![](bg9.png)
The generalization of our reasoning to encode
general pseudo Boolean constraint to CNF is also a
short term perspective. Finally, we plan to conduct
an experimental evaluation of our Pigeon-Hole based
encoding w.r.t. the well-known encodings.
REFERENCES
Aloul, F. A., Ramani, A., Markov, I. L., and Sakallah, K. A.
(2003). Solving difficult instances of boolean satisfi-
ability in the presence of symmetry. IEEE Trans. on
CAD of Integrated Circuits and Systems, 22(9):1117–
1137.
Aloul, F. A., Sakallah, K. A., and Markov, I. L. (2006).
Efficient symmetry breaking for boolean satisfiability.
IEEE Trans. Computers, 55(5):549–558.
As
´
ın, R., Nieuwenhuis, R., Oliveras, A., and Rodr
´
ıguez-
Carbonell, E. (2009). Cardinality networks and their
applications. In 12th International Conference on
Theory and Applications of Satisfiability Testing (SAT
2009), pages 167–180.
Bailleux, O. and Boufkhad, Y. (2003). Efficient cnf encod-
ing of boolean cardinality constraints. In 9th Interna-
tional Conference on Principles and Practice of Con-
straint Programming (CP 2003), pages 108–122.
Bailleux, O., Boufkhad, Y., and Roussel, O. (2009). New
encodings of pseudo-boolean constraints into cnf. In
SAT’2009, pages 181–194.
Benhamou, B. and Sais, L. (1992). Theoretical study of
symmetries in propositional calculus and applications.
In 11th International Conference on Automated De-
duction (CADE’1992), volume 607 of Lecture Notes
in Computer Science, pages 281–294. Springer.
Benhamou, B. and Sais, L. (1994). Tractability through
symmetries in propositional calculus. Journal of Au-
tomated Reasoning, 12(1):89–102.
Cook, S. A. (1976). A short proof of the pigeon hole
principle using extended resolution. SIGACT News,
8(4):28–32.
Crawford, J. (1992). A theorical analysis of reasoning by
symmetry in first order logic. In Proceedings of Work-
shop on Tractable Reasonning, AAAI, pages 17–22.
Crawford, J. M., Ginsberg, M. L., Luks, E. M., and Roy,
A. (1996). Symmetry-breaking predicates for search
problems. In KR, pages 148–159.
E
´
en, N. and S
¨
orensson, N. (2006). Translating pseudo-
boolean constraints into sat. JSAT, 2(1-4):1–26.
Gent, I. P., Petrie, K. E., and Puget, J.-F. (2006). Chapter 10
symmetry in constraint programming. In F. Rossi, P.
v. B. and Walsh, T., editors, Handbook of Constraint
Programming, volume 2 of Foundations of Artificial
Intelligence, pages 329 – 376. Elsevier.
Jabbour, S., Sa
¨
ıs, L., and Salhi, Y. (2014). A pigeon-hole
based encoding of cardinality constraints. In Interna-
tional Symposium on Artificial Intelligence and Math-
ematics, ISAIM 2014, Fort Lauderdale, FL, USA, Jan-
uary 6-8, 2014.
Jabbour, S., Sais, L., Salhi, Y., and Uno, T. (2013). Mining-
based compression approach of propositional formu-
lae. In CIKM, pages 289–298.
Krishnamurthy, B. (1985). Shorts proofs for tricky formu-
las. Acta Informatica, 22:253–275.
Kullmann, O. (1997). On a generalization of extended res-
olution. Discrete Applied Mathematics, 34:73–95.
Plaisted, D. A. and Greenbaum, S. (1986). A structure-
preserving clause form translation. Journal of Sym-
bolic Computation, 2(3):293–304.
Puget, J. (1993). On the satisfiability of symmetrical con-
straint satisfaction problems. In proceedings of ISMIS,
pages 350–361.
Silva, J. P. M. and Lynce, I. (2007). Towards robust cnf
encodings of cardinality constraints. In 13th Interna-
tional Conference on Principles and Practice of Con-
straint Programming (CP 2007), pages 483–497.
Sinz, C. (2005). Towards an optimal cnf encoding of
boolean cardinality constraints. In 11th International
Conference on Principles and Practice of Constraint
Programming (CP 2005), pages 827–831.
Tseitin, G. (1968). On the complexity of derivations in the
propositional calculus. In Slesenko, H., editor, Struc-
tures in Constructives Mathematics and Mathematical
Logic, Part II, pages 115–125.
Walsh, T. (2006). General symmetry breaking con-
straints. In 12th International Conference on Prin-
ciples and Practice of Constraint Programming (CP
2006), pages 650–664.
Warners, J. P. (1996). A linear-time transformation of linear
inequalities into conjunctive normal form. Informa-
tion Processing Letters.
Enhancing Pigeon-Hole based Encoding of Boolean Cardinality Constraints
307