## Abstract

Many constraints restricting the result of some computations over an integer sequence can be compactly represented by counter automata. We improve the propagation of the conjunction of such constraints on the same sequence by synthesising a database of linear and non-linear invariants using their counter-automaton representation. The obtained invariants are formulae parameterised by the sequence length and proven to be true for any long enough sequence. To assess the quality of such linear invariants, we developed a method to verify whether a generated linear invariant is a facet of the convex hull of the feasible points. This method, as well as the proof of non-linear invariants, are based on the systematic generation of constant-size deterministic finite automata that accept all integer sequences whose result verifies some simple condition. We apply such methodology to a set of 44 time-series constraints and obtain 1400 linear invariants from which 70% are facet defining, and 600 non-linear invariants, which were tested on short-term electricity production problems.

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## References

- 1.
Appa, G., Magos, D., & Mourtos, I. (2004). LP relaxations of multiple all_different predicates. In Régin, J-C, & Rueher, M (Eds.)

*Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems, First International Conference, CPAIOR 2004, volume 3011 of LNCS, pages 364–369, Springer*. - 2.
Arafailova, E. (2018). Functional Description of Sequence Constraints and Synthesis of Combinatorial Objects. PhD Thesis, IMT Atlantique LS2N.

- 3.
Arafailova, E., Beldiceanu, N., Douence, R., Carlsson, M., Flener, P., Rodríguez, M. A. F., Pearson, J., & Simonis, H. (2018). Global Constraint Catalog, Volume ii, time-Series Constraints. CoRR, abs/1609, 08925.

- 4.
Arafailova, E., Beldiceanu, N., & Simonis, H. (2017). Generating linear invariants for a conjunction of automata constraints. In Beck, C (Ed.)

*Principles and Practice of Constraint Programming - CP 2017, volume 10416 of LNCS, pages 21–37. Springer*. - 5.
Arafailova, E., Beldiceanu, N., & Simonis, H. (2018). Deriving generic bounds for time-series constraints based on regular expressions characteristics.

*Constraints*,*23*(1), 44–86. - 6.
Arafailova, E., Beldiceanu, N., Carlsson, M., Flener, P., Rodríguez, M. A. F., Pearson, J., & Simonis, H. (2016). Systematic derivation of bounds and glue constraints for time-series constraints. In Rueher, M (Ed.)

*Principles and Practice of Constraint Programming - CP 2016, volume 9892 of LNCS, pages 13–29. Springer*. - 7.
Arafailova, E., Beldiceanu, N., Douence, R., Flener, P., Rodríguez, M A. F., Pearson, J, & Simonis, H. (2016). Time-Series Constraints: Improvements and Application in CP and MIP Contexts. In Quimper, C-G (Ed.)

*Integration of AI and OR Techniques in Constraint Programming - CPAIOR 2016, volume 9676 of LNCS, pages 18–34. Springer*. - 8.
Beldiceanu, N., Carlsson, M., Debruyne, R., & Petit, T. (2005). Reformulation of global constraints based on constraints checkers.

*Constraints*,*10*(4), 339–362. - 9.
Beldiceanu, N., Carlsson, M., Douence, R., & Simonis, H. (2015). Using finite transducers for describing and synthesising structural time-series constraints.

*Constraints*,*21*(1), 22–40. January 2016. Journal fast track of CP 2015: summary on p. 723 of LNCS 9255, Springer. - 10.
Beldiceanu, N., Ifrim, G., Lenoir, A., & Simonis, H. (2013). Describing and generating solutions for the EDF unit commitment problem with the Model Seeker. In Schulte, C (Ed.)

*Principles and Practice of Constraint Programming - CP 2013, volume 8124 of LNCS, pages 733–748. Springer*. - 11.
Beldiceanu, N., Carlsson, M., Rampon, J.-X., & Truchet, C. (2005). Graph invariants as necessary conditions for global constraints. In Beek, P V (Ed.)

*Principles and Practice of Constraint Programming - CP 2005, volume 3709 of LNCS, pages 92–106. Springer*. - 12.
Beldiceanu, N., & Contejean, E. (1994). Introducing global constraints in CHIP.

*Mathl Comput Modelling*,*20*(12), 97–123. - 13.
Beldiceanu, N., Flener, P., Pearson, J., & Hentenryck, P. V. (2014). Propagating regular counting constraints. In Brodley, C E , & Stone, P (Eds.)

*AAAI 2014, pages 2616–2622. AAAI Press*. - 14.
Beldiceanu, N., Mats, C., & Petit, T. (2004). Deriving filtering algorithms from constraint checkers. In Wallace, M (Ed.)

*Principles and Practice of Constraint Programming - CP 2004, volume 3258 of LNCS, pages 107–122. Springer*. - 15.
Boutilier, C., Patrascu, R., Poupart, P., & Schuurmans, D. (2003). Constraint-based optimization with the minimax decision criterion. In Rossi, F (Ed.)

*Principles and Practice of Constraint Programming - CP 2003, volume 2833 of LNCS, pages 168–182. Springer*. - 16.
Boyd, S. P., & Vandenberghe, L. (2004). Convex optimization. Cambridge University Press.

