Quantifying over information change with common knowledge

Large language model (LLM) agents reason and act in dynamic environments but often lack an explicit account of what they know or believe at each step. This paper introduces a lightweight epistemic state management framework inspired by public announcement logic (PAL) and graded modal logic. We treat observations, tool results and internal inferences as announcements that narrow an agent’s knowledge state. Each announcement updates a minimal knowledge base $\varSigma $ that serves as a filter on the agent’s possible worlds, maintaining consistency while avoiding full model enumeration. Our concrete update mechanism is a consistency-preserving knowledge-base operation that draws on PAL’s model-restriction semantics as conceptual motivation. We survey graded and plausibility-based extensions for representing uncertainty and discuss how qualitative plausibility orderings can model belief revision under bounded rationality. The framework enables explicit, inspectable reasoning in LLM agents, improving self-consistency, interpretability and epistemic awareness. We provide implementation sketches with working code and discuss evaluation strategies for benchmarking epistemic reasoning in LLM agents.

Past public announcement operators have been defined in Hoshi and Yap (Synthese 169(2):259–281, 2009) and Yap (Dynamic logic montreal, 2007), to describe an agent’s knowledge before an announcement occurs. These operators rely on branching-time structures that do not mirror the traditional, relativization-based semantics of public announcement logic (PAL), and favor a historical reading of past announcements. In this paper, we introduce reverse public announcement operators that are interpreted on expanded models. Our model expansion adds accessibility links from an epistemic model $$\mathcal {M}$$ to a filtrated submodel of the canonical model for $$\mathbf K _g$$ . Here $$\mathbf K _g$$ is the minimal normal modal logic together with $$\mathbf S5 $$ axioms for the universal operator U. This yields a highly general pre-announcement version of $$\mathcal {M}$$ that makes our operators potentially useful for studying non-standard interpretations of rescinded announcements in PAL. Indeed, we find that our reverse announcement operators cannot be represented by product update, and that they have an intimate connection with the knowledge forgetting of Zhang and Zhou (Artif Intell J 173(16–17):1525–1537, 2009). We show that the logic resulting from adding reverse announcements to PAL is sound and complete.

This paper presents a logical system in which various group-level epistemic actions are incorporated into the object language. That is, we consider the standard modeling of knowledge among a set of agents by multi-modal Kripke structures. One might want to consider actions that take place, such as announcements to groups privately, announcements with suspicious outsiders, etc. In our system, such actions correspond to additional modalities in the object language. That is, we do not add machinery on top of models (as in Fagin et al [1], but we reify aspects of the machinery in the logical language. The first case of the kind of logics we consider appears in Gerbrandy and Groeneveld [3]. They introduced a language in which one can faithfully represent all of the reasoning in examples such as the Muddy Children scenario. In that paper we find operators for updating worlds via announcements to groups of agents who are isolated from all others. We advance this by considering many more actions; and by using a more general semantics. Our logic contains the infinitary operators used in the standard modeling of common knowledge. We present a sound and complete logical system for the logic, and we study its expressive power.

This contribution is a gentle introduction to so-called dynamic epistemic logics, that can describe how agents change their knowledge and beliefs. We start with a concise introduction to epistemic logic, through the example of one, two and finally three players holding cards; and, mainly for the purpose of motivating the dynamics, we also very summarily introduce the concepts of general and common knowledge. We then pay ample attention to the logic of public announcements, wherein agents change their knowledge as the result of, indeed, public announcements. One crucial topic in that setting is that of unsuccessful updates: formulas that become false when announced. The Moore-sentences that were already extensively discussed at the conception of epistemic logic in [15] give rise to such unsuccessful updates. After that, we present a few examples of more complex epistemic updates. Our closing observations are on recent developments that link the ‘standard’ topic of (theory) belief revision [1] to the dynamic epistemic logics introduced here.

In this paper, we introduce public announcement logic in different geometric frameworks. First, we consider topological models, and then extend our discussion to a more expressive model, namely, subset space models. Furthermore, we prove the completeness of public announcement logic in those frameworks. After that, we apply our results to different issues: announcement stabilization, backward induction and persistence.

Public announcement logic is an extension of multi-agent epistemic logic with dynamic operators to model the informational consequences of announcements to the entire group of agents. We propose an extension of public announcement logic with a dynamic modal operator that expresses what is true after any announcement: sφ expresses that φ is true after an arbitrary announcement ψ. As this includes the trivial announcement ⊤, one might as well say that sφ expresses what remains true after any announcement: it therefore corresponds to truth persistence after (definable) relativisation. The dual operation dφ expresses that there is an announcement after which φ. This gives a perspective on Fitch's knowability issues: for which formulas φ does it hold that φ → dKφ? We give various semantic results, and we show completeness for a Hilbert-style axiomatisation of this logic.

