Private Announcements on Topological Spaces (original) (raw)

Announcement as effort on topological spaces

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.

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.

Topological Evidence Logics: Multi-Agent Setting

2019

We introduce a multi-agent topological semantics for evidencebased belief and knowledge, which extends the dense interior semantics developed in [2]. We provide the complete logic of this multi-agent framework together with generic models for a fragment of the language. We also define a new notion of group knowledge which differs conceptually from previous approaches.

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.

The Logic of Public Announcements and Common Knowledge and Private Suspicions

Theoretical Aspects of Rationality and Knowledge, 1998

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 Logic of Public Announcements, Common Knowledge, and Private Suspicions (extended abstract)

2000

The logic of public announcements, common knowledge, and private suspicions

1999

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.

Epistemic Protocols for Distributed Gossiping

arXiv (Cornell University), 2016

Gossip protocols aim at arriving, by means of point-to-point or group communications, at a situation in which all the agents know each other's secrets. We consider distributed gossip protocols which are expressed by means of epistemic logic. We provide an operational semantics of such protocols and set up an appropriate framework to argue about their correctness. Then we analyze specific protocols for complete graphs and for directed rings. Epistemic Protocols for Distributed Gossip 2.1 Syntax We loosely use the syntax of the language CSP (Communicating Sequential Processes) of [11] that extends the guarded command language of [6] by disjoint parallel composition and commands for synchronous communication. CSP was realized in the distributed programming language OCCAM (see INMOS [12]). The main difference is that we use as guards epistemic formulas and as communication primitives calls that do not require synchronization. Also, the syntax of our distributed programs is very limited. In order to define gossip protocols we introduce in turn calls and epistemic guards. Throughout the paper we assume a fixed finite set A of at least three agents. We assume that each agent holds exactly one secret and that there exists a bijection between the set of agents and the set of secrets. We denote by P the set of all secrets (for propositions). Furthermore, it is assumed that each secret carries information identifying the agent to whom that secret belongs. 2.1.1 Calls Each call concerns two agents, the caller (a below) and the agent called (b). We distinguish three modes of communication of a call: push-pull, written as ab or (a, b). During this call the caller and the called agent learn each other's secrets, push, written as a ⊲ b. After this call the called agent learns all the secrets held by the caller, pull, written as a ⊳ b. After this call the caller learns all the secrets held by the called agent. Variables for calls are denoted by c, d. Abusing notation we write a ∈ c to denote that agent a is one of the two agents involved in the call c (e.g., for c := ab we have a ∈ c and b ∈ c). Calls in which agent a is involved are denoted by c a. 2.1.2 Epistemic guards Epistemic guards are defined as formulas in a simple modal language with the following grammar: φ ::= F a p | ¬φ | φ ∧ φ | K a φ , where p ∈ P and a ∈ A. Each secret is viewed as a distinct symbol. We denote the secret of agent a by A, the secret of agent b by B and so on. We denote the set of so defined formulas by L and we refer to its members as epistemic formulas or epistemic guards. We read F a p as 'agent a is familiar with the secret p' (or 'p belongs to the set of secrets a knows about') and K a φ as 'agent a knows that formula φ is true'. So this language is an epistemic language where atoms consist of 'knowing whether' statements about propositional atoms, if we view secrets as Boolean variables. Atomic expressions in L concern only who knows what secrets. As a consequence the language cannot express formally the truth of a secret p. This level of abstraction suffices for the purposes of the current paper. However, expressions F a p could be given a more explicit epistemic reading in terms of 'knowing whether'. That is, 'a is familiar with p' can be interpreted (on a suitable Kripke model) as 'a knows whether the secret p is true or not'. This link is established in [3].

Logics of communication and change

Information and Computation, 2006

Current dynamic epistemic logics for analyzing effects of informational events often become cumbersome and opaque when common knowledge is added for groups of agents. Still, postconditions involving common knowledge are essential to successful multi-agent communication. We propose new systems that extend the epistemic base language with a new notion of 'relativized common knowledge', in such a way that the resulting full dynamic logic of information flow allows for a compositional analysis of all epistemic postconditions via perspicuous 'reduction axioms'. We also show how such systems can deal with factual alteration, rather than just information change, making them cover a much wider range of realistic events. After a warm-up stage of analyzing logics for public announcements, our main technical results are expressivity and completeness theorems for a much richer logic that we call LCC. This is a dynamic epistemic logic whose static base is propositional dynamic logic (PDL), interpreted epistemically. This system is capable of expressing all modelshifting operations with finite action models, while providing a compositional analysis for a wide range of informational events. This makes LCC a serious candidate for a standard in dynamic epistemic logic, as we illustrate by analyzing some complex communication scenarios, including sending successive emails with both 'cc' and 'bcc' lines, and other private announcements to subgroups. Our proofs involve standard modal techniques, combined with a new application of Kleene's Theorem on finite automata, as well as new Ehrenfeucht games of model comparison.

Private Announcements on Topological Spaces (original) (raw)

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