Dynamic Complexity of Parity Exists Queries

Abstract

Given a graph whose nodes may be coloured red, the parity of the number ofred nodes can easily be maintained with first-order update rules in the dynamiccomplexity framework DynFO of Patnaik and Immerman. Can this be generalised toother or even all queries that are definable in first-order logic extended byparity quantifiers? We consider the query that asks whether the number of nodesthat have an edge to a red node is odd. Already this simple query of quantifierstructure parity-exists is a major roadblock for dynamically capturingextensions of first-order logic. We show that this query cannot be maintained with quantifier-free first-orderupdate rules, and that variants induce a hierarchy for such update rules withrespect to the arity of the maintained auxiliary relations. Towards maintainingthe query with full first-order update rules, it is shown thatdegree-restricted variants can be maintained