An Undecidability Proof for the Plan Existence Problem

2026-04-24

AI Summary

This study addresses the planning existence problem in epistemic reasoning: given a modal-logic goal, an initial epistemic state, and a set of epistemic actions, does there exist a sequence of actions that achieves the goal? Focusing on a highly restricted setting where epistemic actions have preconditions of modal depth at most one and no postconditions, the analysis employs modal logic, Kripke semantics, and formal reduction techniques. The work establishes, for the first time, that even under these strong syntactic and semantic constraints, the planning existence problem remains undecidable. This result fills a critical gap in the decidability landscape of epistemic planning and delineates new theoretical boundaries for automated planning and cognitive reasoning systems.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Planning, Routing, and Scheduling: Model-Based Reasoning

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation

Abstract

The plan existence problem asks, given a goal in the form of a formula in modal logic, an initial epistemic state (a pointed Kripke model), and a set of epistemic actions, whether there exists a sequence of actions that can be applied to reach the goal. We prove that even in the case where the preconditions of the epistemic actions have modal depth at most 1, and there are no postconditions, the plan existence problem is undecidable. The (un)decidability of this problem was previously unknown.

Problem

• plan existence problem
• epistemic actions
• modal logic
• undecidability
• Kripke model

Innovation

• plan existence problem
• undecidability
• epistemic planning
• modal logic
• Kripke model