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

- 209 Downloads
- 1 Citations

## 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).

## References

- 1.Aceto, L., Della Monica, D., Ingólfsdóttir, A., Montanari, A., Sciavicco, G.: An algorithm for enumerating maximal models of Horn theories with an application to modal logics. In: Proceedings of the 19th LPAR, LNCS, vol. 8312, pp. 1–17. Springer (2013)Google Scholar
- 2.Aceto, L., Della Monica, D., Ingólfsdóttir, A., Montanari, A., Sciavicco, G.: A complete classification of the expressiveness of interval logics of Allen’s relations over dense linear orders. In: Proceedings of the 20th TIME, pp. 65–72. IEEE Computer Society (2013)Google Scholar
- 3.Aceto, L., Della Monica, D., Ingólfsdóttir, A., Montanari, A., Sciavicco, G.: On the expressiveness of the interval logic of Allen’s relations over finite and discrete linear orders. In: Fermé, E., Leite, J. (eds.) Proceedings of the 14th JELIA, LNAI, vol. 8761, pp. 267–281. Springer (2014)Google Scholar
- 4.Allen, J.F.: Maintaining knowledge about temporal intervals. Commun. ACM
**26**(11), 832–843 (1983)CrossRefzbMATHGoogle Scholar - 5.Allen, J.F.: Towards a general theory of action and time. Artif. Intell.
**23**(2), 123–154 (1984)CrossRefzbMATHGoogle Scholar - 6.Balbiani, P., Goranko, V., Sciavicco, G.: Two-sorted point-interval temporal logics. Electron. Notes Theor. Comput. Sci.
**278**, 31–45 (2011)MathSciNetCrossRefGoogle Scholar - 7.Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar
- 8.Bresolin, D., Della Monica, D., Goranko, V., Montanari, A., Sciavicco, G.: Decidable and undecidable fragments of Halpern and Shoham’s interval temporal logic: towards a complete classification. In: Proceedings of the 15th LPAR, LNCS, vol. 5330, pp. 590–604. Springer (2008)Google Scholar
- 9.Bresolin, D., Della Monica, D., Goranko, V., Montanari, A., Sciavicco, G.: Metric propositional neighborhood interval logics on natural numbers. Softw. Syst. Model.
**12**(2), 245–264 (2013)CrossRefGoogle Scholar - 10.Bresolin, D., Della Monica, D., Goranko, V., Montanari, A., Sciavicco, G.: The dark side of interval temporal logic: marking the undecidability border. Ann. Math. Artif. Intell.
**71**(1–3), 41–83 (2014)MathSciNetCrossRefzbMATHGoogle Scholar - 11.Bresolin, D., Della Monica, D., Montanari, A., Sala, P., Sciavicco, G.: Interval temporal logics over finite linear orders: the complete picture. In: Proceedings of the 20th ECAI, pp. 199–204 (2012)Google Scholar
- 12.Bresolin, D., Della Monica, D., Montanari, A., Sala, P., Sciavicco, G.: Interval temporal logics over strongly discrete linear orders: the complete picture. In: Proceedings of the 3rd GandALF, vol. 96, pp. 155–168. EPTCS (2012)Google Scholar
- 13.Bresolin, D., Goranko, V., Montanari, A., Sala, P.: Tableau-based decision procedures for the logics of subinterval structures over dense orderings. J. Log. Comput.
**20**(1), 133–166 (2010)MathSciNetCrossRefzbMATHGoogle Scholar - 14.Bresolin, D., Goranko, V., Montanari, A., Sciavicco, G.: Propositional interval neighborhood logics: expressiveness, decidability, and undecidable extensions. Ann. Pure Appl. Log.
**161**(3), 289–304 (2009)MathSciNetCrossRefzbMATHGoogle Scholar - 15.Bresolin, D., Montanari, A., Sala, P., Sciavicco, G.: What’s decidable about Halpern and Shoham’s interval logic? The maximal fragment abbl. In: Proceedings of the 26th LICS, pp. 387–396. IEEE Computer Society (2011)Google Scholar
- 16.Bresolin, D., Sala, P., Sciavicco, G.: On begins, meets, and before. Int. J. Found. Comput. Sci.
