Generalized Implicit Definitions on Finite Structures

We propose a natural generalization of the concept of implicit deenitions over nite structures, allowing non-determinism at an intermediate level of a (deterministic) deenition. These generalized implicit deenitions ooer more expressive power than classical implicit dee-nitions. Moreover, their expressive power can be characterized over un-ordered nite structures in terms of the complexity class NP \ co-NP. Finally, we investigate a subclass of these where the non-determinism is restricted to the choice of a unique relation with respect to an implicit linear order, and prove that it captures UP \ co-UP also over the class of all nite structures. These results shed some light on the expressive power of non-deterministic primitives.