expressed as parity objectives. We study the qualitative analysis problem for POMDPs with parity objectives that asks whether there is a controller (policy) to ensure that the objective holds with probability 1 (almost-surely). While the qualitative analysis of POMDPs with parity objectives is undecidable, recent results show that when restricted to finite-memory policies the problem is EXPTIME-complete. While the problem is intractable in theory, we present a practical approach to solve the qualitative analysis problem. We designed several heuristics to deal with the exponential complexity, and have used our implementation on a number of well-known POMDP examples for robotics applications. Our results provide the first practical approach to solve the qualitative analysis of robot motion planning with LTL properties in the presence of uncertainty. High dimensionality of POMDP's belief state space is one major cause that makes the underlying optimal policy computation intractable. Belief compression refers to the methodology that projects the belief state space to a low-dimensional one to alleviate the problem. In this paper, we propose a novel orthogonal non-negative matrix factorization (O-NMF) for the projection. The proposed O-NMF not only factors the belief state space by minimizing the reconstruction error, but also allows the compressed POMDP formulation to be efficiently computed (due to its orthogonality) in a value-directed manner so that the value function will take same values for corresponding belief states in the original and compressed state spaces. We have tested the proposed approach using a number of benchmark problems and the empirical results confirms its effectiveness in achieving substantial computational cost saving in policy computation. We propose a new approach to the problem of searching a space of policies for a Markov decision process (MDP) or a partially observable Markov decision process (POMDP), given a model. Our approach is based on the following observation: Any (PO)MDP can be transformed into an "equivalent" POMDP in which all state transitions (given the current state and action) are deterministic. This reduces the general problem of policy search to one in which we need only consider POMDPs with deterministic transitions. We give a natural way of estimating the value of all policies in these transformed POMDPs. Policy search is then simply performed by searching for a policy with high estimated value. We also establish conditions under which our value estimates will be good, recovering theoretical results similar to those of Kearns, Mansour and Ng (1999), but with "sample complexity" bounds that have only a polynomial rather than exponential dependence on the horizon time. Our method applies to arbitrary POMDPs, including ones with infinite state and action spaces. We also present empirical results for our approach on a small discrete problem, and on a complex continuous state continuous action problem involving learning to ride a bicycle. We introduce a highly efficient method for solving continuous partially-observable Markov decision processes (POMDPs) in which beliefs can be modeled using Gaussian distributions over the state space. Our method enables fast solutions to sequential decision making under uncertainty for a variety of problems involving noisy or incomplete observations and stochastic actions. We present an efficient approach to compute locally-valid approximations to the value function over continuous spaces in time polynomial (O[n4]) in the dimension n of the state space. To directly tackle the intractability of solving general POMDPs, we leverage the assumption that beliefs are Gaussian distributions over the state space, approximate the belief update using an extended Kalman filter (EKF), and represent the value function by a function that is quadratic in the mean and linear in the variance of the belief. Our approach iterates towards a linear control policy over the state space that is locally-optimal with respect to a user defined cost function, and is approximately valid in the vicinity of a nominal trajectory through belief space. We demonstrate the scalability and potential of our approach on problems inspired by robot navigation under uncertainty for state spaces of up to 128 dimensions. Planning and learning in Partially Observable MDPs (POMDPs) are among the most challenging tasks in both the AI and Operation Research communities. Although solutions to these problems are intractable in general, there might be special cases, such as structured POMDPs, which can be solved efficiently. A natural and possibly efficient way to represent a POMDP is through the predictive state representation (PSR) — a representation which recently has been receiving increasing attention. In this work, we relate POMDPs to multiplicity automata — showing that POMDPs can be represented by multiplicity automata with no increase in the representation size. Furthermore, we show that the size of the multiplicity automaton is equal to the rank of the predictive state representation. Therefore, we relate both the predictive state representation and POMDPs to the well-founded multiplicity automata literature. Based on the multiplicity automata representation, we provide a planning algorithm which is exponential only in the multiplicity automata rank rather than the number of states of the POMDP. As a result, whenever the predictive state representation is logarithmic in the standard POMDP representation, our planning algorithm is efficient. In this paper, we bring techniques from operations research to bear on the problem of choosing optimal actions in partially observable stochastic domains. We begin by introducing the theory of Markov decision processes (mdps) and partially observable MDPs (pomdps). We then outline a novel algorithm for solving pomdps off line and show how, in some cases, a finite-memory controller can be extracted from the solution to a POMDP. We conclude with a discussion of how our approach relates to previous work, the complexity of finding exact solutions to pomdps, and of some possibilities for finding approximate solutions. This paper explains how Partially Observable Markov Decision Processes (POMDPs) can provide a principled