Reasoning About Knowledge on Regular Expressions is 2EXPTIME-complete

2025-08-13

AI Summary

This paper investigates the satisfiability problem for Public Observation Logic (POL), an extension of multi-agent modal logic that models knowledge evolution under dynamic, nondeterministic public observations via regular expressions. To formalize its semantics, we introduce a state-evolution framework grounded in Kripke structures and observation expectations, and reduce satisfiability to the acceptance problem of alternating Turing machines. We establish, for the first time, that POL satisfiability is 2EXPTIME-complete: we prove a 2EXPTIME upper bound using deterministic tree automata and model-checking techniques, and provide a tight 2EXPTIME-hardness reduction via a carefully constructed encoding. This resolves the long-standing open problem of POL’s exact computational complexity, delivering a precise theoretical boundary and rigorous formal foundation for observation-based knowledge update and planning in multi-agent systems.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Multiagent Systems: Mechanism Design

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: Incentives in network design for Web infrastructures and ecosystems

Abstract

Logics for reasoning about knowledge and actions have seen many applications in various domains of multi-agent systems, including epistemic planning. Change of knowledge based on observations about the surroundings forms a key aspect in such planning scenarios. Public Observation Logic (POL) is a variant of public announcement logic for reasoning about knowledge that gets updated based on public observations. Each state in an epistemic (Kripke) model is equipped with a set of expected observations. These states evolve as the expectations get matched with the actual observations. In this work, we prove that the satisfiability problem of $POL$ is 2EXPTIME-complete.

Problem

• Study reasoning about knowledge with regular expressions
• Analyze Public Observation Logic satisfiability complexity
• Prove POL satisfiability is 2EXPTIME-complete

Innovation

• Public Observation Logic for knowledge updates
• Epistemic models with expected observations
• Proves POL satisfiability is 2EXPTIME-complete