Connecting Knowledge Compilation Classes Width Parameters

  • Antoine Amarilli
  • Florent Capelli
  • Mikaël MonetEmail author
  • Pierre Senellart
Part of the following topical collections:
  1. Special Issue on Database Theory (2018)


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).


Knowledge compilation Treewidth Pathwidth Circuit Boolean function 



We acknowledge Chandra Chekuri for his helpful comments at, 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.


  1. 1.
    Amarilli, A., Bourhis, P., Jachiet, L., Mengel, S.: A circuit-based approach to efficient enumeration. In: ICALP (2017)Google Scholar
  2. 2.
    Amarilli, A., Bourhis, P., Monet, M., Senellart, P.: Combined tractability of query evaluation via tree automata and cycluits. In: ICDT (2017)Google Scholar
  3. 3.
    Amarilli, A., Bourhis, P., Senellart, P.: Provenance circuits for trees and treelike instances. In: ICALP (2015)Google Scholar
  4. 4.
    Amarilli, A., Bourhis, P., Senellart, P.: Tractable lineages on treelike instances: limits and extensions. In: PODS (2016)Google Scholar
  5. 5.
    Amarilli, A., Monet, M., Senellart, P.: Connecting width and structure in knowledge compilation. In: ICDT (2018)Google Scholar
  6. 6.
    Beame, P., Li, J., Roy, S., Suciu, D.: Lower bounds for exact model counting and applications in probabilistic databases. In: UAI (2013)Google Scholar
  7. 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)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Beame, P., Liew, V.: New limits for knowledge compilation and applications to exact model counting. In: UAI (2015)Google Scholar
  9. 9.
    Bodlaender, H.L.: A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 5(6), 1305?-1317 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bollig, B., Buttkus, M.: On the relative succinctness of sentential decision diagrams. arXiv:1802.04544 (2018)
  12. 12.
    Bollig, B., Wegener, I.: Complexity theoretical results on partitioned (nondeterministic) binary decision diagrams. In: MFCS (1997)Google Scholar
  13. 13.
    Bova, S.: SDDs are exponentially more succinct than OBDDs. In: AAAI (2016)Google Scholar
  14. 14.
    Bova, S., Capelli, F., Mengel, S., Slivovsky, F.: Knowledge compilation meets communication complexity. In: IJCAI (2016)Google Scholar
  15. 15.
    Bova, S., Slivovsky, F.: On compiling structured CNFs to OBDDs. In: International Computer Science Symposium in Russia, pp 80–93. Springer (2015)Google Scholar
  16. 16.
    Bova, S., Szeider, S.: Circuit treewidth, sentential decision, and query compilation. In: PODS (2017)Google Scholar
  17. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Bryant, R.E.: Symbolic Boolean manipulation with ordered binary-decision diagrams. ACM Comput. Surv. 24(3), 293–318 (1992)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Calí, A., Capelli, F., Razgon, I.: Non-FPT lower bounds for structural restrictions of decision DNNF. arXiv:1708.07767v1 (2017)
  20. 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. 21.
    Capelli, F.: Understanding the complexity of #SAT using knowledge compilation. In: LICS (2017)Google Scholar
  22. 22.
    Capelli, F., Mengel, S.: Tractable QBF by knowledge compilation. In: STACS (2019)Google Scholar
  23. 23.
    Capelli, F., Strozecki, Y.: Enumerating models of DNF faster: Breaking the dependency on the formula size. arXiv:1810.04006 (2018)
  24. 24.
    Creignou, N., Olive, F., Schmidt, J.: Enumerating all solutions of a Boolean CSP by non-decreasing weight. In: SAT (2011)Google Scholar
  25. 25.
    Darwiche, A.: Decomposable negation normal form. JACM 48(4), 608–647 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Darwiche, A.: A differential approach to inference in Bayesian networks. JACM 50(3), 280–305 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Darwiche, A.: SDD: A new canonical representation of propositional knowledge bases. In: IJCAI (2011)Google Scholar
  29. 29.
    Darwiche, A., Marquis, P.: A knowledge compilation map. JAIR 17, 229–264 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 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)CrossRefGoogle Scholar
  31. 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)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Grohe, M., Marx, D.: On tree width, bramble size, and expansion. J. Combinatorial Theory Series B 99(1), 218–228 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Jha, A.K., Olteanu, D., Suciu, D.: Bridging the gap between intensional and extensional query evaluation in probabilistic databases. In: EDBT (2010)Google Scholar
  34. 34.
    Jha, A.K., Suciu, D.: On the tractability of query compilation and bounded treewidth. In: ICDT (2012)Google Scholar
  35. 35.
    Jha, A.K., Suciu, D.: Knowledge compilation meets database theory: Compiling queries to decision diagrams. TCS 52(3), 403–?440 (2013)MathSciNetzbMATHGoogle Scholar
  36. 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)Google Scholar
  37. 37.
    Meinel, C., Theobald, T.: Algorithms and Data Structures in VLSI Design: OBDD-foundations and applications. Springer Science & Business Media (2012)Google Scholar
  38. 38.
    Monet, M., Olteanu, D.: Towards deterministic decomposable circuits for safe queries. In: AMW (2018)Google Scholar
  39. 39.
    Nordstrand, J.A.: Exploring graph parameters similar to tree-width and path-width. University of Bergen, Master’s thesis (2017)Google Scholar
  40. 40.
    Pipatsrisawat, K., Darwiche, A.: New compilation languages based on structured decomposability. In: AAAI (2008)Google Scholar
  41. 41.
    Pipatsrisawat, K., Darwiche, A.: A lower bound on the size of decomposable negation normal form. In: AAAI (2010)Google Scholar
  42. 42.
    Pipatsrisawat, T.: Reasoning with propositional knowledge: Frameworks for Boolean satisfiability and knowledge compilation. Ph.D. thesis, University of California (2010)Google Scholar
  43. 43.
    Razgon, I.: On OBDDs for CNFs of bounded treewidth. In: KR (2014)Google Scholar
  44. 44.
    Razgon, I.: No small nondeterministic read-once branching programs for CNFs of bounded treewidth. In: IPEC (2014)Google Scholar
  45. 45.
    Robertson, N., Seymour, P.: Graph minors. X. Obstructions to tree-decomposition. Journal of Combinatorial Theory Series B 52(2), 153–190 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Sauerhoff, M.: Approximation of boolean functions by combinatorial rectangles. Theor. Comput. Sci. 301(1–3), 45–78 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Sherstov, A.A.: Communication complexity theory: Thirty-five years of set disjointness. In: MFCS (2014)Google Scholar
  48. 48.
    Strozecki, Y.: Enumeration complexity and matroid decomposition, p 7. Ph.D. Thesis, Paris (2010)Google Scholar
  49. 49.
    Suciu, D., Olteanu, D., Ré, C., Koch, C.: Probabilistic databases. Morgan & Claypool (2011)Google Scholar
  50. 50.
    Szeider, S.: On fixed-parameter tractable parameterizations of SAT. In: SAT (2004)Google Scholar
  51. 51.
    Wegener, I.: The complexity of Boolean functions. Wiley, New York (1991)zbMATHGoogle Scholar
  52. 52.
    Wegener, I.: Branching programs and binary decision diagrams: Theory and applications. SIAM (2000)Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.LTCI, Télécom ParisTechUniversité Paris-SaclayParisFrance
  2. 2.CRIStALUniversité de Lille, CNRSLilleFrance
  3. 3.Inria, LilleParisFrance
  4. 4.DI ENS, ENS, CNRSPSL UniversityParisFrance

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