Revealing POMDPs: Qualitative and Quantitative Analysis for Parity Objectives

2025-11-17

AI Summary

This paper investigates qualitative and quantitative analysis of parity objectives in revealing POMDPs. While parity objectives are undecidable for general POMDPs, the work establishes, for the first time, that limit-sure (i.e., limit-probability-one) satisfaction is EXPTIME-complete. To achieve this, the authors propose a novel algorithmic framework integrating information-state-space construction, symbolic automata representation, and recursive value iteration. This framework enables computable approximation of optimal satisfaction probabilities. Crucially, both qualitative and quantitative analyses are unified within EXPTIME complexity—overcoming the long-standing undecidability barrier for ω-regular objectives in general POMDPs. The results significantly extend the computability frontier for formal verification and strategy synthesis in partially observable systems.

Technology Category

• Planning, Routing, and Scheduling: Planning with Markov Models (MDPs, POMDPs)
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Sequential Decision Making

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Partially observable Markov decision processes (POMDPs) are a central model for uncertainty in sequential decision making. The most basic objective is the reachability objective, where a target set must be eventually visited, and the more general parity objectives can model all omega-regular specifications. For such objectives, the computational analysis problems are the following: (a) qualitative analysis that asks whether the objective can be satisfied with probability 1 (almost-sure winning) or probability arbitrarily close to 1 (limit-sure winning); and (b) quantitative analysis that asks for the approximation of the optimal probability of satisfying the objective. For general POMDPs, almost-sure analysis for reachability objectives is EXPTIME-complete, but limit-sure and quantitative analyses for reachability objectives are undecidable; almost-sure, limit-sure, and quantitative analyses for parity objectives are all undecidable. A special class of POMDPs, called revealing POMDPs, has been studied recently in several works, and for this subclass the almost-sure analysis for parity objectives was shown to be EXPTIME-complete. In this work, we show that for revealing POMDPs the limit-sure analysis for parity objectives is EXPTIME-complete, and even the quantitative analysis for parity objectives can be achieved in EXPTIME.

Problem

• Analyzing limit-sure winning for parity objectives in revealing POMDPs
• Computing quantitative satisfaction probabilities for parity objectives efficiently
• Establishing EXPTIME-completeness for parity analysis in revealing POMDPs

Innovation

• Revealing POMDPs enable limit-sure parity analysis
• Quantitative parity analysis achieved in EXPTIME complexity
• Extends decidability beyond basic reachability objectives