First-order queries on structures of bounded degree are computable with constant delay

A bounded degree structure is either a relational structure all of whose relations are of bounded degree or a functional structure involving bijective functions only. In this paper, we revisit the complexity of the evaluation problem of not necessarily Boolean first-order queries over structures of bounded degree. Query evaluation is considered here as a dynamical process. We prove that any query on bounded degree structures is $\constantdelaylin$, i.e., can be computed by an algorithm that has two separate parts: it has a precomputation step of linear time in the size of the structure and then, it outputs all tuples one by one with a constant (i.e. depending on the size of the formula only) delay between each. Seen as a global process, this implies that queries on bounded structures can be evaluated in total time $O(f(|\phi|).(|\calS|+|\phi(\calS)|))$ and space $O(f(|\phi|).|\calS|)$ where $\calS$ is the structure, $\phi$ is the formula, $\phi(\calS)$ is the result of the query and $f$ is some function. Among other things, our results generalize a result of \cite{Seese-96} on the data complexity of the model-checking problem for bounded degree structures. Besides, the originality of our approach compared to that \cite{Seese-96} and comparable results is that it does not rely on the Hanf's model-theoretic technic (see \cite{Hanf-65}) and is completely effective.