Answering FO+MOD Queries Under Updates on Bounded Degree Databases

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 to answer a Boolean FO+MOD query and to compute the size of the query result within constant time after every database update. Furthermore, after every update we are able to immediately enumerate the new query result with constant delay between the output tuples. The 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]

Modulo-Counting First-Order Logic on Bounded Expansion Classes

We prove that, on bounded expansion classes, every first-order formula with modulo counting is equivalent, in a linear-time computable monadic lift, to an existential first-order formula. As a consequence, we derive, on bounded expansion classes, that first-order transductions with modulo counting have the same encoding power as existential first-order transductions. Also, modulo-counting first-order model checking and computation of the size of sets definable in modulo-counting first-order logic can be achieved in linear time on bounded expansion classes. As an application, we prove that a class has structurally bounded expansion if and only if is a class of bounded depth vertex-minors of graphs in a bounded expansion class. We also show how our results can be used to implement fast matrix calculus on bounded expansion matrices over a finite field.Comment: submitted to CSGT2022 special issu

Enumeration of MSO Queries on Strings with Constant Delay and Logarithmic Updates

International audienceWe consider the enumeration of MSO queries over strings under updates. For each MSO query we build an index structure enjoying the following properties: The index structure can be constructed in linear time, it can be updated in logarithmic time and it allows for constant delay time enumeration. This improves from the previous known index structures allowing for constant delay enumeration that would need to be reconstructed from scratch, hence in linear time, in the presence of updates. We allow relabeling updates, insertion of individual labels and removal of individual labels