We refine our algebraic axiomatization in [8, 9] of epistemic actions and epistemic update (notions defined in [5, 6] using a relational, Kripke-style semantics), to incorporate a mechanism for dynamic belief revision in a multi-agent setting. Our approach has a number of novel features, when compared with traditional belief revision systems such as AGM [2]. Firstly: while traditional belief revision was

We present two applications to AI of recently introduced high level quantum structures. These structures are the categorical quantum logic of (Abramsky & Coecke 2004) and the quantale quantum logic of (Coecke, Moore, & Stubbe 2001). Firstly, we show how the diagrammatic toolkit of categorical quantum logic, when restricted to its pregroup fragment (Lambek 1999; 2001), simplifies analysis of sentence structure of different languages. Moreover, the quantitative values that arise in these diagrams signify different degrees of complexity of sentences, which turn out to vary for different languages. Secondly, we show how expanding the quantale quantum logic with epistemic modalities provides a powerful system to reason about information update in multi-agent systems. Finally, we indicate how the above two applications to non-quantum domains can themselves be ‘re-quantized’, providing applications to quantum informatics of distributed systems.

We present an algebra and sequent calculus to reason about dynamic epistemic logic, a logic for information update in multi-agent systems. We contribute to it by equipping it with a logical account of resources, a semi-automatic way of reasoning through the algebra and sequent calculus, and finally by generalizing it to non-boolean settings. Dynamic Epistemic Logic (DEL) is a PDL-style logic [14] to reason about epistemic actions and updates in a multi-agent system. It focuses in particular on epistemic programs, i.e. programs that update the information state of agents, and it has applications to modelling and reasoning about informationflow and information exchange between agents. This is a major problem in several fields such as secure communication where one has to deal with the privacy and authentication of communication protocols, software reliability for concurrent programs, Artificial Intelligence where agents are to be provided with reliable tools to reason about their environment and each other’s knowledge, and e-commerce where agents need to have knowledge acquisition strategies over complex networks. The standard approach to information flow in a multi-agent system has been presented in [8] but it does not present a formal description of epistemic programs and their updates. The first attempts to

Abstract. We present a coalgebraic semantics for reasoning about information update in multi-agent systems. The novelty is that we have one structure for both states and actions and thus our models do not involve the ”change-of-model ” phenomena that arise when using Kripke models. However, we prove that the usual models can be constructed from ours by categorical adjunction. The generality and abstraction of our coalgebraic model turns out to be extremely useful in proving preservation properties of update. In particular, we prove that positive knowledge is preserved and acquired as a result of epistemic update. We also prove common and nested knowledge properties of epistemic updates induced by specific epistemic actions such as public and private announcements, lying, and in particular unsafe actions of security protocols. Our model directly gives rise to a coalgebraic logic with both dynamic and epistemic modalities. We prove a soundness and completeness result for this logic, and illustrate the applicability of the logic by deriving knowledge properties of a simple security protocol. 1

6. Conclusion and future plans

We develop a cut-free sequent calculus for a Dynamic Epistemic Logic. The calculus is nested and represents a sub-structural action logic which acts on a propositional logic via a dynamic modality and its left adjoint update. Both logics are positive and have agent-indexed adjoint pairs of epistemic modalities. We prove admissibility (where appropriate) of Weakening and Contraction and Cut, as well as soundness and completeness theorems with regard to the algebraic semantics. To model epistemic protocols, we add assumption rules, prove that the admissibility results are preserved, and derive properties of a toy protocol that has honest and dishonest public and private announcements. Keywords: Dynamic Epistemic Logic, Algebraic Modal Logic, Cut-Admissibility, Epistemic Protocols.