Modal Logic

The course will provide the students with the basic concepts, tools and techniques regarding modal logics, a tool that can be used for reasoning about diverse mathematical structures, and thus to reason about the many phenomena they can represent. The main focus will be relational structures (i.e., graphs), as they play a fundamental modelling role in many disciplines, including theoretical computer science, knowledge representation, multi-agent systems, computational linguistics, formal semantics of natural language, economics and philosophy, among others. In this way, the course will enable the students to carry out advanced research projects in these disciplines.

The course is divided into four parts. The first one recalls the required mathematical preliminaries, including basic mathematical concepts (sets, relations, functions, inductive and recursive definitions), propositional logic (syntax, semantics, logical equivalence, logical consequence) and strategies for proving mathematical statements. The second presents the basic definitions for the modal logic framework, including its formal language and its semantic interpretation on relational structure; it also discusses expressivity and axiomatisation results. The third focuses on two main topics: extended modal languages (including, e.g., the global and the inaccessibility modality as well as nominals and dynamic logic and Boolean modal logic) and the framework's computational aspects (model checking and satisfiability/validity). The fourth explores advanced topics, including alternative semantics (e.g., neighbourhood semantics) as well as modal predicate logic.

The course material will consist of textbook chapters and survey/research articles.

A student who has completed the course should have the following learning outcomes defined in terms of knowledge, skills and general competence:

Knowledge 

After completing the course, the student

• will learn know what modal languages are.
• will be able to evaluate formulas of these languages, and will understand the concepts of satisfiability and validity.
• will be able to discuss different invariance results.
• will understand the notion of bisimulation, and will be able to use it to analyse the expressivity of different modal languages.
• will be able to relate different modal languages to other logical frameworks.

Skills 

The student will be able to

• use modal logics for analysing different phenomena that can be represented by means of relational structures (examples include, among others, the execution of programs and as well as interaction between agents).
• present and discuss research papers that use modal logics for analysing different phenomena (this includes papers on, among others, theoretical aspects of multi-agent systems in AI).
• carry on advanced research projects that reason about a given phenomenon through modal languages.