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

TL;DR

This paper demonstrates that first-order queries on structures of bounded degree can be evaluated efficiently with a linear precomputation and constant delay output, generalizing previous results without relying on Hanf's technique.

Contribution

It introduces a new effective algorithm for evaluating first-order queries on bounded degree structures with constant delay after linear precomputation.

Findings

Queries can be evaluated with linear precomputation and constant delay.

Total evaluation time is proportional to structure size and formula complexity.

The approach does not rely on Hanf's model-theoretic techniques.

Abstract

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.