Answering FO+MOD Queries under Updates on Bounded Degree Databases

Abstract

We investigate the query evaluation problem for fixed queries over fully dynamic databases, where tuples can be inserted or deleted. The task is to design a dynamic algorithm that immediately reports the new result of a fixed query after every database update. We consider queries in first-order logic (FO) and its extension with modulo-counting quantifiers (FO+MOD) and show that they can be efficiently evaluated under updates, provided that the dynamic database does not exceed a certain degree bound. In particular, we construct a data structure that allows us to answer a Boolean FO+MOD query and to compute the size of the result of a non-Boolean query within constant time after every database update. Furthermore, after every database update, we can update the data structure in constant time such that afterwards we are able to test within constant time for a given tuple whether or not it belongs to the query result, to enumerate all tuples in the new query result, and to enumerate the difference between the old and the new query result with constant delay between the output tuples. The preprocessing time needed to build the data structure is linear in the size of the database. Our results extend earlier work on the evaluation of first-order queries on static databases of bounded degree and rely on an effective Hanf normal form for FO+MOD recently obtained by Heimberg, Kuske, and Schweikardt (LICS 2016).