Announcement as effort on topological spaces (original) (raw)

Announcement as effort on topological spaces-Extended version

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.

Arbitrary Announcements on Topological Subset Spaces

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].

Subset Space Public Announcement Logic Revisited

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.

Private Announcements on Topological Spaces

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.

Topological Subset Space Models for Public Announcements

Outstanding Contributions to Logic, 2018

We reformulate a key definition given by Wáng andÅgotnes (2013) to provide semantics for public announcements in subset spaces. More precisely, we interpret the precondition for a public announcement of ϕ to be the "local truth" of ϕ, semantically rendered via an interior operator. This is closely related to the notion of ϕ 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.

A proof-theoretical perspective on Public Announcement Logic

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.

Logic and Topology for Knowledge, Knowability, and Belief

The Review of Symbolic Logic, 2019

In recent work, Stalnaker proposes a logical framework in which belief is realized as a weakened form of knowledge [30]. Building on Stalnaker's core insights, and using frameworks developed in [11] and [4], we employ topological tools to refine and, we argue, improve on this analysis. The structure of topological subset spaces allows for a natural distinction between what is known and (roughly speaking) what is knowable; we argue that the foundational axioms of Stalnaker's system rely intuitively on both of these notions. More precisely, we argue that the plausibility of the principles Stalnaker proposes relating knowledge and belief relies on a subtle equivocation between an " evidence-in-hand " conception of knowledge and a weaker " evidence-out-there " notion of what could come to be known. Our analysis leads to a trimodal logic of knowledge, knowability, and belief interpreted in topological subset spaces in which belief is definable in terms of knowledge and knowability. We provide a sound and complete axiomatization for this logic as well as its uni-modal belief fragment. We then consider weaker logics that preserve suitable translations of Stalnaker's postulates, yet do not allow for any reduction of belief. We propose novel topological semantics for these irreducible notions of belief, generalizing our previous semantics, and provide sound and complete axiomatizations for the corresponding logics.

Topo-Logic as a dynamic-epistemic logic

We extend the 'topologic' framework [14] with dynamic modalities for 'topological public announcements' in the style of Bjorndahl [5]. We give a complete axiomatization for this " Dynamic Topo-Logic " , which is in a sense simpler than the standard axioms of topologic. Our completeness proof is also more direct (making use of a standard canonical model construction). Moreover, we study the relations between this extension and other known logical formalisms, showing in particular that it is co-expressive with the simpler (and older) logic of interior and global modality [10, 4, 15, 1]. This immediately provides an easy decidability proof (both for topologic and for our extension).

Tableaux for Non-normal Public Announcement Logic

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, without employing reduction axioms, 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. 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:

Tableaux for public announcement logic

Journal of Logic and Computation, 2010

Public announcement logic extends multi-agent epistemic logic with dynamic operators to model the informational consequences of announcements to the entire group of agents. In this article we propose a labelled tableau calculus for this logic, and show that it decides satisfiability of formulas in deterministic polynomial space. Since this problem is known to be PSPACE-complete, it follows that our proof method is optimal.