We study the problem of incremental view maintenance (IVM) under inserts to $K$-databases, where $K$ is a commutative semiring without additive inverse. The key observation put forward in this paper is that the complexity of the IVM problem depends fundamentally on the underlying semiring. We introduce a class of conjunctive queries called $p$-hierarchical and show that for any $p$-hierarchical query with fractional hypertree width $\fhtw$ and any insert-only update sequence of length $N$ to an initially empty $K$-database over an arbitrary semiring $K$ without additive inverse, we can construct a data structure that can be updated in amortized $\bigO(N^{\fhtw-1})$ time and can support constant delay enumeration of the query result. In particular, the amortized update time for any $α$-acyclic $p$-hierarchical query is constant. We also give conditional lower bounds showing that any conjunctive query without self-joins that is not $p$-hierarchical cannot be maintained with amortized constant update time and constant enumeration delay under inserts to $K$-databases. Here, $K$ can be the natural semiring and its generalizations to the provenance and covariance semirings or any idempotent and strictly ordered semiring such as the tropical semiring. When put together, our upper and lower bounds imply a dichotomy for the insert-only maintenance of conjunctive queries without self-joins and the aforementioned semirings: A query can be maintained with amortized constant update time and constant enumeration delay if and only if it is $α$-acyclic $p$-hierarchical.

翻译：我们研究了在向$K$-数据库插入元组时增量视图维护（IVM）问题，其中$K$是无加法逆元的交换半环。本文提出的关键观察是：IVM问题的复杂度根本上取决于底层半环。我们引入了一类称为$p$-层次化的合取查询，并证明：对于任意分数超树宽度为$\fhtw$的$p$-层次化查询，以及作用于初始为空的$K$-数据库（基于任意无加法逆元的半环$K$）上的长度为$N$的仅插入更新序列，我们可以构建一种数据结构，其更新操作的平摊时间为$\bigO(N^{\fhtw-1})$，并能支持查询结果的常数延迟枚举。特别地，任何$\alpha$-无环的$p$-层次化查询的平摊更新时间为常数。我们还给出了条件性下界，表明：对于$K$-数据库上的插入操作，任何非$p$-层次化且无自连接操作的合取查询，都无法在保持常数平摊更新时间和常数枚举延迟的条件下进行维护。此处$K$可以是自然半环及其推广形式（如来源半环和协方差半环），也可以是热带半环之类的任意幂等严格有序半环。综合起来，我们的上下界结果揭示了无自连接合取查询在上述半环下的仅插入维护存在二分性：一个查询能被维护以保持常数平摊更新时间和常数枚举延迟，当且仅当它是$\alpha$-无环的$p$-层次化查询。