Modal Logics for Metric Spaces: Open Problems

Modal logics and their models have been used to speak about and represent topological spaces since the 1940s [22, 23, 16, 17]. Examples include Tarski’s programme of algebraisation of topology (“of creating an algebraic apparatus for the treatment of portions of point-set topology,” to be more precise) which involved modal logic S4 [17], and the use of the extension of S4 with the universal modality (and its fragments) for spatial representation and reasoning; see, e.g., [5, 18, 7, 8, 1, 9] and references therein. Metric spaces are even more important mathematical structures that are fundamental for many areas of mathematics and computer science (recent examples include classification in bioinformatics, linguistics, botany, etc. using various similarity measures). A natural research programme is then to find out to which extent modal-like formalisms can be useful for speaking about metric spaces. Such a programme was launched in 2000 [21, 13, 14]. The aim of this note is to attract attention to the most important open problems and new directions of research in this exciting and promising area.