Skip to main content
SpringerLink
Log in

Connecting Knowledge Compilation Classes Width Parameters

  • Published:
Theory of Computing Systems Aims and scope Submit manuscript

A Correction to this article was published on 25 January 2020

This article has been updated

Abstract

The field of knowledge compilation establishes the tractability of many tasks by studying how to compile them to Boolean circuit classes obeying some requirements such as structuredness, decomposability, and determinism. However, in other settings such as intensional query evaluation on databases, we obtain Boolean circuits that satisfy some width bounds, e.g., they have bounded treewidth or pathwidth. In this work, we give a systematic picture of many circuit classes considered in knowledge compilation and show how they can be systematically connected to width measures, through upper and lower bounds. Our upper bounds show that bounded-treewidth circuits can be constructively converted to d-SDNNFs, in time linear in the circuit size and singly exponential in the treewidth; and that bounded-pathwidth circuits can similarly be converted to uOBDDs. We show matching lower bounds on the compilation of monotone DNF or CNF formulas to structured targets, assuming a constant bound on the arity (size of clauses) and degree (number of occurrences of each variable): any d-SDNNF (resp., SDNNF) for such a DNF (resp., CNF) must be of exponential size in its treewidth, and the same holds for uOBDDs (resp., n-OBDDs) when considering pathwidth. Unlike most previous work, our bounds apply to any formula of this class, not just a well-chosen family. Hence, we show that pathwidth and treewidth respectively characterize the efficiency of compiling monotone DNFs to uOBDDs and d-SDNNFs with compilation being singly exponential in the corresponding width parameter. We also show that our lower bounds on CNFs extend to unstructured compilation targets, with an exponential lower bound in the treewidth (resp., pathwidth) when compiling monotone CNFs of constant arity and degree to DNNFs (resp., nFBDDs).

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

Change history

  • 25 January 2020

    The article title in the original publication contains an error. The correct title is presented in this Erratum. The online version of the original article can be found at: <ExternalRef><RefSource>https://doi.org/10.1007/s00224-019-09930-2</RefSource><RefTarget Address="10.1007/s00224-019-09930-2" TargetType="DOI"/></ExternalRef>

Notes

  1. This observation is due to Stefan Mengel and is adapted from the recent article [22].

References

  1. Amarilli, A., Bourhis, P., Jachiet, L., Mengel, S.: A circuit-based approach to efficient enumeration. In: ICALP (2017)

  2. Amarilli, A., Bourhis, P., Monet, M., Senellart, P.: Combined tractability of query evaluation via tree automata and cycluits. In: ICDT (2017)

  3. Amarilli, A., Bourhis, P., Senellart, P.: Provenance circuits for trees and treelike instances. In: ICALP (2015)

  4. Amarilli, A., Bourhis, P., Senellart, P.: Tractable lineages on treelike instances: limits and extensions. In: PODS (2016)

  5. Amarilli, A., Monet, M., Senellart, P.: Connecting width and structure in knowledge compilation. In: ICDT (2018)

  6. Beame, P., Li, J., Roy, S., Suciu, D.: Lower bounds for exact model counting and applications in probabilistic databases. In: UAI (2013)

  7. Beame, P., Li, J., Roy, S., Suciu, D.: Exact model counting of query expressions: Limitations of propositional methods. ACM Trans. Database Syst. (TODS) 42(1), 1 (2017)

    Article  MathSciNet  Google Scholar 

  8. Beame, P., Liew, V.: New limits for knowledge compilation and applications to exact model counting. In: UAI (2015)

  9. Bodlaender, H.L.: A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 5(6), 1305?-1317 (1996)

    Article  MathSciNet  Google Scholar 

  10. Bodlaender, H.L., Drange, P.G., Dregi, M.S., Fomin, F.V., Lokshtanov, D., Pilipczuk, M.: A ck n 5-approximation algorithm for treewidth. SIAM J. Comput. 45(2), 317?-378 (2016)