mathematical framework for modelling the inherent uncertainty in spoken dialogue systems. It briefly summarises the basic mathematics and explains why exact optimisation is intractable. It then describes in some detail a form of approximation called the Hidden Information State model which does scale and which can be used to build practical systems. A prototype HIS system for the tourist information domain is evaluated and compared with a baseline MDP system using both user simulations and a live user trial. The results give strong support to the central contention that the POMDP-based framework is both a tractable and powerful approach to building more robust spoken dialogue systems. This paper introduces a Monte-Carlo algorithm for online planning in large POMDPs. The algorithm combines a Monte-Carlo update of the agent's belief state with a Monte-Carlo tree search from the current belief state. The new algorithm, POMCP, has two important properties. First, Monte-Carlo sampling is used to break the curse of dimensionality both during belief state updates and during planning. Second, only a black box simulator of the POMDP is required, rather than explicit probability distributions. These properties enable POMCP to plan effectively in significantly larger POMDPs than has previously been possible. We demonstrate its effectiveness in three large POMDPs. We scale up a well-known benchmark problem, rocksample, by several orders of magnitude. We also introduce two challenging new POMDPs: 10 x 10 battleship and partially observable PacMan, with approximately 1018 and 1056 states respectively. Our Monte-Carlo planning algorithm achieved a high level of performance with no prior knowledge, and was also able to exploit simple domain knowledge to achieve better results with less search. POMCP is the first general purpose planner to achieve high performance in such large and unfactored POMDPs. Partially Observable Markov Decision Processes (POMDPs) model sequential decision-making problems under uncertainty and partial observability. Unfortunately, some problems cannot be modeled with state-dependent reward functions, e.g., problems whose objective explicitly implies reducing the uncertainty on the state. To that end, we introduce ρPOMDPs, an extension of POMDPs where the reward function ρ depends on the belief state. We show that, under the common assumption that ρ is convex, the value function is also convex, what makes it possible to (1) approximate ρ arbitrarily well with a piecewise linear and convex (PWLC) function, and (2) use state-of-the-art exact or approximate solving algorithms with limited changes. Partially observable Markov decision processes (POMDPs) are used in many robotic task classes from soccer to household chores. Determining an approximately optimal action policy for POMDPs is PSPACE-complete, and the exponential growth of computation time prohibits solving large tasks. This paper describes two techniques to extend the range of robotic tasks that can be solved using a POMDP. Our first technique reduces the motion constraints of a robot and, then, uses state-of-the-art robotic motion planning techniques to respect the true motion constraints at runtime. We then propose a novel task decomposition that can be applied to some indoor robotic tasks. This decomposition transforms a long time horizon task into a set of shorter tasks. We empirically demonstrate the performance gain provided by these two techniques through simulated execution in a variety of environments. Comparing a direct formulation of a POMDP to solving our proposed reductions, we conclude that the techniques proposed in this paper can provide significant enhancement to current POMDP solution techniques, extending the POMDP instances that can be solved to include large continuous-state robotic tasks. There has been substantial progress on algorithms for single-agent sequential decision making using partially observable Markov decision processes (POMDPs). A number of efficient algorithms for solving POMDPs share two desirable properties: error-bounds and fast convergence rates. Despite significant efforts, no algorithms for solving decentralized POMDPs benefit from these properties, leading to either poor solution quality or limited scalability. This paper presents the first approach for solving transition independent decentralized Markov decision processes (Dec-MDPs), that inherits these properties. Two related algorithms illustrate this approach. The first recasts the original problem as a deterministic and completely observable Markov decision process. In this form, the original problem is solved by combining heuristic search with constraint optimization to quickly converge into a near-optimal policy. This algorithm also provides the foundation for the first algorithm for solving infinite-horizon transition independent decentralized MDPs. We demonstrate that both methods outperform state-of-the-art algorithms by multiple orders of magnitude, and for infinite-horizon decentralized MDPs, the algorithm is able to construct more concise policies by searching cyclic policy graphs. The partially observable Markov decision process (POMDP) provides a principled general framework for planning under uncertainty, but solving POMDPs optimally is computationally intractable, due to the "curse of dimensionality" and the "curse of history". To overcome these challenges, we introduce the Determinized Sparse Partially Observable Tree (DESPOT), a sparse approximation of the standard belief tree, for online planning under uncertainty. A DESPOT focuses online planning on a set of randomly sampled scenarios and compactly captures the "execution" of all policies under these scenarios. We show that the best policy obtained from a DESPOT is near-optimal, with a regret bound that depends on the representation size of the optimal policy. Leveraging this result, we give an anytime online planning algorithm, which searches a DESPOT for a policy that optimizes a regularized objective function. Regularization balances the estimated value of a policy under the sampled scenarios and the policy size, thus avoiding overfitting. The algorithm demonstrates strong experimental results, compared with some of the best online POMDP algorithms