Abstract.We present a general framework for applying machine-learning algorithms to the verification of Markov decision processes (MDPs). The primary goal of these techniques is to improve performance by avoiding an exhaustive exploration of the state space. Our framework focuses on probabilistic reachability, which is a core property for verification, and is illustrated through two distinct instantiations. The first assumes that full knowledge of the MDP is available, and performs a heuristic-driven partial exploration of the model, yielding precise lower and upper bounds on the required probability. The second tackles the case where we may only sample the MDP, and yields probabilistic guarantees, again in terms of both the lower and upper bounds, which provides efficient stopping criteria for the approximation. The latter is the first extension of statistical model-checking for unbounded properties in MDPs. In contrast with other related approaches, we do not restrict our attention to time-bounded (finite-horizon) or discounted properties, nor assume any particular properties of the MDP. We also show how our techniques extend to LTL objectives. We present experimental results showing the performance of our framework on several examples.

We consider two-player turn-based games with zero-reachability and zero-safety objectives generated by extended vector addition systems with states. Although the problem of deciding the winner in such games is undecidable in general, we identify several decidable and even tractable subcases of this problem obtained by restricting the number of counters and/or the sets of target configurations.

We study the computational complexity of central analysis problems for One-Counter Markov Decision Processes (OC-MDPs), a class of finitely-presented, countable-state MDPs. OC-MDPs extend finite-state MDPs with an unbounded counter. The counter can be incremented, decremented, or not changed during each state transition, and transitions may be enabled or not depending on both the current state and on whether the counter value is 0 or not. Some states are "random", from where the next transition is chosen according to a given probability distribution, while other states are "controlled", from where the next transition is chosen by the controller. Different objectives for the controller give rise to different computational problems, aimed at computing optimal achievable objective values and optimal strategies. OC-MDPs are in fact equivalent to a controlled extension of (discrete-time) Quasi-Birth-Death processes (QBDs), a purely stochastic model heavily studied in queueing theory and applied probability. They can thus be viewed as a natural "adversarial" extension of a classic stochastic model. They can also be viewed as a natural probabilistic/controlled extension of classic onecounter automata. OC-MDPs also subsume (as a very restricted special case) a recently studied MDP model called "solvency games" that model a risk-averse gambling scenario. Basic computational questions for OC-MDPs include "termination" questions and "limit" questions, such as the following: does the controller have a strategy to ensure that the counter (which may, for example, count the number of jobs in the queue) will hit value 0 (the empty queue) almost surely (a.s.)? Or that the counter will have lim sup value ∞, a.s.? Or, that it will hit value 0 in a selected terminal state, a.s.? Or, in case such properties are not satisfied almost surely, compute their optimal probability over all strategies. We provide new upper and lower bounds on the complexity of such problems. Specifically, we show that several quantitative and almost-sure limit problems can be answered in polynomial time, and that almost-sure termination problems (without selection of desired terminal states) can also be answered in polynomial time. On the other hand, we show that the almost-sure termination problem with selected terminal states is PSPACE-hard and we provide an exponential time algorithm for this problem. We also characterize classes of strategies that suffice for optimality in several of these settings. Our upper bounds combine a number of techniques from the theory of MDP reward models, the theory of random walks, and a variety of automata-theoretic methods.given probability. Such questions are similar in spirit to questions asked in the rich literature on "adversarial queueing theory" (see, e.g., [4]), although this is a somewhat different setting. These considerations lead naturally to the extension of QBDs with control, and thus to OC-MDPs. Indeed, MDP variants of QBDs have already been studied in the stochastic modeling literature, see [27,19]. However, ...

We consider qualitative and quantitative model-checking problems for probabilistic pushdown automata (pPDA) and various temporal logics. We prove that the qualitative and quantitative model-checking problem for ω-regular properties and pPDA is in 2-EXPSPACE and 3-EXPTIME, respectively. We also prove that model-checking the qualitative fragment of the logic PECTL * for pPDA is in 2-EXPSPACE, and model-checking the qualitative fragment of PCTL for pPDA is in EXPSPACE. Furthermore, model-checking the qualitative fragment of PCTL is shown to be EXPTIME-hard even for stateless pPDA. Finally, we show that PCTL model-checking is undecidable for pPDA, and PCTL + model-checking is undecidable even for stateless pPDA.

Abstract. We show that a subclass of infinite-state probabilistic programs that can be modeled by probabilistic one-counter automata (pOC) admits an efficient quantitative analysis. In particular, we show that the expected termination time can be approximated up to an arbitrarily small relative error with polynomially many arithmetic operations, and the same holds for the probability of all runs that satisfy a given ω-regular property. Further, our results establish a powerful link between pOC and martingale theory, which leads to fundamental observations about quantitative properties of runs in pOC. In particular, we provide a "divergence gap theorem", which bounds a positive non-termination probability in pOC away from zero.

We show that a subclass of infinite-state probabilistic programs that can be modeled by probabilistic one-counter automata (pOC) admits an efficient quantitative analysis. In particular, we show that the expected termination time can be approximated up to an arbitrarily small relative error with polynomially many arithmetic operations, and the same holds for the probability of all runs that satisfy a given ω-regular property. Further, our results establish a powerful link between pOC and martingale theory, which leads to fundamental observations about quantitative properties of runs in pOC. In particular, we provide a "diver-gence gap theorem", which bounds a positive non-termination probability in pOC away from zero.

We consider a class of infinite-state stochastic games generated by stateless pushdown automata (or, equivalently, 1-exit recursive state machines), where the winning objective is specified by a regular set of target configurations and a qualitative probability constraint '>0' or '=1'. The goal of one player is to maximize the probability of reaching the target set so that the constraint is satisfied, while the other player aims at the opposite. We show that the winner in such games can be determined in P for the '>0' constraint, and in NP ∩ co-NP for the '=1' constraint. Further, we prove that the winning regions for both players are regular, and we design algorithms which compute the associated finite-state automata. Finally, we show that winning strategies can be synthesized effectively.

Abstract. We study Markov decision processes (MDPs) with multiple limit-average (or mean-payoff) functions. We consider two different objectives, namely, expectation and satisfaction objectives. Given an MDP with k k k limit-average functions, in the expectation objective the goal is to maximize the expected limit-average value, and in the satisfaction objective the goal is to maximize the probability of runs such that the limit-average value stays above a given vector. We show that under the expectation objective, in contrast to the case of one limit-average function, both randomization and memory are necessary for strategies even for ε ε ε-approximation, and that finite-memory randomized strategies are sufficient for achieving Pareto optimal values. Under the satisfaction objective, in contrast to the case of one limit-average function, infinite memory is necessary for strategies achieving a specific value (i.e. randomized finite-memory strategies are not sufficient), whereas memoryless randomized strategies are sufficient for ε ε ε-approximation, for all ε > 0 ε > 0 ε > 0. We further prove that the decision problems for both expectation and satisfaction objectives can be solved in polynomial time and the trade-off curve (Pareto curve) can be ε ε ε-approximated in time polynomial in the size of the MDP and, and exponential in the number of limitaverage functions, for all ε > 0 ε > 0 ε > 0. Our analysis also reveals flaws in previous work for MDPs with multiple mean-payoff functions under the expectation objective, corrects the flaws, and allows us to obtain improved results. ACM CCS: [Mathematics