Skip to main content
SpringerLink
Log in

Propagation complete encodings of smooth DNNF theories

  • Published:
Constraints Aims and scope Submit manuscript

Abstract

We investigate conjunctive normal form (CNF) encodings of a function represented with a decomposable negation normal form (DNNF). Several encodings of DNNFs and decision diagrams were considered by (Abío et al., 2016). The authors differentiate between encodings which implement consistency or domain consistency by unit propagation from encodings which are unit refutation complete or propagation complete. The difference is that in the former case we do not care about propagation strength of the encoding with respect to the auxiliary variables while in the latter case we treat all variables (the main and the auxiliary ones) in the same way. The currently known encodings of DNNF theories implement domain consistency. Building on these encodings we generalize the result of (Abío et al., 2016) on a propagation complete encoding of decision diagrams and present a propagation complete encoding of a DNNF and its generalization for variables with finite domains.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Abío, I., Gange, G., Mayer-Eichberger, V., & Stuckey, P. J. (2016). On CNF encodings of decision diagrams. In C.-G. Quimper (Ed.) Integration of AI and OR techniques in constraint programming (pp. 1–17). Cham: Springer International Publishing.

  2. Abío, I., Nieuwenhuis, R., Oliveras, A., Rodríguez-Carbonell, E., & Stuckey, P. J. (2013). To encode or to propagate? The best choice for each constraint in SAT. In Christian Schulte (Ed.) Principles and practice of constraint programming (pp. 97–106). Berlin: Springer.

  3. Abío, I., & Stuckey, P. J. (2012). Conflict directed lazy decomposition. In M. Milano (Ed.) Principles and practice of constraint programming (pp. 70–85). Berlin: Springer.

  4. Babaki, B., & Pesant, G. (2020). Parallel planning using a lazy clause generation solver. In 32nd IEEE international conference on tools with artificial intelligence, ICTAI 2020, Baltimore, MD, USA, November 9-11, 2020, (pp. 272–276). IEEE.

  5. Babka, M., Balyo, T., Čepek, O., Gurský, Š., Kučera, P., & Vlček, V. (2013). Complexity issues related to propagation completeness. Artificial Intelligence, 203(0), 19–34.

    Article  MathSciNet  Google Scholar 

  6. Bacchus, F. (2007). GAC via unit propagation. In C. Bessière (Ed.) Principles and practice of constraint programming – CP 2007, volume 4741 of lecture notes in computer science (pp. 133–147. Berlin: Springer).

  7. Barrett, A. (2004). From hybrid systems to universal plans via domain compilation. In Principles of knowledge representation and reasoning: proceedings of the ninth international conference (KR2004) (pp. 654–661).

  8. Bessiere, C., Katsirelos, G., Narodytska, N., & Walsh, T. (2009). Circuit complexity and decompositions of global constraints. In Proceedings of the twenty-first international joint conference on artificial intelligence (IJCAI-09) (pp. 412–418).

  9. Biere, A., Heule, M., van Maaren, H., & Walsh, T. (2009). Handbook of satisfiability, volume 185 of frontiers in artificial intelligence and applications. Amsterdam: IOS Press.

  10. Bordeaux, L., & Bieliková, J. M. -S. (2012). Knowledge compilation with empowerment. In M. Bieliková, G. Friedrich, G. Gottlob, S. Katzenbeisser, & G. Turan (Eds.) SOFSEM 2012: Theory and practice of computer science, volume 7147 of lecture notes in computer science (pp. 612–624). Berlin: Springer.

  11. Bova, S., Capelli, F., Mengel, S., & Slivovsky, F. (2016). Knowledge compilation meets communication complexity. In Proceedings of the twenty-fifth international joint conference on artificial intelligence, IJCAI’16, (pp. 1008–1014). AAAI Press.

