Epistemic Group Attitudes
This project aims to develop and investigate new notions of group knowledge in multi-agent systems. The two classic forms of group knowledge are common knowledge and distributed knowledge. We will devise novel variants of these notions, which are important for distributed computing, and study their logical and mathematical properties.
One part of our project deals with eventual common knowledge (ECK), which is a form of common knowledge that is relevant for the epistemic analysis of many applications. However, ECK needs to be better understood from a mathematical and logical perspective.
We will develop axiomatic systems for ECK and investigate its meta-logical properties, such as compactness, finite model property, etc. We will then study the relationship between task solvability and the properties of the corresponding form of ECK. In particular, we will establish the precise connection of ECK and the Firing Rebels with Relay problem, which is a particularly interesting case in distributed computing.
The second part of the project is concerned with novel variants of distributed knowledge. Simplicial models are an essential tool in the area of distributed computing. Recently, it has been observed that simplicial models also provide an interesting semantics for multi-agent epistemic logic. These models focus on the local states of the agents instead of the global state of the system (represented as a possible world). This simplicial semantics makes it possible to represent new forms of distributed knowledge.
Current simplicial models support notions of knowledge. However, in the presence of Byzantine agents, only forms of belief can be obtained, but not knowledge. We will adapt simplicial models to support belief and belief dynamics. Further, we will develop and study an epistemic logic for simplicial sets. This is the most general form of simplicial models. Yet, their logical analysis is still missing
Proof and Model Theory of Intuitionistic Temporal Logic
Intuitionistic logic enjoys a myriad of interpretations based on computation, information or topology, making it a natural framework to reason about dynamic processes in which these phenomena play a crucial role. Yet there is a large gap to be filled regarding our understanding of the computational behaviour of intuitionistic temporal logics. The aim of this project is to cement our understanding of intuitionistic temporal logics by developing their model theory based on dynamic topological systems, and their proof theory based on prominent paradigms such as Gentzen-style calculi as well as cyclic proofs. Further information can be found here:
https://users.ugent.be/~dfernnde/itl.html
Modalities in Substructural Logics: Theory, Methods and Applications
Modal logics are a family of formal systems based on classical logic which aim at improving the expressive power of the classical calculus allowing to reason about “modes of truth”. The aim of the present proposal is to put forward a systematic study of substructural modal logics, understood as those modal logics in which the modal operators are based upon the general ground of substructural logics, weaker deductive systems than classical logic. Our aim is also to explore the applications of substructural modal logics outside the bounds of mathematical logic and, in particular, in the areas of knowledge representation; legal reasoning; data privacy and security; logical analysis of natural language.
Explicit reasons
This project is concerned with reasons why one believes something, reasons why one knows something, and reasons why one ought to do something. We develop formal languages in which reasons can be represented explicitly and investigate the logical properties of explicit reasons. To achieve this, we rely on the framework of justification logic.
In particular, we present non-normal deontic logics with justifications. Further, we develop a semiring framework for justifications, and we engineer a possible world semantics for justifications that supports additional structure like graded justifications or probability distributions on justifications.
