First-Order Query Evaluation with Cardinality Conditions

TL;DR

This paper investigates the complexity of evaluating first-order logic with cardinality conditions, showing fixed-parameter tractability on certain classes of structures and identifying a fragment suitable for practical applications.

Contribution

It introduces the FOC1(P) fragment of FOC(P) and proves its fixed-parameter tractability on nowhere dense classes, extending the applicability of efficient query evaluation.

Findings

Fixed-parameter tractability of FOC(P) fails for unbounded structures like trees and strings.

FOC1(P) remains expressive enough for SQL COUNT applications.

Query evaluation for FOC1(P) is almost linear on nowhere dense classes.

Abstract

We study an extension of first-order logic that allows to express cardinality conditions in a similar way as SQL's COUNT operator. The corresponding logic FOC(P) was introduced by Kuske and Schweikardt (LICS'17), who showed that query evaluation for this logic is fixed-parameter tractable on classes of structures (or databases) of bounded degree. In the present paper, we first show that the fixed-parameter tractability of FOC(P) cannot even be generalised to very simple classes of structures of unbounded degree such as unranked trees or strings with a linear order relation. Then we identify a fragment FOC1(P) of FOC(P) which is still sufficiently strong to express standard applications of SQL's COUNT operator. Our main result shows that query evaluation for FOC1(P) is fixed-parameter tractable with almost linear running time on nowhere dense classes of structures. As a corollary, we also obtain a fixed-parameter tractable algorithm for counting the number of tuples satisfying a query over nowhere dense classes of structures.