On Classical Decidable Logics extended with Percentage Quantifiers and Arithmetics

During the last decades, a lot of effort was put into identifying decidable fragments of first-order logic. Such efforts gave birth, among the others, to the two-variable fragment, the guarded fragment, depending on the type of restriction imposed on formulae from the language. Despite the success of the mentioned logics in formal verification and knowledge representation, such first-order fragments are too weak to express even the simplest statistical constraints, required for modelling of influence networks or in statistical reasoning. In this work we investigate the extensions of these classical decidable logics with percentage quantifiers, specifying how frequent a formula is satisfied in the indented model. We show, surprisingly, that all the mentioned decidable fragments become undecidable under such extension, sharpening the existing results in the literature. Our negative results are supplemented by decidability of the two-variable guarded fragment with even more expressive counting, namely Presburger constraints. Our results can be applied to infer decidability of various modal and description logics, e.g. Presburger Modal Logics with Converse or ALCI, with expressive cardinality constraints.