Abstract
For Boolean satisfiability problems, the structure of the solution space is characterized by the solution graph, where the vertices are the solutions, and two solutions are connected iff they differ in exactly one variable. Motivated by research on heuristics and the satisfiability threshold, in 2006, Gopalan et al. studied connectivity properties of the solution graph and related complexity issues for CSPs [3]. They found dichotomies for the diameter of connected components and for the complexity of the st-connectivity question, and conjectured a trichotomy for the connectivity question. Their results were refined by Makino et al. [7]. Recently, we were able to establish the trichotomy [15].
Here, we consider connectivity issues of satisfiability problems defined by Boolean circuits and propositional formulas that use gates, resp. connectives, from a fixed set of Boolean functions. We obtain dichotomies for the diameter and the connectivity problems: on one side, the diameter is linear and both problems are in P, while on the other, the diameter can be exponential and the problems are PSPACE-complete.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Böhler, E., Creignou, N., Reith, S., Vollmer, H.: Playing with boolean blocks, part i: Posts lattice with applications to complexity theory. In: SIGACT News (2003)
Fu, Z., Malik, S.: Extracting logic circuit structure from conjunctive normal form descriptions. In: 20th International Conference on VLSI Design, Held Jointly with 6th International Conference on Embedded Systems, pp. 37–42. IEEE (2007)
Gopalan, P., Kolaitis, P.G., Maneva, E., Papadimitriou, C.H.: The connectivity of boolean satisfiability: Computational and structural dichotomies. SIAM J. Comput. 38(6), 2330–2355 (2009), http://dx.doi.org/10.1137/07070440X
Ito, T., Demaine, E.D., Harvey, N.J.A., Papadimitriou, C.H., Sideri, M., Uehara, R., Uno, Y.: On the complexity of reconfiguration problems. Theor. Comput. Sci. 412(12-14), 1054–1065 (2011), http://dx.doi.org/10.1016/j.tcs.2010.12.005
Kamiński, M., Medvedev, P., Milanič, M.: Shortest paths between shortest paths and independent sets. In: Iliopoulos, C.S., Smyth, W.F. (eds.) IWOCA 2010. LNCS, vol. 6460, pp. 56–67. Springer, Heidelberg (2011)
Lewis, H.R.: Satisfiability problems for propositional calculi. Mathematical Systems Theory 13(1), 45–53 (1979)
Makino, K., Tamaki, S., Yamamoto, M.: On the boolean connectivity problem for horn relations. In: Marques-Silva, J., Sakallah, K.A. (eds.) SAT 2007. LNCS, vol. 4501, pp. 187–200. Springer, Heidelberg (2007)
Maneva, E., Mossel, E., Wainwright, M.J.: A new look at survey propagation and its generalizations. Journal of the ACM (JACM) 54(4), 17 (2007)
Mézard, M., Mora, T., Zecchina, R.: Clustering of solutions in the random satisfiability problem. Physical Review Letters 94(19), 197205 (2005)
Michael, T.: On the applicability of post’s lattice. Information Processing Letters 112(10), 386–391 (2012)
Post, E.L.: The Two-Valued Iterative Systems of Mathematical Logic(AM-5), vol. 5. Princeton University Press (1941)
Reith, S., Wagner, K.W.: The complexity of problems defined by Boolean circuits (2000)
Schaefer, T.J.: The complexity of satisfiability problems. In: STOC 1978, pp. 216–226 (1978)
Schnoor, H.: Algebraic techniques for satisfiability problems. Ph.D. thesis, Universität Hannover (2007)
Schwerdtfeger, K.W.: A computational trichotomy for connectivity of boolean satisfiability. ArXiv CoRR abs/1312.4524 (2013), extended version of a paper submitted to the JSAT Journal, http://arxiv.org/abs/1312.4524
Schwerdtfeger, K.W.: The connectivity of boolean satisfiability: Dichotomies for formulas and circuits. ArXiv CoRR abs/1312.6679 (2013), extended version of this paper, http://arxiv.org/abs/1312.6679
Vollmer, H.: Introduction to Circuit Complexity: A Uniform Approach. Springer-Verlag New York, Inc. (1999)
Wu, C.A., Lin, T.H., Lee, C.C., Huang, C.Y.R.: Qutesat: a robust circuit-based sat solver for complex circuit structure. In: Proceedings of the Conference on Design, Automation and Test in Europe, EDA Consortium, pp. 1313–1318 (2007)
Zverovich, I.E.: Characterizations of closed classes of boolean functions in terms of forbidden subfunctions and post classes. Discrete Appl. Math. 149(1-3), 200–218 (2005), http://dx.doi.org/10.1016/j.dam.2004.06.028
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Schwerdtfeger, K. (2014). The Connectivity of Boolean Satisfiability: Dichotomies for Formulas and Circuits. In: Hirsch, E.A., Kuznetsov, S.O., Pin, JÉ., Vereshchagin, N.K. (eds) Computer Science - Theory and Applications. CSR 2014. Lecture Notes in Computer Science, vol 8476. Springer, Cham. https://doi.org/10.1007/978-3-319-06686-8_27
Download citation
DOI: https://doi.org/10.1007/978-3-319-06686-8_27
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-06685-1
Online ISBN: 978-3-319-06686-8
eBook Packages: Computer ScienceComputer Science (R0)