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We initiate the study of finite characterizations and exact learnability ofmodal languages. A finite characterization of a modal formula w.r.t. a set offormulas is a finite set of finite models (labelled either positive ornegative) which distinguishes this formula from every other formula from thatset. A modal language L admits finite characterisations if every L-formula hasa finite characterization w.r.t. L. This definition can be applied not only tothe basic modal logic K, but to arbitrary normal modal logics. We show that anormal modal logic admits finite characterisations (for the full modallanguage) iff it is locally tabular. This shows that finite characterizationswith respect to the full modal language are rare, and hence motivates the studyof finite characterizations for fragments of the full modal language. Our mainresult is that the positive modal language without the truth-constants $\top$and $\bot$ admits finite characterisations. Moreover, we show that this resultis essentially optimal: finite characterizations no longer exist when thelanguage is extended with the truth constant $\bot$ or with all but verylimited forms of negation.

Characterising Modal Formulas with Examples