# A complete classification of the expressiveness of interval logics of Allen’s relations: the general and the dense cases

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## Abstract

Interval temporal logics take time intervals, instead of time points, as their primitive temporal entities. One of the most studied interval temporal logics is Halpern and Shoham’s modal logic of time intervals HS, which associates a modal operator with each binary relation between intervals over a linear order (the so-called Allen’s interval relations). In this paper, we compare and classify the expressiveness of all fragments of HS on the class of all linear orders and on the subclass of all dense linear orders. For each of these classes, we identify a complete set of definabilities between HS modalities, valid in that class, thus obtaining a complete classification of the family of all 4096 fragments of HS with respect to their expressiveness. We show that on the class of all linear orders there are exactly 1347 expressively different fragments of HS, while on the class of dense linear orders there are exactly 966 such expressively different fragments.

## Notes

### Acknowledgments

We thank the anonymous referees for their careful reading of our original journal submission and their insightful comments, which led to several improvements. The authors acknowledge the support from the Spanish fellowship program ‘*Ramon y Cajal*’ *R*YC-2011-07821 and the Spanish MEC project *TIN2009-14372-C03-01* (G. Sciavicco), the project *Processes and Modal Logics* (Project No. 100048021) of the Icelandic Research Fund (L. Aceto, D. Della Monica, and A. Ingólfsdóttir), the project *Decidability and Expressiveness for Interval Temporal Logics* (Project No. 130802-051) of the Icelandic Research Fund in partnership with the European Commission Framework 7 Programme (People) under ‘Marie Curie Actions’ (D. Della Monica), and the Italian GNCS project *Automata, Games, and Temporal Logics for the verification and synthesis of controllers in safety-critical systems* (A. Montanari).

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