Abstract
This paper discusses an algorithm to solve polynomial constraints over finite domains, namely constraints which are expressed in terms of equalities, inequalities and disequalities of polynomials with integer coefficients whose variables are associated with finite domains. The proposed algorithm starts with a preliminary step intended to rewrite all constraints to a canonical form. Then, the modified Bernstein form of obtained polynomials is used to recursively restrict the domains of variables, which are assumed to be initially approximated by a bounding box. The proposed algorithm proceeds by subdivisions, and it ensures that each variable is eventually associated with the inclusion-maximal finite domain in which the set of constraints is satisfiable. If arbitrary precision integer arithmetic is available, no approximation is introduced in the solving process because the coefficients of the modified Bernstein form are integer numbers.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Bergenti, F., Monica, S., Rossi, G.: Polynomial constraint solving over finite domains with the modified Bernstein form. In: Fiorentini, C., Momigliano, A. (eds.) Proceedings of the 31st Italian Conference on Computational Logic, vol. 1645. CEUR Workshop Proceedings, pp. 118–131. RWTH Aachen (2016)
Apt, K.: Principles of Constraint Programming. Cambridge University Press, Cambridge (2003)
Triska, M.: The finite domain constraint solver of SWI-prolog. In: Schrijvers, T., Thiemann, P. (eds.) FLOPS 2012. LNCS, vol. 7294, pp. 307–316. Springer, Heidelberg (2012). doi:10.1007/978-3-642-29822-6_24
Wielemaker, J., Schrijvers, T., Triska, M., Lager, T.: SWI-prolog. Theory Pract. Logic Program. 12(1–2), 67–96 (2012)
Bernstein, S.N.: Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités. Commun. Soc. Math. de Kharkov 2:XIII(1), 1–2 (1912)
Lorentz, G.G.: Bernstein Polynomials. University of Toronto Press, Toronto (1953)
Sánchez-Reyes, J.: Algebraic manipulation in the Bernstein form made simple via convolutions. Comput. Aided Des. 35, 959–967 (2003)
Farouki, R.T., Rajan, V.T.: Algorithms for polynomials in Bernstein form. Comput. Aided Geom. Des. 5(1), 1–26 (1988)
Nataraj, P., Arounassalame, M.: A new subdivision algorithm for the Bernstein polynomial approach to global optimization. Int. J. Autom. Comput. 4(4), 342–352 (2007)
Mourrain, B., Pavone, J.: Subdivision methods for solving polynomial equations. J. Symbolic Comput. 44(3), 292–306 (2009)
Garloff, J.: Convergent bounds for the range of multivariate polynomials. In: Nickel, K. (ed.) IMath 1985. LNCS, vol. 212, pp. 37–56. Springer, Heidelberg (1986). doi:10.1007/3-540-16437-5_5
Garloff, J.: The Bernstein algorithm. Interval Comput. 2, 154–168 (1993)
Farouki, R.T.: The Bernstein polynomial basis: a centennial retrospective. Comput. Aided Geom. Des. 29(6), 379–419 (2012)
Garloff, J., Smith, A.P.: Solution of systems of polynomial equations by using Bernstein expansion. In: Alefeld, G., Rohn, J., Rump, S., Yamamoto, T. (eds.) Symbolic Algebraic Methods and Verification Methods, pp. 87–97. Springer, Vienna (2001)
Ray, S., Nataraj, P.: An efficient algorithm for range computation of polynomials using the Bernstein form. J. Global Optim. 45, 403–426 (2009)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing AG
About this paper
Cite this paper
Bergenti, F., Monica, S., Rossi, G. (2016). A Subdivision Approach to the Solution of Polynomial Constraints over Finite Domains Using the Modified Bernstein Form. In: Adorni, G., Cagnoni, S., Gori, M., Maratea, M. (eds) AI*IA 2016 Advances in Artificial Intelligence. AI*IA 2016. Lecture Notes in Computer Science(), vol 10037. Springer, Cham. https://doi.org/10.1007/978-3-319-49130-1_14
Download citation
DOI: https://doi.org/10.1007/978-3-319-49130-1_14
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-49129-5
Online ISBN: 978-3-319-49130-1
eBook Packages: Computer ScienceComputer Science (R0)