    Article  MathSciNet  Google Scholar 

  11. Bollig, B., Buttkus, M.: On the relative succinctness of sentential decision diagrams. arXiv:1802.04544 (2018)

  12. Bollig, B., Wegener, I.: Complexity theoretical results on partitioned (nondeterministic) binary decision diagrams. In: MFCS (1997)

  13. Bova, S.: SDDs are exponentially more succinct than OBDDs. In: AAAI (2016)

  14. Bova, S., Capelli, F., Mengel, S., Slivovsky, F.: Knowledge compilation meets communication complexity. In: IJCAI (2016)

  15. Bova, S., Slivovsky, F.: On compiling structured CNFs to OBDDs. In: International Computer Science Symposium in Russia, pp 80–93. Springer (2015)

  16. Bova, S., Szeider, S.: Circuit treewidth, sentential decision, and query compilation. In: PODS (2017)

  17. Bryant, R.E.: On the complexity of VLSI implementations and graph representations of Boolean functions with application to integer multiplication. IEEE Trans. Comput. 40(2), 205–213 (1991)

    Article  MathSciNet  Google Scholar 

  18. Bryant, R.E.: Symbolic Boolean manipulation with ordered binary-decision diagrams. ACM Comput. Surv. 24(3), 293–318 (1992)

    Article  MathSciNet  Google Scholar 

  19. Calí, A., Capelli, F., Razgon, I.: Non-FPT lower bounds for structural restrictions of decision DNNF. arXiv:1708.07767v1 (2017)

  20. Capelli, F.: Structural restrictions of CNF-formulas: Applications to model counting and knowledge compilation. Ph.D. thesis, Université Paris-Diderot (2016)

    Google Scholar 

  21. Capelli, F.: Understanding the complexity of #SAT using knowledge compilation. In: LICS (2017)

  22. Capelli, F., Mengel, S.: Tractable QBF by knowledge compilation. In: STACS (2019)

  23. Capelli, F., Strozecki, Y.: Enumerating models of DNF faster: Breaking the dependency on the formula size. arXiv:1810.04006 (2018)

  24. Creignou, N., Olive, F., Schmidt, J.: Enumerating all solutions of a Boolean CSP by non-decreasing weight. In: SAT (2011)

  25. Darwiche, A.: Decomposable negation normal form. JACM 48(4), 608–647 (2001)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  27. Darwiche, A.: A differential approach to inference in Bayesian networks. JACM 50(3), 280–305 (2003)

    Article  MathSciNet  Google Scholar 

  28. Darwiche, A.: SDD: A new canonical representation of propositional knowledge bases. In: IJCAI (2011)

  29. Darwiche, A., Marquis, P.: A knowledge compilation map. JAIR 17, 229–264 (2002)

    Article  MathSciNet  Google Scholar 

  30. Devadas, S.: Comparing two-level and ordered binary decision diagram representations of logic functions. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 12(5), 722–723 (1993)

    Article  Google Scholar 

  31. Fierens, D., den Broeck, G.V., Renkens, J., Shterionov, D., Gutmann, B., Thon, I., Janssens, G., Raedt, L.D.: Inference and learning in probabilistic logic programs using weighted Boolean formulas. TPLP 15(3), 358–401 (2015)

    MathSciNet  MATH  Google Scholar 

  32. Grohe, M., Marx, D.: On tree width, bramble size, and expansion. J. Combinatorial Theory Series B 99(1), 218–228 (2009)

    Article  MathSciNet  Google Scholar 

  33. Jha, A.K., Olteanu, D., Suciu, D.: Bridging the gap between intensional and extensional query evaluation in probabilistic databases. In: EDBT (2010)

  34. Jha, A.K., Suciu, D.: On the tractability of query compilation and bounded treewidth. In: ICDT (2012)

  35. Jha, A.K., Suciu, D.: Knowledge compilation meets database theory: Compiling queries to decision diagrams. TCS 52(3), 403–?440 (2013)

