Modal Logic

The course will provide the students with the basic concepts, tools and techniques regarding modal logic, a tool that can be used for reasoning about diverse mathematical structures, and thus to reason about the different 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 first and main part of the course will focus on the basic aspects of the basic modal language over relational structures. These basic aspects include topics such as syntax, semantics, axiomatisations, invariance, expressivity, standard translation, correspondence theory, computability and complexity, all of them crucial not only for understanding the advantages and limitations of the basic modal system, but also for understanding how it relates to other logical frameworks. Having studied the modal logic framework in depth, the second part of the course focuses on extended modal languages, which increase the power of the basic modal language through additional modal operators. Each of these new operators is studied in depth, discussing how it affects the profile (axiomatisations, invariance, expressivity and so on) of the basic setting. The third and final part of the course studies the use of modal languages for describing and reasoning about different structures (e.g., neighbourhood models).

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 what modal languages are.
• be able to evaluate formulas of these languages in different structures.
• understand the concepts of satisfiability and validity.
• be able to discuss critical aspects such as expressivity and computational complexity.
• be able to discuss the languages' relationship with other logical frameworks.

Skills 

The student will be able to

• use modal logics for analysing different phenomena that can be represented through different mathematical structures (examples include, among others, the execution of programs and as well as diverse forms of 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 given phenomena through modal languages.