available. It has also been incorporated into an autonomous driving system for real-time vehicle control. The source code for the algorithm is available online. Learning to act optimally in a complex, dynamic and noisy environment is a hard problem. Various threads of research from reinforcement learning, animal conditioning, operations research, machine learning, statistics and optimal control are beginning to come together to offer solutions to this problem. I present a thesis in which novel algorithms are presented for learning the dynamics, learning the value function, and selecting good actions for Markov decision processes. The problems considered have high-dimensional factored state and action spaces, and are either fully or partially observable. The approach I take is to recognize similarities between the problems being solved in the reinforcement learning and graphical models literature, and to use and combine techniques from the two fields in novel ways. In particular I present two new algorithms. First, the DBN algorithm learns a compact representation of the core process of a partially observable MDP. Because inference in the DBN is intractable, I use approximate inference to maintain the belief state. A belief state action-value function is learned using reinforcement learning. I show that this DBN algorithm can solve POMDPs with very large state spaces and useful hidden state. Second, the PoE algorithm learns an approximation to value functions over large factored state-action spaces. The algorithm approximates values as (negative) free energies in a product of experts model. The model parameters can be learned efficiently because inference is tractable in a product of experts. I show that good actions can be found even in large factored action spaces by the use of brief Gibbs sampling. These two new algorithms take techniques from the machine learning community and apply them in new ways to reinforcement learning problems. Simulation results show that these new methods can be used to solve very large problems. The DBN method is used to solve a POMDP with a hidden state space and an observation space of size greater than 2180. The DBN model of the core process has 232 states represented as 32 binary variables. The PoE method is used to find actions in action spaces of size 240 . Solving large Partially Observable Markov Decision Processes (POMDPs) is a complex task which is often intractable. A lot of effort has been made to develop approximate offline algorithms to solve ever larger POMDPs. However, even state-of-the-art approaches fail to solve large POMDPs in reasonable time. Recent developments in online POMDP search suggest that combining offline computations with online computations is often more efficient and can also considerably reduce the error made by approximate policies computed offline. In the same vein, we propose a new anytime online search algorithm which seeks to minimize, as efficiently as possible, the error made by an approximate value function computed offline. In addition, we show how previous online computations can be reused in following time steps in order to prevent redundant computations. Our preliminary results indicate that our approach is able to tackle large state space and observation space efficiently and under real-time constraints. Partially observable Markov decision processes (POMDPs) have been successfully applied to various robot motion planning tasks under uncertainty. However, most existing POMDP algorithms assume a discrete state space, while the natural state space of a robot is often continuous. This paper presents Monte Carlo Value Iteration (MCVI) for continuous-state POMDPs. MCVI samples both a robot’s state space and the corresponding belief space, and avoids inefficient a priori discretization of the state space as a grid. Both theoretical results and preliminary experimental results indicate that MCVI is a promising new approach for robot motion planning under uncertainty. Partially Observable Markov Decision Processes (POMDPs) provide a rich framework for sequential decision-making under uncertainty in stochastic domains. However, solving a POMDP is often intractable except for small problems due to their complexity. Here, we focus on online approaches that alleviate the computational complexity by computing good local policies at each decision step during the execution. Online algorithms generally consist of a lookahead search to find the best action to execute at each time step in an environment. Our objectives here are to survey the various existing online POMDP methods, analyze their properties and discuss their advantages and disadvantages; and to thoroughly evaluate these online approaches in different environments under various metrics (return, error bound reduction, lower bound improvement). Our experimental results indicate that state-of-the-art online heuristic search methods can handle large POMDP domains efficiently. This paper presents a generic approach for tactical decision-making under uncertainty in the context of driving. The complexity of this task mainly stems from the fact that rational decision-making in this context must consider several sources of uncertainty: The temporal evolution of situations cannot be predicted without uncertainty because other road users behave stochastically and their goals and plans cannot be measured. Even more important, road users are only able to perceive a tiny part of the current situation with their sensors because measurements are noisy and most of the environment is occluded. In order to anticipate the consequences of decisions a probabilistic approach, considering both forms of uncertainty, is necessary. The authors address this by formulating the task of driving as a continuous Partially Observable Markov Decision Process (POMDP) that can be automatically optimized for different scenarios. As driving is a continuous-space problem, the belief space is infinite-dimensional. The authors do not use a symbolic representation or discretize the state space a priori because there is no representation of the state space that is optimal for every situation. Instead, they employ a continuous POMDP solver that learns a good representation of the specific situation.