# Generalized Support and Formal Development of Constraint Propagators

Constraint programming is a family of techniques for solving combinatorial problems, where the problem is modelled as a set of decision variables (typically with finite domains) and a set of constraints that express relations among the decision variables. One key concept in constraint programming is propagation: reasoning on a constraint or set of constraints to derive new facts, typically to remove values from the domains of decision variables. Specialised propagation algorithms (propagators) exist for many classes of constraints. The concept of support is pervasive in the design of propagators. Traditionally, when a domain value ceases to have support, it may be removed because it takes part in no solutions. Arc-consistency algorithms such as AC2001 make use of support in the form of a single domain value. GAC algorithms such as GAC-Schema use a tuple of values to support each literal. We generalize these notions of support in two ways. First, we allow a set of tuples to act as support. Second, the supported object is generalized from a set of literals (GAC-Schema) to an entire constraint or any part of it. We design a methodology for developing correct propagators using generalized support. A constraint is expressed as a family of support properties, which may be proven correct against the formal semantics of the constraint. Using Curry-Howard isomorphism to interpret constructive proofs as programs, we show how to derive correct propagators from the constructive proofs of the support properties. The framework is carefully designed to allow efficient algorithms to be produced. Derived algorithms may make use of dynamic literal triggers or watched literals for efficiency. Finally, two case studies of deriving efficient algorithms are given.

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