Logic and Topology for Knowledge, Knowability, and Belief - Extended Abstract

In recent work, Stalnaker proposes a logical framework in which belief is realized as a weakened form of knowledge. Building on Stalnaker's core insights, and using frameworks developed by Bjorndahl and Baltag et al., 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.Comment: In Proceedings TARK 2017, arXiv:1707.08250. The full version of this paper, including the longer proofs, is at arXiv:1612.0205

Symbolic Model Checking for Dynamic Epistemic Logic

Dynamic Epistemic Logic (DEL) can model complex information scenarios in a way that appeals to logicians. However, existing DEL implementations are ad-hoc, so we do not know how the framework really performs. For this purpose, we want to hook up with the best available model-checking and SAT techniques in computational logic. We do this by first providing a bridge: a new faithful representation of DEL models as so-called knowledge structures that allow for symbolic model checking. Next, we show that we can now solve well-known benchmark problems in epistemic scenarios much faster than with existing DEL methods. Finally, we show that our method is not just a matter of implementation, but that it raises significant issues about logical representation and update

Symbolic Model Checking for Dynamic Epistemic Logic

Dynamic Epistemic Logic (DEL) can model complex information scenarios in a way that appeals to logicians. However, existing DEL implementations are ad-hoc, so we do not know how the framework really performs. For this purpose, we want to hook up with the best available model-checking and SAT techniques in computational logic. We do this by first providing a bridge: a new faithful representation of DEL models as so-called knowledge structures that allow for symbolic model checking. Next, we show that we can now solve well-known benchmark problems in epistemic scenarios much faster than with existing DEL methods. Finally, we show that our method is not just a matter of implementation, but that it raises significant issues about logical representation and update

Subset space logic with arbitrary announcements

National audienceIn 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

DGD Gallery: Storage, sharing, and publication of digital research data

We describe a project, called the "Discretization in Geometry and Dynamics Gallery", or DGD Gallery for short, whose goal is to store geometric data and to make it publicly available. The DGD Gallery offers an online web service for the storage, sharing, and publication of digital research data.Comment: 19 pages, 8 figures, to appear in "Advances in Discrete Differential Geometry", ed. A. I. Bobenko, Springer, 201