  12. Chen, J. (2010). A new SAT encoding of the at-most-one constraint. In Proc. constraint modelling and reformulation.

  13. Darwiche, A. (1999). Compiling knowledge into decomposable negation normal form. In Proceedings of the 16th international joint conference on artifical intelligence - volume 1, IJCAI’99, (pp. 284–289), San Francisco: Morgan Kaufmann Publishers Inc.

  14. Darwiche, A. (2001). On the tractable counting of theory models and its application to truth maintenance and belief revision. Journal of Applied Non-Classical Logics, 11(1-2), 11–34.

    Article  MathSciNet  Google Scholar 

  15. Darwiche, A. (2011). SDD: A new canonical representation of propositional knowledge bases. In Proceedings of the twenty-second international joint conference on artificial intelligence - volume volume two, IJCAI’11, (pp. 819–826). AAAI Press.

  16. Darwiche, A., & Marquis, P. (2002). A knowledge compilation map. Journal of Artificial Intelligence Research, 17, 229–264.

    Article  MathSciNet  Google Scholar 

  17. de Uña, D., Gange, G., Schachte, P., & Stuckey, P. J. (2019). Compiling CP subproblems to MDDs and d-DNNFs. Constraints, 24(1), 56–93.

    Article  MathSciNet  Google Scholar 

  18. del Val, A. (1994). Tractable databases: How to make propositional unit resolution complete through compilation. In Knowledge representation and reasoning, (pp. 551–561).

  19. Elliott, P., & Williams, B. (2006). DNNF-based belief state estimation. In Proceedings of the national conference on artificial intelligence, (vol. 21, pp. 36). Menlo Park, CA, Cambridge, MA; London; AAAI Press; MIT Press, 1999.

  20. Frisch, A. M., & Giannaros, P. A. (2010). SAT encodings of the at-most-k constraint. some old, some new, some fast, some slow. In Proc. of the tenth int. workshop of constraint modelling and reformulation.

  21. Frisch, A. M., Peugniez, T. J., & Doggett, A. J. (2005). Solving non-boolean satisfiability problems with stochastic local search: A comparison of encodings. Journal of Automated Reasoning, 35(1-3), 143–179.

    Article  MathSciNet  Google Scholar 

  22. Gange, G., Harabor, D., & Stuckey, P. J. (2019). Lazy CBS: implicit conflict-based search using lazy clause generation. In J. Benton, .r Lipovetzky, E. Onaindia, D. E. Smith, & S. Srivastava (Eds.) Proceedings of the twenty-ninth international conference on automated planning and scheduling, ICAPS 2018, Berkeley, CA, USA, July 11-15, 2019 (pp. 155–162). AAAI Press.

  23. Gange, G., & Stuckey, P. J. (2012). Explaining propagators for s-DNNF circuits. In N. Beldiceanu, N. Jussien, & É. Pinson (Eds.) Integration of AI and OR techniques in contraint programming for combinatorial optimzation problems, (pp. 195–210). Springer: Berlin.

  24. Gange, G.s, Stuckey, P. J., & Szymanek, R. (2011). MDD Propagators with explanation. Constraints, 16(4), 407.

    Article  MathSciNet  Google Scholar 

  25. Gange, G., Stuckey, P. J., & Van Hentenryck, P. (2013). Explaining propagators for edge-valued decision diagrams. In International conference on principles and practice of constraint programming, (pp. 340–355). Springer.

  26. Gent, I. P., & Nightingale, P. (2004). A new encoding of AllDifferent into SAT. International Workshop on Modelling and Reformulating Constraint Satisfaction, 3, 95–110.

    Google Scholar 

  27. Gwynne, M., & Kullmann, O. (2013). Generalising and unifying SLUR and unit-refutation completeness. In P. van Emde Boas, Frans C. A. Groen, G. F. Italiano, J. Nawrocki, & H. Sack (Eds.) SOFSEM 2013: theory and practice of computer science, (pp. 220–232), Berlin: Springer.

