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

Abstract

A relational structure is d -degree-bounded, for some integer d , if each element of the domain belongs to at most d tuples. In this paper, we revisit the complexity of the evaluation problem of not necessarily Boolean first-order ( FO ) queries over d -degree-bounded structures. Query evaluation is considered here as a dynamical process. We prove that any FO query on d -degree-bounded structures belongs to the complexity class constant-Delay lin , that is, can be computed by an algorithm that has two separate parts: it has a precomputation step of time linear in the size of the structure and then, it outputs all solutions (i.e., tuples that satisfy the formula) one by one with a constant delay (i.e., depending on the size of the formula only) between each. Seen as a global process, this implies that queries on d -degree-bounded structures can be evaluated in total time f (|φ|).(| S | + |φ( S )|) and space g (|φ|).| S | where S is the structure, φ is the formula, φ( S ) is the result of the query and f , g are some fixed functions. Among other things, our results generalize a result of Seese on the data complexity of the model-checking problem for d -degree-bounded structures. Besides, the originality of our approach compared to related results is that it does not rely on the Hanf's model-theoretic technique and is simple and informative since it essentially rests on a quantifier elimination method.