PReMo : An Analyzer for P robabilistic Re cursive Mo dels

This paper describes PReMo, a tool for analyzing Recursive Markov Chains, and their controlled/game extensions: (1-exit) Recursive Markov Decision Processes and Recursive Simple Stochastic Games

Qualitative Reachability in Stochastic BPA Games

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 PTIME for the `>0' constraint, and both in NP and coNP 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.Comment: Submitted to Information and Computation. 48 pages, 3 figure

Approximating the Termination Value of One-Counter MDPs and Stochastic Games

One-counter MDPs (OC-MDPs) and one-counter simple stochastic games (OC-SSGs) are 1-player, and 2-player turn-based zero-sum, stochastic games played on the transition graph of classic one-counter automata (equivalently, pushdown automata with a 1-letter stack alphabet). A key objective for the analysis and verification of these games is the termination objective, where the players aim to maximize (minimize, respectively) the probability of hitting counter value 0, starting at a given control state and given counter value. Recently, we studied qualitative decision problems ("is the optimal termination value = 1?") for OC-MDPs (and OC-SSGs) and showed them to be decidable in P-time (in NP and coNP, respectively). However, quantitative decision and approximation problems ("is the optimal termination value ? p", or "approximate the termination value within epsilon") are far more challenging. This is so in part because optimal strategies may not exist, and because even when they do exist they can have a highly non-trivial structure. It thus remained open even whether any of these quantitative termination problems are computable. In this paper we show that all quantitative approximation problems for the termination value for OC-MDPs and OC-SSGs are computable. Specifically, given a OC-SSG, and given epsilon > 0, we can compute a value v that approximates the value of the OC-SSG termination game within additive error epsilon, and furthermore we can compute epsilon-optimal strategies for both players in the game. A key ingredient in our proofs is a subtle martingale, derived from solving certain LPs that we can associate with a maximizing OC-MDP. An application of Azuma's inequality on these martingales yields a computable bound for the "wealth" at which a "rich person's strategy" becomes epsilon-optimal for OC-MDPs.Comment: 35 pages, 1 figure, full version of a paper presented at ICALP 2011, invited for submission to Information and Computatio

Recursive Concurrent Stochastic Games

We study Recursive Concurrent Stochastic Games (RCSGs), extending our recent analysis of recursive simple stochastic games to a concurrent setting where the two players choose moves simultaneously and independently at each state. For multi-exit games, our earlier work already showed undecidability for basic questions like termination, thus we focus on the important case of single-exit RCSGs (1-RCSGs). We first characterize the value of a 1-RCSG termination game as the least fixed point solution of a system of nonlinear minimax functional equations, and use it to show PSPACE decidability for the quantitative termination problem. We then give a strategy improvement technique, which we use to show that player 1 (maximizer) has \epsilon-optimal randomized Stackless & Memoryless (r-SM) strategies for all \epsilon > 0, while player 2 (minimizer) has optimal r-SM strategies. Thus, such games are r-SM-determined. These results mirror and generalize in a strong sense the randomized memoryless determinacy results for finite stochastic games, and extend the classic Hoffman-Karp strategy improvement approach from the finite to an infinite state setting. The proofs in our infinite-state setting are very different however, relying on subtle analytic properties of certain power series that arise from studying 1-RCSGs. We show that our upper bounds, even for qualitative (probability 1) termination, can not be improved, even to NP, without a major breakthrough, by giving two reductions: first a P-time reduction from the long-standing square-root sum problem to the quantitative termination decision problem for finite concurrent stochastic games, and then a P-time reduction from the latter problem to the qualitative termination problem for 1-RCSGs.Comment: 21 pages, 2 figure

Polynomial Time Algorithms for Multi-Type Branching Processes and Stochastic Context-Free Grammars

We show that one can approximate the least fixed point solution for a multivariate system of monotone probabilistic polynomial equations in time polynomial in both the encoding size of the system of equations and in log(1/\epsilon), where \epsilon > 0 is the desired additive error bound of the solution. (The model of computation is the standard Turing machine model.) We use this result to resolve several open problems regarding the computational complexity of computing key quantities associated with some classic and heavily studied stochastic processes, including multi-type branching processes and stochastic context-free grammars

One-Counter Stochastic Games

We study the computational complexity of basic decision problems for one-counter simple stochastic games (OC-SSGs), under various objectives. OC-SSGs are 2-player turn-based stochastic games played on the transition graph of classic one-counter automata. We study primarily the termination objective, where the goal of one player is to maximize the probability of reaching counter value 0, while the other player wishes to avoid this. Partly motivated by the goal of understanding termination objectives, we also study certain "limit" and "long run average" reward objectives that are closely related to some well-studied objectives for stochastic games with rewards. Examples of problems we address include: does player 1 have a strategy to ensure that the counter eventually hits 0, i.e., terminates, almost surely, regardless of what player 2 does? Or that the liminf (or limsup) counter value equals infinity with a desired probability? Or that the long run average reward is >0 with desired probability? We show that the qualitative termination problem for OC-SSGs is in NP intersection coNP, and is in P-time for 1-player OC-SSGs, or equivalently for one-counter Markov Decision Processes (OC-MDPs). Moreover, we show that quantitative limit problems for OC-SSGs are in NP intersection coNP, and are in P-time for 1-player OC-MDPs. Both qualitative limit problems and qualitative termination problems for OC-SSGs are already at least as hard as Condon's quantitative decision problem for finite-state SSGs.Comment: 20 pages, 1 figure. This is a full version of a paper accepted for publication in proceedings of FSTTCS 201

