relation representing the Boolean disjunction (a relation containing all tuples of two elements that contains at least a 1), this relation is placed as a constraint on and , because their values in the table above are , again, and . Since all these values satisfy the constraint, the constraint is placed. On the other hand, a constraint with this relation is not placed on and , since the restriction of the table above to these two variables contains as a third row, and this evaluation violates that constraint. The universal gadget of order is the constraint satisfaction problem containing all constraints that can be placed in order to obtain the table above. The solutions of the universal gadget include the rows of this table, but can contain other rows. If the solutions are exactly the rows of the table, every relation can be expressed by projecting on a subset of the variables. However, even if the solutions include other rows, some relations can still be expressed. A property of the universal gadget is that it is able to express, by projection, every relation that can be expressed by projection from an arbitrary constraint satisfaction problem based on the same language. More precisely, the universal gadget of order expresses all relations of rows that can be expressed in the constraint language. Given a specific relation, its expressibility in the language can be checked by considering an arbitrary list of variables whose columns in the table above (the "ideal" solutions to the universal gadget) form that relation. The relation can be expressed in the language if and only if the solutions of the universal gadget coincides with the relation when projected over such a list of variables. In other words, one can check expressibility by selecting variables "as if" the solutions of the universal gadget were like in the table, and then check whether the restriction of the "real" solutions is actually the same as the relation. In the example above, the expressibility of the relation in the table on the right can be checked by looking whether the solutions of the universal gadget, when restricted to the variables and , are exactly the rows of this table. Solutions as functions in the universal gadget A necessary condition for tractability can be expressed in terms of the universal gadget. The solutions of such a gadget can be tabulated as follows: a