Reasoning about Dynamic Epistemic Logic

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 formalize such programs and updates were done by Plaza [19], Gerbrandy and Groeneveld [12], and Gerbrandy [10, 11]. However, they only studied a restricted class of epistemic programs. A general notion of epistemic programs and updates for DEL was introduced in [5]. In our papers [2, 3], we introduced an algebraic semantics based on the notion of epistemic systems and a sequent calculus for a version of DEL, but the completeness of the sequent calculus was still an open problem. In this paper, we summarize the material in [2, 3] and present an updated version of the sequent calculus for which we have proved the completeness theorem with regard to the algebraic semantics. Our work contributes to DEL in three ways. First, it introduces a logical account of actions and agents as dynamic and epistemic resources in situations of information exchange. In these situations each new repetition of the same announcement might add new information to the agents. Thus it makes a difference whether or not unlimited “supplies” of these actions are available. We consider epistemic action as dynamic resources , which are similar to the usual use-only-once resources of linear logic [13]. We will also deal with epistemic resources to capture the presence of agents in a given situation (or availability of agents as computing resources for other agents). These resources capture the cases where presence of agents makes a difference in the validity of some deductions and execution of some actions by other agents. In other words, some deductions are only valid (and some actions are only executable) in the presence of certain agents, i.e. valid not in the real world, but in the world as it appears to these agents. Note that agents and actions are not only resources but also “consumers of resources”; actions need certain preconditions to be executable and agents need certain contexts to be able do their reasoning.