Efficient Algorithms for Approximating the Limit Value and Parity Value in Concurrent Stochastic Games

Core Concepts

The authors present improved complexity results and algorithms for approximating the limit value of stateful-discounted objectives and the parity value in concurrent stochastic games.

Abstract

The content discusses concurrent stochastic games, which are two-player zero-sum games played on finite-state graphs for an infinite number of steps. In each step, both players simultaneously and independently choose an action, and the next state is obtained according to a stochastic transition function.
The authors consider two types of objectives:

Stateful-discounted objectives: Similar to the classical discounted-sum objectives, but each state is associated with a different discount factor.
Parity objectives: A canonical representation for ω-regular objectives, where each state is associated with a priority, and the objective is to minimize the minimum priority visited infinitely often.

The main computational problems are the value-approximation problems, which aim to compute an approximation of the value within an arbitrary additive error.
The authors establish the following results:

The value-approximation problem for the limit value of stateful-discounted objectives and the parity value are in TFNP[NP], improving the previous EXPSPACE and PSPACE upper bounds, respectively.
The authors present algorithms that improve the dependency on the number of actions in the exponent from linear to logarithmic. In particular, if the number of states is constant, the algorithms run in polynomial time.

The technical contributions include:

A bound on the roots of multi-variate polynomials with integer coefficients, which is used to establish a connection between the stateful-discounted value and the limit value.
New characterizations of the stateful-discounted value and the limit value, leading to the improved complexity and algorithmic results.

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Key Insights Distilled From

Concurrent Stochastic Games with Stateful-discounted and Parity Objectives: Complexity and Algorithms

by Ali Asadi,Kr... at arxiv.org 05-07-2024

https://arxiv.org/pdf/2405.02486.pdf

Deeper Inquiries

How can the techniques developed in this work be extended to other types of objectives in concurrent stochastic games, such as mean-payoff or total-reward objectives

The techniques developed in this work for stateful-discounted and parity objectives in concurrent stochastic games can be extended to other types of objectives, such as mean-payoff or total-reward objectives, by adapting the mathematical characterizations and algorithms to suit the specific objective functions. For mean-payoff objectives, which involve maximizing the average reward per step, the stateful-discounted approach can be modified to consider the average rewards over time. This may involve adjusting the discount factors or the reward functions to capture the mean-payoff criteria. Similarly, for total-reward objectives that aim to maximize the cumulative reward over the entire play, the algorithms can be tailored to optimize the total rewards obtained in the game. By incorporating the appropriate definitions and constraints specific to each objective type, the techniques developed in this work can be applied effectively to a broader range of objectives in concurrent stochastic games.

What are the practical implications of the improved complexity and algorithmic results presented in this work

The improved complexity and algorithmic results presented in this work have significant practical implications for the analysis and verification of real-world reactive systems. The ability to approximate the value of concurrent stochastic games with stateful-discounted and parity objectives within an arbitrary additive error in polynomial time, especially when the number of states is constant, can streamline the verification process for complex systems. This can lead to more efficient model checking and validation of system designs, ensuring their correctness and reliability. The algorithms developed in this work can be utilized in various applications, such as autonomous systems, cybersecurity protocols, and communication networks, where the behavior of multiple interacting components needs to be analyzed under uncertainty and probabilistic transitions. By providing faster and more accurate solutions to value approximation problems in concurrent stochastic games, the results of this work can enhance the scalability and effectiveness of formal verification techniques in verifying the correctness of reactive systems.

How can they impact the analysis and verification of real-world reactive systems

There are connections between the techniques used in this work and the theory of real-closed fields or the theory of reals, particularly in the context of computational complexity and algorithm design. The approach of reducing the value approximation problems in concurrent stochastic games to the theory of reals, as demonstrated in this work, showcases the relevance of real number computations in solving complex decision problems efficiently. By leveraging the properties of real-closed fields and the theory of reals, researchers can further explore the computational boundaries and optimize the algorithms for solving value approximation problems in various domains. Investigating these connections can lead to novel insights into the computational complexity of decision-making processes and provide avenues for developing more efficient algorithms for analyzing stochastic systems with different types of objectives.