A sufficient condition for backtrack-bounded search
Journal of the ACM (JACM)
An optimal k-consistency algorithm
Artificial Intelligence
Gro¨bner bases: a computational approach to commutative algebra
Gro¨bner bases: a computational approach to commutative algebra
RISC-CLP (real): logic programming with non-linear constraints over the reals
Constraint logic programming
Tractable constraints on ordered domains
Artificial Intelligence
Using the Groebner basis algorithm to find proofs of unsatisfiability
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
A Variant of the Buchberger Algorithm for Integer Programming
SIAM Journal on Discrete Mathematics
Bucket elimination: a unifying framework for reasoning
Artificial Intelligence
Symbolic-interval cooperation in constraint programming
Proceedings of the 2001 international symposium on Symbolic and algebraic computation
Buchberger Algorithm and Integer Programming
AAECC-9 Proceedings of the 9th International Symposium, on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
The Construction of Multivariate Polynomials with Preassigned Zeros
EUROCAM '82 Proceedings of the European Computer Algebra Conference on Computer Algebra
Gröbner-Bases, Gaussian elimination and resolution of systems of algebraic equations
EUROCAL '83 Proceedings of the European Computer Algebra Conference on Computer Algebra
Constraint Processing
Implementing a Test for Tractability
Constraints
Handbook of Constraint Programming (Foundations of Artificial Intelligence)
Handbook of Constraint Programming (Foundations of Artificial Intelligence)
MiniZinc: towards a standard CP modelling language
CP'07 Proceedings of the 13th international conference on Principles and practice of constraint programming
FGb: a library for computing Gröbner bases
ICMS'10 Proceedings of the Third international congress conference on Mathematical software
Journal of Symbolic Computation
Local consistency and SAT-solvers
Journal of Artificial Intelligence Research
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In this paper we investigate the use of a system of multivariate polynomials to represent the restrictions imposed by a collection of constraints. One advantage of using polynomials to represent constraints is that it allows many different forms of constraints to be treated in a uniform way. Systems of polynomials have been widely studied, and a number of general techniques have been developed, including algorithms that generate an equivalent system with certain desirable properties, called a Gröbner Basis. General algorithms to compute a Gröbner Basis have doubly exponential complexity, but we observe that for the systems we use to describe constraint problems over finite domains a Gröbner Basis can be computed more efficiently than this. We also describe a family of algorithms, related to the calculation of Gröbner Bases, that can be used on any system of polynomials to compute an equivalent system in polynomial time. We show that these modified algorithms can simulate the effect of the local-consistency algorithms used in constraint programming and hence solve certain broad classes of constraint problems in polynomial time. Finally we discuss the use of adaptive consistency techniques for systems of polynomials.