On Classical Decidable Logics extended with Percentage Quantifiers and Arithmetics

TL;DR

This paper explores the extension of classical decidable first-order logic fragments with percentage quantifiers, revealing that such extensions generally lead to undecidability, except for certain guarded fragments with Presburger constraints.

Contribution

It demonstrates that adding percentage quantifiers to well-known decidable fragments causes undecidability, except for the guarded fragment with Presburger counting, advancing understanding of logical expressiveness.

Findings

Most classical decidable fragments become undecidable with percentage quantifiers.

The two-variable guarded fragment with Presburger constraints remains decidable.

Results inform the decidability of modal and description logics with cardinality constraints.

Abstract

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 and the guarded fragment, depending on the type of restriction imposed on formulae from the language. Despite the success of the mentioned logics in areas like 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 frequently 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.