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

  • Authors:

  • Arnaud Durand;Etienne Grandjean

  • Affiliations:

  • Université Paris 7, Paris cedex, France;Université de Caen, Caen cedex, France

  • Venue:
  • ACM Transactions on Computational Logic (TOCL)
  • Year:
  • 2007

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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-Delaylin, 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.