- 17.
Bshouty, N. H., Drachsler-Cohen, D., Vechev, M.T., & Yahav, E. (2017). Learning disjunctions of predicates. In Satyen K, & Ohad S (Eds.)

*Proceedings of the 30th Conference on Learning Theory, COLT 2017, Amsterdam, The Netherlands, 7–10 July 2017*, (Vol. 65 pp. 346–369). - 18.
Bshouty, N. H., Goldberg, P. W., Goldman, S. A., & Mathias, H. D. (1999). Exact learning of discretized geometric concepts.

*SIAM J Comput*,*28*(2), 674–699. - 19.
Charnley, J., Colton, S., & Miguel, I. (2006). Automatic generation of implied constraints. In

*ECAI 2006, volume 141 of Frontiers in AI and Applications, pages 73–77. IOS Press*. - 20.
Chen, Z., & Ameur, F. (1999). The learnability of unions of two rectangles in the two-dimensional discretized space.

*Journal of Computer and System Sciences*,*59*(1), 70–83. - 21.
COSYTEC, CHIP Reference Manual Release 5.1 edition, 1997.

- 22.
Crochemore, M., Hancart, C., & Lecroq, T. (2007). Algorithms on strings. Cambridge University Press.

- 23.
Dincbas, M., Simonis, H., & Hentenryck, P. V. (1988). Solving the car-sequencing problem in constraint logic programming. In

*ECAI*(pp. 290–295). - 24.
Rodríguez, M. A. F., Flener, P., & Pearson, J. (2015). Implied constraints for Automaton constraints. In Gottlob, G, Sutcliffe, G, & Voronkov, A (Eds.)

*Global Conference on Artificial Intelligence, GCAI 2015, volume 36 of EPiC Series in Computing, pages 113–126. EasyChair*. - 25.
Rodríguez, M. A. F., Flener, P., & Pearson, J. (2017). Automatic generation of descriptions of time-series constraints. In Brodsky, A (Ed.)

*29th IEEE International Conference on Tools with Artificial Intelligence, ICTAI 2017, pages 102–109. IEEE Computer Society*. - 26.
Decision theory: An Introduction to the Mathematics of Rationality. (1986) French, S (Ed.), Halsted Press, New York.

- 27.
Frisch, A., Miguel, I., & Walsh, T. (2001). Extensions to proof planning for generating implied constraints. In

*9th Symp. on the Integration of Symbolic Computation and Mechanized Reasoning*. - 28.
Graham, R. L. (1972). An efficient algorithm for determining the convex hull of a finite planar set.

*Information Processing Letters*,*1*(4), 132–133. - 29.
Hansen, P., & Caporossi, G. (2000). Autographix: an automated system for finding conjectures in graph theory.

*Electronic Notes in Discrete Mathematics*,*5*, 158–161. - 30.
Hooker, J.N. (2011). Integrated Methods for Optimization, 2nd edn., Springer Publishing Company Incorporated, Berlin.

- 31.
Lee, J. (2002). All-different polytopes.

*J Comb Optim*,*6*(3), 335–352. - 32.
Menana, J. (Oct 2011). Automata and constraint programming for personnel scheduling problems. PhD Thesis, Universite de Nanteś.

- 33.
Menana, J., & Demassey, S. (2009). Sequencing and counting with the multicost-regular constraint. In Hoeve, W. J. V., & Hooker, J. N. (Eds.)

*Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems, 6th International Conference, CPAIOR 2009, volume 5547 of LNCS, pages 178–192, Springer*. - 34.
Pesant, G. (2001). A filtering algorithm for the stretch constraint. In Walsh, T (Ed.)

*Principles and Practice of Constraint Programming - CP 2001, 7th International Conference, CP 2001, volume 2239 of LNCS, pages 183–195. Springer*. - 35.
Leonard J. S. (1951). The theory of statistical decision.

*Journal of the American Statistical Association*,*46*(253), 55–67. - 36.
Veanes, M., Hooimeijer, P., Livshits, B., Molnar, D., & Bjørner, N. D. (2012). Symbolic finite state transducers: algorithms and applications. In Field, J, & Hicks, M (Eds.)

*Proceedings of the 39th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL 2012, pages 137–150. ACM*.

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This is an extended version of the CP 2017 article [4]. Ekaterina Arafailova is supported by the EU H2020 programme under grant 640954 for project GRACeFUL. Nicolas Beldiceanu is partially supported by the GRACeFUL project and by the Gaspard Monge Program for Optimisation and Operations Research (PGMO). Helmut Simonis is supported by Science Foundation Ireland (SFI) under grant SFI/10/IN.1/I3032; the Insight Centre for Data Analytics is supported by SFI under grant SFI/12/RC/2289.

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Arafailova, E., Beldiceanu, N. & Simonis, H. Invariants for time-series constraints.
*Constraints* **25, **71–120 (2020). https://doi.org/10.1007/s10601-020-09308-z

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### Keywords

- Register automaton
- Time-series constraints
- Linear invariant
- Non-linear invariant
- Parameterised invariant
- Finite automaton