**23**(3), 559–583 (2012)MathSciNetCrossRefzbMATHGoogle Scholar - 17.Conradie, W., Sciavicco, G.: On the expressive power of first order-logic extended with Allen’s relations in the strict case. In: Proceedings of the 14th CAEPIA, LNCS, vol. 7023, pp. 173–182. Springer (2011)Google Scholar
- 18.Della Monica, D., Goranko, V., Montanari, A., Sciavicco, G.: Expressiveness of the interval logics of Allen’s relations on the class of all linear orders: complete classification. In: Proceedings of the 22nd IJCAI, pp. 845–850 (2011)Google Scholar
- 19.Monica, D., Goranko, V., Montanari, A., Sciavicco, G.: Interval temporal logics: a journey. Bull. Eur. Assoc. Theor. Comput. Sci.
**105**, 73–99 (2011)MathSciNetzbMATHGoogle Scholar - 20.Emerson, E.A.: Temporal and modal logic. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, vol. B: Formal Models and Semantics, pp. 995–1072. MIT Press, Cambridge (1990)Google Scholar
- 21.Gennari, R., Tonelli, S., Vittorini, P.: An AI-based process for generating games from flat stories. In: Proceedings of the 33rd SGAI, pp. 337–350 (2013)Google Scholar
- 22.Halpern, J., Shoham, Y.: A propositional modal logic of time intervals. J. ACM
**38**(4), 935–962 (1991)MathSciNetCrossRefzbMATHGoogle Scholar - 23.Hennessy, M., Milner, R.: Algebraic laws for nondeterminism and concurrency. J. ACM
**32**(1), 137–161 (1985)MathSciNetCrossRefzbMATHGoogle Scholar - 24.Laban, S., El-Desouky, A.: RISMA: A rule-based interval state machine algorithm for alerts generation, performance analysis and monitoring real-time data processing. In: Proceedings of the EGU General Assembly 2013, Geophysical Research Abstracts, vol. 15 (2013)Google Scholar
- 25.Marcinkowski, J., Michaliszyn, J.: The undecidability of the logic of subintervals. Fundam. Inform.
**131**(2), 217–240 (2014)MathSciNetzbMATHGoogle Scholar - 26.Montanari, A., Puppis, G., Sala, P.: Maximal decidable fragments of Halpern and Shoham’s modal logic of intervals. In: Proceedings of the 37th ICALP, LNCS, vol. 6199, pp. 345–356. Springer (2010)Google Scholar
- 27.Montanari, A., Puppis, G., Sala, P., Sciavicco, G.: Decidability of the interval temporal logic \({\sf AB} \overline{{\sf B}}\) over the natural numbers. In: Proceedings of the 31st STACS, pp. 597–608 (2010)Google Scholar
- 28.Moszkowski, B.: Reasoning About Digital Circuits. Technical report. stan-cs-83-970, Dept. of Computer Science, Stanford University, Stanford, CA (1983)Google Scholar
- 29.Pratt-Hartmann, I.: Temporal prepositions and their logic. Artif. Intell.
**166**(1–2), 1–36 (2005)MathSciNetCrossRefzbMATHGoogle Scholar - 30.Rosenstein, J.: Linear Orderings. Academic Press, Waltham (1982)zbMATHGoogle Scholar
- 31.Stirling, C.: Modal and temporal properties of processes. Springer, Berlin (2001)CrossRefzbMATHGoogle Scholar
- 32.Terenziani, P., Snodgrass, R.T.: Reconciling point-based and interval-based semantics in temporal relational databases: a treatment of the telic/atelic distinction. IEEE Trans. Knowl. Data Eng.
**16**(5), 540–551 (2004)CrossRefGoogle Scholar - 33.Venema, Y.: Expressiveness and completeness of an interval tense logic. Notre Dame J. Form. Log.
**31**(4), 529–547 (1990)MathSciNetCrossRefzbMATHGoogle Scholar - 34.Zhou, C., Hansen, M.R.: Duration Calculus: A Formal Approach to Real-Time Systems. EATCS Monographs in Theoretical Computer Science. Springer, Berlin (2004)Google Scholar