Qualitative Multi-Objective Reachability for Ordered Branching MDPs

We study qualitative multi-objective reachability problems for Ordered Branching Markov Decision Processes (OBMDPs), or equivalently context-free MDPs, building on prior results for single-target reachability on Branching Markov Decision Processes (BMDPs). We provide two separate algorithms for "almost-sure" and "limit-sure" multi-target reachability for OBMDPs. Specifically, given an OBMDP, $\mathcal{A}$, given a starting non-terminal, and given a set of target non-terminals $K$ of size $k = |K|$, our first algorithm decides whether the supremum probability, of generating a tree that contains every target non-terminal in set $K$, is $1$. Our second algorithm decides whether there is a strategy for the player to almost-surely (with probability $1$) generate a tree that contains every target non-terminal in set $K$. The two separate algorithms are needed: we show that indeed, in this context, "almost-sure" $\not=$ "limit-sure" for multi-target reachability, meaning that there are OBMDPs for which the player may not have any strategy to achieve probability exactly $1$ of reaching all targets in set $K$ in the same generated tree, but may have a sequence of strategies that achieve probability arbitrarily close to $1$. Both algorithms run in time $2^{O(k)} \cdot |\mathcal{A}|^{O(1)}$, where $|\mathcal{A}|$ is the total bit encoding length of the given OBMDP, $\mathcal{A}$. Hence they run in polynomial time when $k$ is fixed, and are fixed-parameter tractable with respect to $k$. Moreover, we show that even the qualitative almost-sure (and limit-sure) multi-target reachability decision problem is in general NP-hard, when the size $k$ of the set $K$ of target non-terminals is not fixed.Comment: 47 page

One-Counter Stochastic Games

We study the computational complexity of basic decision problems for one-counter simple stochastic games (OC-SSGs), under various objectives. OC-SSGs are 2-player turn-based stochastic games played on the transition graph of classic one-counter automata. We study primarily the termination objective, where the goal of one player is to maximize the probability of reaching counter value 0, while the other player wishes to avoid this. Partly motivated by the goal of understanding termination objectives, we also study certain ``limit\u27\u27 and ``long run average\u27\u27 reward objectives that are closely related to some well-studied objectives for stochastic games with rewards. Examples of problems we address include: does player 1 have a strategy to ensure that the counter eventually hits 0, i.e., terminates, almost surely, regardless of what player 2 does? Or that the $liminf$ (or $limsup$) counter value equals $infty$ with a desired probability? Or that the long run average reward is $>0$ with desired probability? We show that the qualitative termination problem for OC-SSGs is in $NP$ intersect $coNP$, and is in P-time for 1-player OC-SSGs, or equivalently for one-counter Markov Decision Processes (OC-MDPs). Moreover, we show that quantitative limit problems for OC-SSGs are in $NP$ intersect $coNP$, and are in P-time for 1-player OC-MDPs. Both qualitative limit problems and qualitative termination problems for OC-SSGs are already at least as hard as Condon\u27s quantitative decision problem for finite-state SSGs

Multi-Objective Model Checking of Markov Decision Processes

We study and provide efficient algorithms for multi-objective model checking problems for Markov Decision Processes (MDPs). Given an MDP, M, and given multiple linear-time (\omega -regular or LTL) properties \varphi\_i, and probabilities r\_i \epsilon [0,1], i=1,...,k, we ask whether there exists a strategy \sigma for the controller such that, for all i, the probability that a trajectory of M controlled by \sigma satisfies \varphi\_i is at least r\_i. We provide an algorithm that decides whether there exists such a strategy and if so produces it, and which runs in time polynomial in the size of the MDP. Such a strategy may require the use of both randomization and memory. We also consider more general multi-objective \omega -regular queries, which we motivate with an application to assume-guarantee compositional reasoning for probabilistic systems. Note that there can be trade-offs between different properties: satisfying property \varphi\_1 with high probability may necessitate satisfying \varphi\_2 with low probability. Viewing this as a multi-objective optimization problem, we want information about the "trade-off curve" or Pareto curve for maximizing the probabilities of different properties. We show that one can compute an approximate Pareto curve with respect to a set of \omega -regular properties in time polynomial in the size of the MDP. Our quantitative upper bounds use LP methods. We also study qualitative multi-objective model checking problems, and we show that these can be analysed by purely graph-theoretic methods, even though the strategies may still require both randomization and memory.Comment: 21 pages, 2 figure