Topological Subset Space Models for Public Announcements

2013

We reformulate a key definition given by Wang and Agotnes (2013) to provide semantics for public announcements in subset spaces. More precisely, we interpret the precondition for a public announcement of {\phi} to be the "local truth" of {\phi}, semantically rendered via an interior operator. This is closely related to the notion of {\phi} being "knowable". We argue that these revised semantics improve on the original and offer several motivating examples to this effect. A key insight that emerges is the crucial role of topological structure in this setting. Finally, we provide a simple axiomatization of the resulting logic and prove completeness.

Subset space semantics for public announcement logic in the spirit of the effort modality have been proposed by Wang and Ågotnes [18] and by Bjorn-dahl [6]. They propose to model the public announcement modality by shrinking the epistemic range with respect to which a postcondition of the announcement is evaluated, instead of by restricting the model to the set of worlds satisfying the announcement. Thus we get an " elegant, model-internal mechanism for interpreting public announcements " [6, p.12]. In this work, we extend Bjorndahl's logic PAL int of public announcement, which is modelled on topological spaces using subset space semantics and adding the interior operator, with an arbitrary announcement modality, and we provide topological subset space semantics for the corresponding arbitrary announcement logic APAL int , and demonstrate completeness of the logic by proving that it is equal in expressivity to the logic without arbitrary announcements, employing techniques from [2, 13].

We propose a multi-agent logic of knowledge, public and arbitrary announcements, that is interpreted on topological spaces in the style of subset space semantics. The arbitrary announcement modality functions similarly to the effort modality in subset space logics, however, it comes with intuitive and semantic differences. We provide axiomatizations for three logics based on this setting, and demonstrate their completeness.

We propose a multi-agent logic of knowledge, public announcements and arbitrary announcements, interpreted on topological spaces in the style of subset space semantics. The arbitrary announcement modality functions similarly to the effort modality in subset space logics, however, it comes with intuitive and semantic differences. We provide axiomatizations for three logics based on this setting, with S5 knowledge modality, and demonstrate their completeness. We moreover consider the weaker axiomatizations of three logics with S4 type of knowledge and prove soundness and completeness results for these systems.

In this work, we present a multi-agent logic of knowledge and change of knowledge interpreted on topological structures. Our dynamics are of the so-called semi-private character where a group G of agents is informed of some piece of information ϕ, while all the other agents observe that group G is informed, but are uncertain whether the information provided is ϕ or ¬ϕ. This article follows up on our prior work [31] where the dynamics were public events. We provide a complete axiomatization of our logic, and give two detailed examples of situations with agents learning information through semi-private announcements.

Lecture Notes in Computer Science, 2013

In this paper we introduce public announcements to Subset Space Logic (SSL). In order to do this we have to change the original semantics for SSL a little and consider a weaker version of SSL without the cross axiom. We present an axiomatization, prove completeness and show that this logic is PSPACE-complete. Finally, we add the arbitrary announcement modality which expresses "true after any announcement", prove several semantic results, and show completeness for a Hilbert-style axiomatization of this logic. Proposition 23. Let ϕ be a formula. For all Γ ∈ S u , we have for all finite sequences (ψ 1 ,. .. , ψ n) of formulas, F s , θ s , ([Γ ] ≡ u , f (Γ)) |= [ψ 1 ]. .. [ψ n ]ϕ iff M u , Γ |= [ψ 1 ]. .. [ψ n ]ϕ.

Lecture Notes in Computer Science, 2015

This paper presents a tableau calculus for two semantic interpretations of public announcements over monotone neighbourhood models: the intersection and the subset semantics, developed by Ma and Sano. We show that both calculi are sound and complete with respect to their corresponding semantic interpretations and, moreover, we establish that the satisfiability problem of this public announcement extensions is NP-complete in both cases. The tableau calculi has been implemented in Lotrecscheme. 2 Preliminaries This section recalls some basic concepts from [17]. We work on the single agent case, but the results obtained can be easily extended to multi-agent scenarios. Throughout this paper, let Prop be a countable set of atomic propositions. The language L EL extends the classical propositional language with formulas of the form 2ϕ, read as "the agent knows that ϕ". Formally, ϕ ::= p | ¬ϕ | ϕ ∧ ψ | 2ϕ with p ∈ Prop. Other propositional connectives (∨, → and ↔) are defined as usual. The dual of 2 is defined as 3ϕ := ¬2¬ϕ. A monotone neighborhood frame is a pair F = (W, τ) where W ∅ is the domain, a set of possible worlds, and τ : W → ℘(℘(W)) is a neighborhood function satisfying the following monotonicity condition: for all w ∈ W and all X, Y ⊆ W, X ∈ τ(w) and X ⊆ Y implies Y ∈ τ(w). A monotone neighborhood model (MNM) M = (F , V) is a monotone neighborhood frame F together with a valuation function V : Prop → ℘(W). Given a M = (W, τ, V) and a L EL-formula ϕ, the notion of ϕ being true at a state w in the model M (written M, w | = ϕ) is defined inductively as follows:

Annals of Pure and Applied Logic, 2014

In the present paper, we start studying epistemic updates using the standard toolkit of duality theory. We focus on public announcements, which are the simplest epistemic actions, and hence on Public Announcement Logic (PAL) without the common knowledge operator. As is well known, the epistemic action of publicly announcing a given proposition is semantically represented as a transformation of the model encoding the current epistemic setup of the given agents; the given current model being replaced with its submodel relativized to the announced proposition. We dually characterize the associated submodelinjection map as a certain pseudo-quotient map between the complex algebras respectively associated with the given model and with its relativized submodel. As is well known, these complex algebras are complete atomic BAOs (Boolean algebras with operators). The dual characterization we provide naturally generalizes to much wider classes of algebras, which include, but are not limited to, arbitrary BAOs and arbitrary modal expansions of Heyting algebras (HAOs). Thanks to this construction, the benefits and the wider scope of applications given by a point-free, intuitionistic theory of epistemic updates are made available. As an application of this dual characterization, we axiomatize the intuitionistic analogue of PAL, which we refer to as IPAL, prove soundness and completeness of IPAL w.r.t. both algebraic and relational models, and show that the well known Muddy Children Puzzle can be formalized in IPAL.

2008

Public announcement logic (PAL) is a paradigm case of dynamic epistemic logic, which models how agents’ epistemic states change when pieces of information are communicated publicly. PAL extends epistemic logic with the operator [A], where the intended reading of [A]φ is “After a public announcement that A, φ holds.” This logic has recently received two improvements. One improvement, studied in [1], is to extend PAL with a generalized public announcement operator that allows quantification over public announcements. The other, studied in [5, 6], is a semantic setting to model “announcement protocols” to restrict the announcable sequences of formulas, while whatever is true is assumed to be announcable in PAL itself. The purpose of the present paper is to merge these two kinds of improvements. We consider the extension of public announcement logic with the generalized public announcement operator in the semantic setting of restricted announcement protocols.

Knowledge is strictly connected with the practice of communication: obviously, our comprehension of the world depends not only on what is known, but also on what eventually we may come to know in the process of information flow. In this perspective knowledge can change and it is considered as a dynamic rather than a static notion. A satisfactory account to knowledge change was an important task in the last years, and Dynamic Epistemic Logic (DEL) is one of the most prominent and recent approaches to this problem.