François Schwarzentruber

Seeing, knowledge and common knowledge

We provide a multi-agent spatially grounded epistemic logical framework to reason about the knowledge of perception (agent a sees agent b) whose potential applications are video games and robotics. Contrary to the classical epistemic modal logic, we prove that in some configurations the logic with the common knowledge operator is as expressive as the logic without the common knowledge operator. We give some complexity results about the model-checking.

Big Brother Logic: visual-epistemic reasoning in stationary multi-agent systems

We consider multi-agent scenarios where each agent controls a surveillance camera in the plane, with fixed position and angle of vision, but rotating freely. The agents can thus observe the surroundings and each other. They can also reason about each other’s observation abilities and knowledge derived from these observations. We introduce suitable logical languages for reasoning about such scenarios which involve atomic formulae stating what agents can see, multi-agent epistemic operators for individual, distributed and common knowledge, as well as dynamic operators reflecting the ability of cameras to turn around in order to reach positions satisfying formulae in the language. We also consider effects of public announcements. We introduce several different but equivalent versions of the semantics for these languages, discuss their expressiveness and provide translations in PDL style. Using these translations we develop algorithms and obtain complexity results for model checking and satisfiability testing for the basic logic BBL that we introduce here and for some of its extensions. Notably, we show that even for the extension with common knowledge, model checking and satisfiability testing remain in PSPACE. We also discuss the sensitivity of the set of validities to the admissible angles of vision of the agents’ cameras. Finally, we discuss some further extensions: adding obstacles, positioning the cameras in 3D or enabling them to change positions. Our work has potential applications to automated reasoning, formal specification and verification of observational abilities and knowledge of multi-robot systems.

Talking About Graphs

We introduce a general formal language in order to talk about properties of Kripke models. We then show how such languages can be defined in LoTREC and how to check that a formula is true in a given world of a given Kripke model. The latter is called model checking. Beyond model checking we also introduce the reasoning tasks of satisfiability checking, validity checking, and model building. We show that all other tasks can be reduced to the latter, which we focus on in the rest of the book: we show how to build models for a series of logics. These logics are grouped into families, according to the techniques the tableaux implementation in LoTREC requires.

Modelling with Graphs

We introduce Kripke models. With the help of a series of examples we present the main modal concepts that can be interpreted by means of possible worlds semantics: the concepts of event, action, program, time, belief, knowledge, and obligation. In the end of the chapter we show how Kripke models can be built by means of LoTREC and give a formal definition of Kripke models.

Modal Logics with Transitive Closure

This last chapter is about the model construction problem in classes of models having relations that are transitive closures of other relations. The main such logics are linear-time temporal logic LTL and propositional dynamic logic PDL. These logics require both blocking and model checking.

Logics with Simple Constraints on Models

This chapter is about the model construction problem in some classes of models: models satisfying the conditions of reflexivity, seriality, symmetry, and combinations thereof. The corresponding logics are KT, KD, KB, KTB, and KDB. We also consider models whose accessibility relation is confluent (logic K.2) or is an equivalence relation (logic KT45, alias S5). While these conditions are only about a single relation, we also study properties involving two accessibility relations: inclusion and permutation.

Logics with Transitive Accessibility Relations

This chapter is about the model construction problem in classes of models satisfying the constraint of transitivity. We present the modal logics of the class of models where the accessibility relation is transitive (K4), transitive and serial (KD4), and transitive and reflexive (KT4, alias S4). For these logics, the model construction procedure may loop, which contrasts with the simple logics of Chap. 4. Termination can be ensured by means of blocking techniques: basically, the construction is stopped when the labels of a node are identical to those of some ancestor node. In the end of the chapter we present another general termination theorem guaranteeing that the tableau construction does not loop and which applies to all these logics.

The Basics of the Model Construction Method

This chapter is about the basic modal logic K and its multimodal version K n . We also present the basic description logic ALC, which can be viewed as a notational variant of K n . The implementation of reasoning methods for all these logics can be done by means of the most basic rules, that are combined by the most basic strategies: fair strategies.

Kripke’s Worlds

Preface.- 1 Modelling things with graphs.- 2 Talking about graphs.- 3 The basics of the model construction method.- 4 Logics with simple constraints on models.- 5 Logics with transitive accessibility relations.- 6 Model Checking.- 7 Modal logics with transitive closure.- Bibliography.- Index.