Epistemic Skills: Reasoning about Knowledge and Oblivion

TL;DR

The paper develops weighted epistemic logics with implicit epistemic skills to model knowledge updates, knowability, and forgettability in both individual and group contexts, introducing operators for upskilling, downskilling, reskilling, and learning. It provides a precise syntax (including $\mathcal{L}_{CDEF+-=\equiv\boxplus\boxminus\Box}$) and a robust semantics on weighted Kripke models, and it connects the framework to rough-set concepts for data-driven abstraction. The authors establish detailed complexity results: model checking is in P for logics without quantifiers (and remains P with updates), while quantified variants push to PSPACE-complete, and satisfiability ranges from PSPACE-complete (without common knowledge) to EXPTIME-complete (with common knowledge) depending on the presence of updates and quantifiers, with a suite of reductions underpinning these bounds. They also introduce variants (fuzzy sets and lattices) and a refined treatment of de re/de dicto readings, demonstrating the framework’s flexibility and potential applicability to data-intensive and dynamic epistemic settings.

Abstract

This paper presents a class of epistemic logics that captures the dynamics of acquiring knowledge and descending into oblivion, while incorporating concepts of group knowledge. The approach is grounded in a system of weighted models, introducing an ``epistemic skills'' metric to represent the epistemic capacities tied to knowledge updates. Within this framework, knowledge acquisition is modeled as a process of upskilling, whereas oblivion is represented as a consequence of downskilling. The framework further enables exploration of ``knowability'' and ``forgettability,'' defined as the potential to gain knowledge through upskilling and to lapse into oblivion through downskilling, respectively. Additionally, it supports a detailed analysis of the distinctions between epistemic de re and de dicto expressions. The computational complexity of the model checking and satisfiability problems is examined, offering insights into their theoretical foundations and practical implications.

Figures (5)

• Figure 1: Illustration of the model $M$. Curly brackets are omitted from set labels for brevity. Edges labeled with the empty set, such as between $w_1$ and $w_4$, indicate universal distinguishability---except by totally incompetent agents (those with an empty skill set)---and are not depicted in the diagram.
• Figure 2: Illustration of $G_0 = ( \{d_1, d_2, d_3, d_4 \}, \{ \{d_1, d_3\}, \{d_1, d_4\}, \{d_2, d_4\}, \{d_3, d_4\} \} )$.
• Figure 3: Illustration of the induced model $M_{G_0}$, with $G_0$ illustrated in Figure \ref{['fig:UEG']}.
• Figure 4: Roadmap of proofs for the complexity of satisfiability problems for logics between $\text{\normalfont L}\xspace$ and $\text{\normalfont L$_{DEF}$}\xspace$. Logics under study are in elliptical frames, while known PSPACE-complete satisfiability problems are in rectangular frames. A solid arrow from one logic to another represents the satisfiability problem for the former logic as a subproblem of the satisfiability problem for the latter. A dashed arrow labeled "PTIME" from one logic to another indicates a polynomial-time reduction from the satisfiability problem for the former to that for the latter. References: K$^D_n$ from FHMV1995 (subscript denotes the number of agents); KB$_1$ is folklore, with a proof in Sahlqvist1975 (named "KB," citing a 1992 manuscript).
• Figure 5: Roadmap of proofs for the complexity of satisfiability problems for logics with common knowledge, excluding update and quantifying modalities. Boxed nodes display known complexity results. A solid arrow from one logic to another indicates that the satisfiability problem for the former logic is a subproblem for the latter. A dashed arrow labeled "PTIME" denotes a polynomial-time reduction from the satisfiability problem for the source logic to that of the target logic. The EXPTIME completeness of S5$^C_2$ is from FHMV1995. The EXPTIME upper bound for CPDL is from PT1991. The EXPTIME completeness of K$^U_2$ is from Spaan1993.

Theorems & Definitions (46)

• Definition 2.2
• Definition 2.3
• Proposition 2.4
• proof
• Lemma 3.2
• Definition 3.3
• Proposition 3.4
• Lemma 3.5
• proof
• Lemma 3.6