Characterising Modal Formulas with Examples

Abstract

We study the existence of finite characterisations for modal formulas. A finite characterisation of a modal formula φ is a finite collection of positive and negative examples that distinguishes φ from every other, non-equivalent modal formula, where an example is a finite pointed Kripke structure. This definition can be restricted to specific frame classes and to fragments of the modal language: a modal fragment \(\mathcal {L} \) admits finite characterisations with respect to a frame class \(\mathcal {F} \) if every formula \(\varphi \in \mathcal {L} \) has a finite characterisation with respect to \(\mathcal {L} \) consisting of examples that are based on frames in \(\mathcal {F} \) . Finite characterisations are useful for illustration, interactive specification, and debugging of formal specifications, and their existence is a precondition for exact learnability with membership queries. We show that the full modal language admits finite characterisations with respect to a frame class \(\mathcal {F} \) only when the modal logic of \(\mathcal {F} \) is locally tabular. We then study which modal fragments, freely generated by some set of connectives, admit finite characterisations. Our main result is that the positive modal language without the truth-constants ⊤ and ⊥ admits finite characterisations w.r.t. the class of all frames. This result is essentially optimal: finite characterizability fails when the language is extended with the truth constant ⊤ or ⊥ or with all but very limited forms of negation.