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.

翻译：如果图表中的节点可能有红色颜色,那么红色节点数的对等性很容易以动态复杂框架中的一阶更新规则来维持, Patnaik 和 Immerman 的 DynFO 和 Immerman 的 DynFO 的动态复杂框架中的一阶更新规则来维持。 这能否被概括到在一阶逻辑中可定义的由对等限定符延伸的所有其它查询中? 我们考虑的问题是, 问对红色节点有边际的对红节点数是否有偏向于红色节点的偏向。 这个简单的问题已经是动态捕获一阶逻辑扩展的主要障碍 。 我们显示, 这个查询无法用无限定度第一阶更新规则来维持, 变式会诱导出这种更新规则的等级, 与维持的辅助关系的一致性有关 。 在保留全阶点更新规则时, 显示受度限制的变体是可以维持的 。