The Complexity of Limited Belief Reasoning -- The Quantifier-Free Case

TL;DR

This paper studies the computational complexity of limited belief reasoning, showing it is tractable at fixed levels but becomes PSPACE-complete when the belief level varies, and analyzes how parameters influence complexity.

Contribution

It provides a detailed complexity analysis of belief reasoning with levels, including parameterized complexity, highlighting the computational challenges involved.

Findings

Reasoning is tractable at fixed belief levels.

Complexity becomes PSPACE-complete when belief level is variable.

Parameterization reveals how belief level and symbols affect complexity.

Abstract

The classical view of epistemic logic is that an agent knows all the logical consequences of their knowledge base. This assumption of logical omniscience is often unrealistic and makes reasoning computationally intractable. One approach to avoid logical omniscience is to limit reasoning to a certain belief level, which intuitively measures the reasoning "depth." This paper investigates the computational complexity of reasoning with belief levels. First we show that while reasoning remains tractable if the level is constant, the complexity jumps to PSPACE-complete -- that is, beyond classical reasoning -- when the belief level is part of the input. Then we further refine the picture using parameterized complexity theory to investigate how the belief level and the number of non-logical symbols affect the complexity.