Dynamic Complexity of Parity Exists Queries

TL;DR

This paper investigates the dynamic complexity of parity-related queries in graphs, demonstrating limitations of quantifier-free update rules and exploring conditions under which certain parity queries can be maintained.

Contribution

It shows that the parity-exists query cannot be maintained with quantifier-free rules and introduces a hierarchy based on relation arity, also identifying cases where degree restrictions enable maintenance.

Findings

Quantifier-free update rules cannot maintain the parity-exists query.

A hierarchy of update rules is established based on auxiliary relation arity.

Degree-restricted variants of the query can be maintained with full first-order rules.

Abstract

Given a graph whose nodes may be coloured red, the parity of the number of red nodes can easily be maintained with first-order update rules in the dynamic complexity framework DynFO of Patnaik and Immerman. Can this be generalised to other or even all queries that are definable in first-order logic extended by parity quantifiers? We consider the query that asks whether the number of nodes that have an edge to a red node is odd. Already this simple query of quantifier structure parity-exists is a major roadblock for dynamically capturing extensions of first-order logic. We show that this query cannot be maintained with quantifier-free first-order update rules, and that variants induce a hierarchy for such update rules with respect to the arity of the maintained auxiliary relations. Towards maintaining the query with full first-order update rules, it is shown that degree-restricted variants can be maintained.