    MathSciNet  MATH  Google Scholar 

  36. Lauritzen, S.L., Spiegelhalter, D.J.: Local computations with probabilities on graphical structures and their application to expert systems. J. Royal Statistical Society Series B (1988)

  37. Meinel, C., Theobald, T.: Algorithms and Data Structures in VLSI Design: OBDD-foundations and applications. Springer Science & Business Media (2012)

  38. Monet, M., Olteanu, D.: Towards deterministic decomposable circuits for safe queries. In: AMW (2018)

  39. Nordstrand, J.A.: Exploring graph parameters similar to tree-width and path-width. University of Bergen, Master’s thesis (2017)

    Google Scholar 

  40. Pipatsrisawat, K., Darwiche, A.: New compilation languages based on structured decomposability. In: AAAI (2008)

  41. Pipatsrisawat, K., Darwiche, A.: A lower bound on the size of decomposable negation normal form. In: AAAI (2010)

  42. Pipatsrisawat, T.: Reasoning with propositional knowledge: Frameworks for Boolean satisfiability and knowledge compilation. Ph.D. thesis, University of California (2010)

    Google Scholar 

  43. Razgon, I.: On OBDDs for CNFs of bounded treewidth. In: KR (2014)

  44. Razgon, I.: No small nondeterministic read-once branching programs for CNFs of bounded treewidth. In: IPEC (2014)

  45. Robertson, N., Seymour, P.: Graph minors. X. Obstructions to tree-decomposition. Journal of Combinatorial Theory Series B 52(2), 153–190 (1991)

    Article  MathSciNet  Google Scholar 

  46. Sauerhoff, M.: Approximation of boolean functions by combinatorial rectangles. Theor. Comput. Sci. 301(1–3), 45–78 (2003)

    Article  MathSciNet  Google Scholar 

  47. Sherstov, A.A.: Communication complexity theory: Thirty-five years of set disjointness. In: MFCS (2014)

  48. Strozecki, Y.: Enumeration complexity and matroid decomposition, p 7. Ph.D. Thesis, Paris (2010)

    Google Scholar 

  49. Suciu, D., Olteanu, D., Ré, C., Koch, C.: Probabilistic databases. Morgan & Claypool (2011)

  50. Szeider, S.: On fixed-parameter tractable parameterizations of SAT. In: SAT (2004)

  51. Wegener, I.: The complexity of Boolean functions. Wiley, New York (1991)

    MATH  Google Scholar 

  52. Wegener, I.: Branching programs and binary decision diagrams: Theory and applications. SIAM (2000)

Download references

Acknowledgments

We acknowledge Chandra Chekuri for his helpful comments at https://cstheory.stackexchange.com/a/38943/, as well as Stefan Mengel for pointing us to a notion of width for d-SDNNFs and suggesting a strengthening of our complexity upper bound in Theorem 4.2.

Author information

Authors and Affiliations

  1. LTCI, Télécom ParisTech, Université Paris-Saclay, Paris, France

    Antoine Amarilli, Mikaël Monet & Pierre Senellart

  2. CRIStAL, Université de Lille, CNRS, Lille, France

    Florent Capelli

  3. Inria, Lille, Paris, France

    Florent Capelli, Mikaël Monet & Pierre Senellart

  4. DI ENS, ENS, CNRS, PSL University, Paris, France

    Pierre Senellart

Authors
  1. Antoine Amarilli
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Florent Capelli
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mikaël Monet
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Pierre Senellart
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mikaël Monet.

Additional information

Publisher’s Note

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

This article is part of the Topical Collection on Special Issue on Database Theory (2018)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Amarilli, A., Capelli, F., Monet, M. et al. Connecting Knowledge Compilation Classes Width Parameters. Theory Comput Syst 64, 861–914 (2020). https://doi.org/10.1007/s00224-019-09930-2

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00224-019-09930-2

Keywords

Search

Navigation