Abstract
This paper generalizes the notion of the backbone of a CNF formula to capture also equations between literals. Each such equation applies to remove a variable from the original formula thus simplifying the formula without changing its satisfiability, or the number of its satisfying assignments. We prove that for a formula with n variables, the generalized backbone is computed with at most n + 1 satisfiable calls and exactly one unsatisfiable call to the SAT solver. We illustrate the integration of generalized backbone computation to facilitate the encoding of finite domain constraints to SAT. In this context generalized backbones are computed for small groups of constraints and then propagated to simplify the entire constraint model. A preliminary experimental evaluation is provided.
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References
Codish, M., Miller, A., Prosser, P., Stuckey, P.J.: Breaking symmetries in graph representation. In: Rossi, F. (ed.) IJCAI. IJCAI/AAAI (2013)
Eén, N., Biere, A.: Effective preprocessing in SAT through variable and clause elimination. In: Bacchus, F., Walsh, T. (eds.) SAT 2005. LNCS, vol. 3569, pp. 61–75. Springer, Heidelberg (2005)
Garnick, D.K., Kwong, Y.H.H., Lazebnik, F.: Extremal graphs without three-cycles or four-cycles. Journal of Graph Theory 17(5), 633–645 (1993)
Heule, M.J.H., Järvisalo, M., Biere, A.: Efficient CNF simplification based on binary implication graphs. In: Sakallah, K.A., Simon, L. (eds.) SAT 2011. LNCS, vol. 6695, pp. 201–215. Springer, Heidelberg (2011)
Heule, M., van Maaren, H.: Aligning CNF- and equivalence-reasoning. In: Hoos, H.H., Mitchell, D.G. (eds.) SAT 2004. LNCS, vol. 3542, pp. 145–156. Springer, Heidelberg (2005)
Janota, M.: SAT Solving in Interactive Configuration. PhD thesis, University College Dublin (November 2010)
Kilby, P., Slaney, J.K., Thiébaux, S., Walsh, T.: Backbones and backdoors in satisfiability. In: Veloso, M.M., Kambhampati, S. (eds.) AAAI, pp. 1368–1373. AAAI Press / The MIT Press (2005)
Kirkpatrick, S., Toulouse, G.: Configuration space analysis of traveling salesman problems. J. Phys. (France) 46, 1277–1292 (1985)
Li, C.-M.: Equivalent literal propagation in the DLL procedure. Discrete Applied Mathematics 130(2), 251–276 (2003)
Manolios, P., Papavasileiou, V.: Pseudo-boolean solving by incremental translation to SAT. In: Bjesse, P., Slobodová, A. (eds.) FMCAD, pp. 41–45. FMCAD Inc. (2011)
Manthey, N.: Coprocessor 2.0 - a flexible CNF simplifier - (tool presentation). In: Cimatti, A., Sebastiani, R. (eds.) SAT 2012. LNCS, vol. 7317, pp. 436–441. Springer, Heidelberg (2012)
Marques-Silva, J., Janota, M., Lynce, I.: On computing backbones of propositional theories. In: Coelho, H., Studer, R., Wooldridge, M. (eds.) ECAI. Frontiers in Artificial Intelligence and Applications, vol. 215, pp. 15–20. IOS Press (2010), Extended version: http://sat.inesc-id.pt/~mikolas/bb-aicom-preprint.pdf
Metodi, A., Codish, M.: Compiling finite domain constraints to SAT with BEE. TPLP 12(4-5), 465–483 (2012)
Metodi, A., Codish, M., Lagoon, V., Stuckey, P.J.: Boolean equi-propagation for optimized SAT encoding. In: Lee, J. (ed.) CP 2011. LNCS, vol. 6876, pp. 621–636. Springer, Heidelberg (2011)
Metodi, A., Codish, M., Stuckey, P.J.: Boolean equi-propagation for concise and efficient SAT encodings of combinatorial problems. J. Artif. Intell. Res. (JAIR) 46, 303–341 (2013)
Monasson, R., Zecchina, R., Kirkpatrick, S., Selman, B., Troyansky, L.: Determining computational complexity for characteristic phase transitions. Nature 400, 133–137 (1998)
Schneider, J.J.: Searching for backbones – an efficient parallel algorithm for the traveling salesman problem. Comput. Phys. Commun. (1996)
Sheeran, M., Stålmarck, G.: A tutorial on stålmarck’s proof procedure for propositional logic. Formal Methods in System Design 16(1), 23–58 (2000)
Stålmark, G.: A system for determining propositional logic theorem by applying values and rules to triplets that are generated from a formula. US Patent 5,276,897; Canadian Patent 2,018,828; European Patent 0403 545; Swedish Patent 467 076 (1994)
Zhang, W.: Phase transitions and backbones of the asymmetric traveling salesman problem. J. Artif. Intell. Res. (JAIR) 21, 471–497 (2004)
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Codish, M., Fekete, Y., Metodi, A. (2013). Backbones for Equality. In: Bertacco, V., Legay, A. (eds) Hardware and Software: Verification and Testing. HVC 2013. Lecture Notes in Computer Science, vol 8244. Springer, Cham. https://doi.org/10.1007/978-3-319-03077-7_1
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DOI: https://doi.org/10.1007/978-3-319-03077-7_1
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