Section: Scientific Foundations

Modal logics for distributed systems

Modal logics are a family of logics that were developed originally to reason about different modalities occurring in natural language, such as for example the modality of knowledge (epistemic logic), the modalities of obligation and permission (deontic logic) and the modality of time (temporal logic). Temporal logics (CTL, LTL, $\mu$-calculus...) are the most prominent (modal) logics used in computer science nowadays, especially in the field of verification.

Epistemic logic and distributed systems.

In the 1980's, epistemic logic was propounded by computer scientists such as Fagin, Halpern, Moses and Vardi to address problems in distributed systems, resulting in the TARK conference series (Theoretical Aspects of Rationality and Knowledge) and the books  [48] , [54] . This interest in epistemic logic was due to their observation that the notion of knowledge plays a central role in the informal reasoning used in the design of distributed protocols. This lead these authors to “hope that a theory of knowledge, communication and action will prove rich enough to provide general foundations for a unified theoretical treatment of distributed systems”  [50] . The research pursued in DistribCom follows this line of thought, although we also strive to feed and confront our theoretical developments with actual problems stemming from diverse areas of application of distributed systems.

In  [48] , the behavior of a distributed system is represented by a set of runs, each run being a possible execution of the distributed system, determined by a given protocol. Processors are called agents and their partial observation of the system is represented at any point in the run by indistinguishability relations between local states of different runs (the local state of a processor represents the state of this processor at a moment of time). This model was used to show for example that the specific notion of common knowledge of epistemic logic is necessary to reach agreement and to coordinate actions  [50] . Dynamic Epistemic Logic (DEL) is another logical framework that can be used to represent and reason about distributed systems (connections between these two logical frameworks were made in  [64] ). DEL deals with the representation of global states of synchronous distributed systems. The global state of the system at a moment in time is represented directly by means of an epistemic model. Events occurring in this distributed system are represented by means of event models and their effects on the local states of agents (processors) are represented by means of a product update.

The contributions in this sub-module are described in Section 6.4 .

Deontic logic and privacy in distributed systems.

We also use deontic logic in combination with epistemic logic for the formalization of privacy regulations. We intend to use this formalization to reason about privacy in the composition of web-services. The combination of these two modal logics can be used to express statements such as “it is forbidden for agent 1 to know that agent 2 sent message $m$” or “if agent 1 is an administrator of the system, then it is permitted for him to know information $i$”. This provides a formal language very close to the natural language used in actual privacy regulations by law legislators. In the long run, we expect this formal language to be used at the level of interfaces of the web-service in order to:

1. check that the privacy policy declared by the web-service on its interface is indeed compliant (coherent) with respect to the privacy regulations expressed by law makers;

2. check that the web-service does enforce and apply the privacy policy it has declared on its interface.

The contributions in this sub-module are described in Section 6.8 .