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.

Subset Space Logic with Arbitrary 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.

Public Announcement Logics with Constrained Protocols

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.

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.

A Runs-and-Systems Semantics for Logics of Announcements

Lecture Notes in Computer Science, 2010

Logics of announcements are logics of knowledge to reason about agents that communicate by broadcasting interpreted messages. These logics are typically given a semantics in terms of updatable Kripke structures, which tend to be abstract. We revisit the semantics of logics of announcements and develop a concrete semantics using runs and systems. The advantage is that we can devise models that capture scenarios without having to express properties of those scenarios within the logic itself. In this concrete setting, we study honesty as well as belief in the presence of announcements that are not broadcast to all agents in a system.

A hybrid public announcement logic with distributed knowledge

Electronic Notes in Theoretical Computer Science 273(2011): 33-50. Post-Proceedings of the International Workshop on Hybrid Logic and Applications (HyLo 2010). 2011, 2011

In this paper the machinery of Hybrid Logic and the logic of public announcements are merged. In order to bring the two logics together properly the underlying hybrid logic has been changed such that nominals only partially denote states. The hybrid logic contains nominals, satisfaction operators, the downarrow binder as well as the global modality. Following this, an axiom system for the Hybrid Public Announcement Logic is presented and using reduction axioms general completeness (in the usual style of Hybrid Logic) is proved. The general completeness allows for an easy way of adding distributed knowledge. Furthermore, it turns out that distributed knowledge is definable using satisfaction operators and the downarrow binder.

Algebraic semantics and model completeness for Intuitionistic Public Announcement Logic

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.

A Tableau Method for Public Announcement Logics

Lecture Notes in Computer Science, 2007

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 a labelled tableau-calculus for this logic. We also present an extension of the calculus for a logic of arbitrary announcements.

A hybrid public announcement logic with distributed knowledge-extended version

Manuscript available from the author, 2011

In this paper the machinery of Hybrid Logic and the logic of public announcements are merged. In order to bring the two logics together properly the underlying hybrid logic has been changed such that nominals only partially denote states. The hybrid logic contains nominals, satisfaction operators, the downarrow binder as well as the global modality. Following this, an axiom system for the Hybrid Public Announcement Logic is presented and using reduction axioms general completeness (in the usual style of Hybrid Logic) is proved. The general completeness allows for an easy way of adding distributed knowledge. Furthermore it turns out that distributed knowledge is definable using satisfaction operators and the downarrow binder. The standard way of adding distributed knowledge using reduction axioms is also discussed and generalized to other modalities sharing properties with the distributed knowledge modality.

What can we achieve by arbitrary announcements?

Proceedings of the 11th conference on Theoretical aspects of rationality and knowledge - TARK '07

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: ϕ expresses that ϕ is true after an arbitrary announcement ψ. As this includes the trivial announcement , one might as well say that ϕ expresses what remains true after any announcement: it therefore corresponds to truth persistence after (definable) relativisation. The dual operation ♦ϕ 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 → ♦Kϕ? We give various semantic results, and we show completeness for a Hilbert-style axiomatisation of this logic.

Generic Models for Topological Evidence Logics

2018

This thesis studies several aspects of the topological semantics for evidencebased belief and knowledge introduced by Baltag, Bezhanishvili, Özgün, and Smets (2016). Building on this work, we introduce a notion of generic models, topological spaces whose logic is precisely the sound and complete logic of topological evidence models. We provide generic models for the different fragments of the language. Moreover, we give a multi-agent framework which generalises that of single-agent topological evidence models. We provide the complete logic of this framework together with some generic models for a fragment of the language. Finally, we define a notion of group knowledge which differs conceptually from previous approaches.

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.