A Logic of General Attention Using Edge-Conditioned Event Models (Extended Version)

TL;DR

The paper develops a general logic of attention within dynamic epistemic logic, addressing the limits of prior DEL approaches that only handle atomic propositions and face exponential growth. It introduces edge-conditioned event models (ECM), a succinct and expressive intermediate formalism that unifies standard event models and generalized arrow updates, and proves that ECMs can be exponentially more succinct for attention scenarios, while preserving full expressivity. The authors extend attention from atomic propositions to arbitrary formulas by introducing the general attention language L_GA with modalities A_a φ, and define attention models and corresponding event models for revelations of formula sets Γ. Through rigorous translations, update-equivalence results, and axiomatisations, the framework enables reasoning about complex attentional biases, social learning, and attention-driven dynamics with potential AI applications in bias detection and robust learning in multi-agent systems. This work lays groundwork for analyzing how attentional focus shapes belief revision and how agents can reason about others' attention, with implications for AI safety and socially aware reasoning.

Abstract

In this work, we present the first general logic of attention. Attention is a powerful cognitive ability that allows agents to focus on potentially complex information, such as logically structured propositions, higher-order beliefs, or what other agents pay attention to. This ability is a strength, as it helps to ignore what is irrelevant, but it can also introduce biases when some types of information or agents are systematically ignored. Existing dynamic epistemic logics for attention cannot model such complex attention scenarios, as they only model attention to atomic formulas. Additionally, such logics quickly become cumbersome, as their size grows exponentially in the number of agents and announced literals. Here, we introduce a logic that overcomes both limitations. First, we generalize edge-conditioned event models, which we show to be as expressive as standard event models yet exponentially more succinct (generalizing both standard event models and generalized arrow updates). Second, we extend attention to arbitrary formulas, allowing agents to also attend to other agents' beliefs or attention. Our work treats attention as a modality, like belief or awareness. We introduce attention principles that impose closure properties on that modality and that can be used in its axiomatization. Throughout, we illustrate our framework with examples of AI agents reasoning about human attentional biases, demonstrating how such agents can discover attentional biases.

Figures (7)

• Figure 1: A pointed Kripke model $(\mathcal{M},w)$ for $\mathcal{L}_{\text{PA}}$. In the figure, $p$ stands for "the applicant has published several papers in top-tier journals", $q$ for "the applicant has made significant contributions to diversity". There are two agents, Ann ($a$) and the AI agent ($b$), i.e. $Ag=\{a,b\}$. We use the following conventions. For a set of agents $Ag'$, $A_{Ag'}p := \bigwedge_{a\in Ag'} A_ap$. Worlds are represented either by a sequence of literals true at the world, or by such a sequence of literals where some of the atoms are followed by question marks: $p_1?,\dots,p_n?, \ell(q_1),\dots,\ell(q_m)$. This is shorthand for the set of $2^n$ worlds corresponding to all possible truth-value assignments of the atoms $p_1,p_2,...,p_n$, where $\ell(q_1), ...,\ell(q_m)$ are true at each of these worlds. When a world appears inside a dashed box, all the literals in the label of that box are also true at the world. The actual world is underlined. The accessibility relations are represented by labelled arrows. An arrow from (or to) the border of a box means that there is an arrow from (or to) all the events inside the box.
• Figure 2: Event model $\mathcal{F}(p\land q)$ from belardinelli2023attention. Solid edges are for Ann ($a$), dotted for the AI agent ($b$). The figure adopts the same conventions as in the cited paper: An event is represented by a list of literals, corresponding to (some of the) literals that appear in the conjunctive precondition of the event itself. The formulas in the label of a dashed box are to be included as conjuncts in the precondition of all events inside the box. The convention for the edges is the same as in Fig. \ref{['figure: propositional attention']}. See belardinelli2023attention for a detailed explanation of the figure.
• Figure 3: Edge-conditioned event model for propositional attention $\mathcal{H}(p\land q)$. Events are represented by conjunctive formulas corresponding to the event's own precondition. When for all agents $i\in Ag$, we have a (conditioned) edge $(e{:}\varphi_i,f{:}\psi_i) \in Q_i$, we add an arrow from $e$ to $f$ labelled by $i{:}(\varphi_i,\psi_i)$. This means that agent $i$ has an edge from $e$ to $f$ with source condition $\varphi_i$ and target condition $\psi_i$. For example, the arrow from event $p\wedge q$ to event $p$ labelled by $i{:}(A_ip\wedge\neg A_iq,A_ip)$ corresponds to the edge $(p\wedge q{:} A_ip\wedge\neg A_iq,p{:}A_ip) \in Q_i$, for all $i\in Ag$. This edge models an agent who paid attention to $p$, but not to $q$, and therefore only learns $p$ and that she paid attention to $p$.
• Figure 4: The pointed Kripke model $(\mathcal{M}\otimes\mathcal{H}(p\land q),(w,e))$ for $\mathcal{L}_{\text{PA}^+}$, with $\mathcal{H}(p\land q)$ given in Fig. \ref{['fig2:comparison-right']}. We use the same conventions as before and omit worlds that are inaccessible by all agents.
• Figure 5: Attention model $(\mathcal{M}'\otimes\mathcal{R}(\{B_ap, \neg B_aq\}),(w',e))$ for $\mathcal{L}_{\text{GA}}$, where $(\mathcal{M}',w')$ is given in Example \ref{['ex: updated general attention']}. We use the same conventions as before and additionally we let $B_a \{p,q\} := B_ap\wedge B_aq$. While earlier $A_ap$ occurring at $w$ represented that $A_a p \in V(w)$, here it represents that $p \in \mathcal{A}_a(w)$ (similarly for $\neg A_a p$).

Theorems & Definitions (44)

• Definition 1: Kripke model
• Definition 2: Standard event model
• Definition 3: Standard product update
• Definition 4: Satisfaction
• Example 1
• Definition 5: Event model for propositional attention $\mathcal{F}(\varphi)$ belardinelli2023attention
• Definition 6: Edge-conditioned event models
• Definition 7: Edge-conditioned product update
• Theorem 1: Soundness and completeness
• proof : Proof Sketch