  28. Hölldobler, S., & Nguyen, V. H. (2013). An efficient encoding of the at-most-one constraint. In Technical report MSU-CSE-00-2 knowledge representation and reasoning group 2013-04 Technische universität Dresden, 01062 Dresden, Germany.

  29. Huang, J (2006). Complan: A conformant probabilistic planner. In Proceedings of the 16th international conference on planning and scheduling (ICAPS).

  30. Järvisalo, M., & Junttila, T. A. (2009). Limitations of restricted branching in clause learning Constraints An. International Journal, 14(3), 325–356.

    MathSciNet  MATH  Google Scholar 

  31. Jung, J. C., Barahona, P., Katsirelos, G., & Walsh, T. (2008). Two encodings of DNNF theories. In ECAI’08 Workshop on inference methods based on graphical structures of knowledge.

  32. Kučera, P., & Savický, P. (2021). Backdoor decomposable monotone circuits and propagation complete encodings. Proceedings of the AAAI Conference on Artificial Intelligence, 35(5), 3832–3840.

    Google Scholar 

  33. Nguyen, V.-H., & Mai, S. T. (2015). A new method to encode the at-most-one constraint into SAT. In H. Q. Thang, L. A. Phuong, L. D. Raedt, Y. Deville, M. Bui, T. T. D. Linh, N. Thi-Oanh, D. V. Sang, & N. B. Ngoc (Eds.) Proceedings of the sixth international symposium on information and communication technology, Hue City, Vietnam, December 3-4, 2015, (pp. 46–53). ACM.

  34. Palacios, H., Bonet, B., Darwiche, A., & Geffner, H. (2005). Pruning conformant plans by counting models on compiled d-DNNF representations. In ICAPS, 5, 141–150.

    Google Scholar 

  35. Palacios, H., & Geffner, H. (2005). Mapping conformant planning into SAT through compilation and projection. In Conference of the spanish association for artificial intelligence (pp. 311–320). Springer.

  36. Quimper, C. -G., & Walsh, T. (2007). Decomposing global grammar constraints. In C. Bessière (Ed.) Principles and practice of constraint programming – CP 2007: 13th international conference, CP 2007, Providence, RI, USA, September 23-27, 2007. Proceedings, (pp. 590–604), Berlin: Heidelberg.

  37. Schlipf, J. S., Annexstein, F. S., Franco, John V., & Swaminathan, R. P. (1995). On finding solutions for extended Horn formulas. Information Processing Letters, 54(3), 133–137.

    Article  MathSciNet  Google Scholar 

  38. Schumann, A., & Sachenbacher, M. (2010). Computing energy-optimal tests using DNNF graphs. In Proceedings of the twenty-first international workshop on the principles of diagnosis.

  39. Sinz, C. (2005). Towards an optimal CNF encoding of boolean cardinality constraints. In P. van Beek (Ed.) Principles and practice of constraint programming — CP 2005: 11th international conference, CP 2005, Sitges, Spain, October 1-5, 2005. Proceedings, (pp. 827–831). Berlin: Springer.

  40. Voronov, A., Åkesson, K., & Ekstedt, F. (2011). Enumeration of valid partial configurations. In Proceedings of workshop on configuration, IJCAI 2011, (Vol. 755, pp. 25–31).

Download references

Acknowledgements

Both authors gratefully acknowledge the support by Grant Agency of the Czech Republic (grant No. GA19–19463S).

Author information

Authors and Affiliations

  1. Department of Theoretical Computer Science and Mathematical Logic, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic

    Petr Kučera

  2. Institute of Computer Science, The Czech Academy of Sciences, Prague, Czech Republic

    Petr Savický

Authors
  1. Petr Kučera
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Petr Savický
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Petr Kučera.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kučera, P., Savický, P. Propagation complete encodings of smooth DNNF theories. Constraints 27, 327–359 (2022). https://doi.org/10.1007/s10601-022-09331-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10601-022-09331-2

Search

Navigation