Multi-Objective Model Checking of Markov Decision Processes (0810.5728v2)

Abstract: 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.

• The paper develops a polynomial-time algorithm to determine and generate strategies meeting specified probability thresholds in MDPs.
• It computes an approximate Pareto curve to effectively balance trade-offs among different linear-time objectives.
• The study integrates graph-based qualitative and LP-based quantitative methods to robustly address multi-objective model checking.

Multi-Objective Model Checking of Markov Decision Processes

This paper addresses the problem of multi-objective model checking in Markov Decision Processes (MDPs), a critical tool for stochastic optimization and modeling systems with probabilistic and nondeterministic behaviors. The research presented here focuses on the development of efficient algorithms to decide the existence and, if applicable, produce strategies for controllers that can satisfy multiple linear-time properties each with a specified probability threshold.

Problem and Approach

The problem is formalized as follows: Given an MDP $M$, multiple linear-time properties $\varphi_i$, and desired probability thresholds $r_i$, the objective is to determine if there exists a strategy $\sigma$ such that the probability of each property $\varphi_i$ is satisfied by the trajectories of $M$ under strategy $\sigma$ with at least probability $r_i$.

The paper proposes an algorithm that runs in polynomial time in the size of the MDP, determining whether such a strategy exists and, if so, producing it. Importantly, these strategies may necessitate the use of both randomization and memory.

Key Contributions

1. Algorithm Development: The authors develop an algorithm that is efficient in determining multi-objective satisfaction and generating corresponding strategies. This algorithm handles the complexity of the trade-offs between different properties effectively.
2. Pareto Curve Computation: The paper introduces methods for computing an approximate Pareto curve for maximizing the probabilities of different properties. The authors show that this approximation can be computed in polynomial time, providing a practical tool for analyzing the trade-offs between different objectives.
3. Qualitative and Quantitative Analysis: The paper explores both qualitative (graph-theoretic) and quantitative (LP-based) methods to address strategy construction for satisfying multi-objective properties. For qualitative queries, graph-theoretic approaches yield strategies, while for quantitative queries, the problem is reduced to a linear programming problem.
4. Extended Problem Consideration: The research further generalizes to multi-objective queries through boolean combinations of quantitative predicates, thereby extending the application potential.

Implications and Future Directions

The implications of this work are profound for both theoretical exploration and practical applications. Theoretically, the methods provide a framework for addressing multi-objective properties in probabilistic verification. Practically, this research can impact the development of controllers in systems where multiple objectives must be satisfied simultaneously, such as in serving multiple clients in a networked environment.

The potential future developments include further investigation into symmetric assume-guarantee compositional reasoning and exploring more efficient or scalable methods to handle larger or more complex MDPs. Additionally, extending algorithmic techniques to other probabilistic models and leveraging these approaches in real-world system designs remain promising areas for future research.

In summary, this paper offers substantial advancement in the toolset available for model checking in probabilistic environments, providing algorithms and methods with both theoretical integrity and practical utility for multi-objective optimization in MDPs.