On the succinctness of some modal logics

In this paper we compare standard ways to perform belief change with attempts to model such change with dynamic modal operators. We address both belief expansion, belief contraction, belief revision, and update. Public announcement logic is an implementation of a belief expansion operator in a dynamic epistemic setting. The postulates of success and minimal change cannot be satisfied in their original AGM formulation. Dynamic doxastic logic provides various implementations of a belief revision operator in a dynamic epistemic setting. We provide an example of the application of such a dynamic doxastic semantics, which can also be seen as a realization of a proposal by Spohn. The purpose of this paper is to provide an overview of different approaches to dynamic logics for belief change, including some of our own contributions to the area, in a way that is accessible to an interdisciplinary audience. It does not contain new technical results not published elsewhere. However, the comparison of different approaches provides, we hope, new insights not seen before. A final version of this paper may include new technical results.

We develop a proof-theoretic semantics—in particular, a base-extension semantics—for multi-agent $S5$ modal logic (and hence also for the usual unindexed $S5$). Following the inferentialist interpretation of logic, this gives us a semantics in which validity is based on proof rather than truth. In base-extension semantics, the validity of formulae is generated by provability in a ‘base’ of atomic rules and an inductive definition of the validity of the connectives. Base-extension semantics for many interesting logics has been explored by several authors and, in particular, a base-extension semantics for the modal logics $K$, $KT$, $K4$, and $S4$ has been developed by the present authors. Here, we give a base-extension semantics for multi-agent $S5$ with $\square _{a}$, for an agent $a$, as our primary operators, framed as the knowledge operator $K_{a}$. Similarly to Kripke semantics, we make use of relational structure between bases, allowing us to establish a correspondence between certain bases and worlds. We use this to establish the appropriate soundness and completeness results. We conclude by discussing how this semantics can be extended to Dynamic Epistemic Logics (DEL) starting with Public Announcement Logic (PAL). This note directly supplements [4] and does not repeat the discussion in it.

We study the symbolic model checking problem against public announcement protocol logic (PAPL), featuring protocols with public announcements, arbitrary public announcements and group announcements. Technically, symbolic models are Kripke models whose accessibility relations are presented as programs described in a dynamic logic style with propositional assignments. We highlight the relevance of such symbolic models and show that the symbolic model checking problem against PAPL is A$_{\textrm{pol}}$Exptime-complete as soon as announcement protocols allow for either arbitrary announcements or iteration of public announcements. However, when both options are discarded, the complexity drops to Pspace-complete.

Arbitrary Public Announcement Logic with Common Knowledge (APALC) is an extension of Public Announcement Logic with common knowledge modality and quantifiers over announcements. We show that the satisfiability problem of APALC on S5-models, as well as that of two other related logics with quantification and common knowledge, is Σ 1 1 -hard. This implies that neither the validities nor the satisfiable formulas of APALC are recursively enumerable. Which, in turn, implies that APALC is not finitely axiomatisable.

Commentary: Alcohol and Motor Vehicle–Related Crashes—Driver Attitudes Need Further Intervention

In this chapter, I present the assertion account of common knowledge in the framework of a multi-agent system for the epistemic logic of knowledge and assertion: the propositional content of a formula \(\upvarphi \) is common knowledge to a group of agents G iff everyone in G knows that \(\upvarphi \) is true and that \(\upvarphi \) is asserted. Three current accounts of common knowledge, including the iterated account, the fixed-point account, and shared environment approach, will be examined. I argue that common knowledge arises from communication which results from overtly observable interactions among agents in a group. I then propose that assertion plays a substantial role in communication, and a fortiori, in the acquisition of common knowledge, given the knowledge account of assertion—one must assert \(\upvarphi \) only if one knows \(\upvarphi \). I point out some semantic implications of the knowledge account of assertion in multi-agent systems, specifically, the transmission of individual knowledge to others, the transition of individual knowledge to common knowledge, and the luminosity of common knowledge. The assertion account of common knowledge is then proposed and justified by a class of Kripke models (referred to as TWC-models) appropriate for a multi-agent system of epistemic logic of common knowledge and assertion. The construction of TWC-models will be specified, and the related semantic rules will be given.

Common knowledge and only knowing capture two intuitive and natural notions that have proven to be useful in a variety of settings, for example to reason about coordination or agreement between agents, or to analyse the knowledge of knowledge-based agents. While these two epistemic operators have been extensively studied in isolation, the approaches made to encode their complex interplay failed to capture some essential properties of only knowing. We propose a novel solution by defining a notion of μ-biworld for countable ordinals μ, which approximates not only the worlds that an agent deems possible, but also those deemed impossible. This approach allows us to define a multi-agent epistemic logic with common knowledge and only knowing operators, and a three-valued model semantics for it. Moreover, we show that we only really need biworlds of depth at most ω²+1. Based on this observation, we define a Kripke semantics on a canonical Kripke structure and show that this semantics coincides with the model semantics. Finally, we discuss issues arising when combining negative introspection or truthfulness with only knowing and show how positive introspection can be integrated into our logic.

Local properties in modal logic