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Homomorphism problem

The following problem is known as the homomorphism problem:

Every constraint satisfaction problem (CSP) has a translation into the homomorphism problem. Therefore, the complexity of CSP can be studied using the methods of finite model theory.

Another application is in database theory, where a relational model of a database is essentially the same thing as a relational structure. It turns out that a conjunctive query on a database can be described by another structure in the same signature as the database model. A homomorphism from the relational model to the structure representing the query is the same thing as a solution to the query. This shows that the conjunctive query problem is also equivalent to the homomorphism problem.