On the Equivalence of Constraint Satisfaction Problems

A solution of a Constraint Satisfaction Problem (CSP) is an assignment of values to all its variables such that all its constraints are satis ed. Usually two CSPs are considered equivalent if they have the same solution set. We nd this de nition limiting, and develop a more general de nition based on the concept of mutual reducibility. In this extended scheme it is reasonable to consider a pair of CSPs equivalent even if they have di erent solutions. The basic idea behind the extended scheme is that two CSPs can be considered equivalent whenever they contain the same \amount of information", i.e. whenever it is possible to obtain the solution of one of them from the solution of the other one, and viceversa. In this way, both constraint and variable redundancy are allowed in CSPs belonging to the same equivalence class. As an example of the usefulness of this new notion of equivalence, we formally prove that binary and non-binary CSPs are equivalent (in the new sense). Such a proof is not possible with the usual notion of equivalence. Two di erent algorithms, currently used for transforming any non-binary CSP into an equivalent binary one, are described. It turns out that only one of them produces a binary CSP equivalent to the given non-binary problem, while the other one can achieve the transformation only at the cost of adding some new arbitrary information.