Variable Forgetting in Reasoning about Knowledge

In this paper, we investigate knowledge reasoning within a simple framework called knowledge structure. We use variable forgetting as a basic operation for one agent to reason about its own or other agents\ knowledge. In our framework, two notions namely agents\ observable variables and the weakest sufficient condition play important roles in knowledge reasoning. Given a background knowledge base and a set of observable variables for each agent, we show that the notion of an agent knowing a formula can be defined as a weakest sufficient condition of the formula under background knowledge base. Moreover, we show how to capture the notion of common knowledge by using a generalized notion of weakest sufficient condition. Also, we show that public announcement operator can be conveniently dealt with via our notion of knowledge structure. Further, we explore the computational complexity of the problem whether an epistemic formula is realized in a knowledge structure. In the general case, this problem is PSPACE-hard; however, for some interesting subcases, it can be reduced to co-NP. Finally, we discuss possible applications of our framework in some interesting domains such as the automated analysis of the well-known muddy children puzzle and the verification of the revised Needham-Schroeder protocol. We believe that there are many scenarios where the natural presentation of the available information about knowledge is under the form of a knowledge structure. What makes it valuable compared with the corresponding multi-agent S5 Kripke structure is that it can be much more succinct.

💡 Research Summary

The paper introduces a compact formalism called a “knowledge structure” for reasoning about knowledge in multi‑agent systems. Traditional multi‑agent S5 Kripke models are expressive but suffer from a combinatorial explosion of worlds, making them impractical for large systems. To address this, the authors adopt variable forgetting—a logical operation that eliminates a set of variables while preserving all consequences concerning the remaining variables—as the core tool for knowledge representation.

A knowledge structure consists of (i) a global background knowledge base Γ, which encodes all static facts and constraints of the system, and (ii) for each agent i, a set O_i of observable variables that the agent can directly perceive. The observable set captures the limited information available to an agent in realistic settings.

Variable forgetting is defined as follows: given a formula φ and a set of variables V, the forgetting of V in φ produces the weakest formula ψ (the “weakest sufficient condition”, WSC) such that ψ entails φ under Γ and ψ mentions none of the variables in V. Intuitively, ψ expresses exactly what can be guaranteed about φ without referring to the forgotten variables.

Using this notion, the epistemic operator K_i α (agent i knows α) is defined as the WSC of α with respect to O_i under Γ. Formally, K_i α holds iff there exists a ψ that mentions only variables in O_i, Γ ∪ ψ ⊨ α, and ψ is the weakest such formula. This captures the idea that i can infer α solely from the information it can observe.

Common knowledge C_G α for a group G is obtained by extending the WSC concept to the union of all agents’ observable variables O_G = ⋃_{i∈G} O_i and iterating the forgetting operation to reflect the mutual awareness required for common knowledge. The resulting generalized WSC coincides with the standard fixed‑point definition of common knowledge but can be computed via variable elimination.

The public announcement operator