In the setting of dynamic complexity, the goal of a dynamic program is to maintain the result of a fixed query for an input database that is subject to changes, possibly using additional auxiliary relations. In other words, a dynamic program updates a materialized view whenever a base relation is changed. The update of query result and auxiliary relations is specified using first-order logic or, equivalently, relational algebra. The original framework by Patnaik and Immerman only considers changes to the database that insert or delete single tuples. This article extends the setting to definable changes , also specified by first-order queries on the database, and generalizes previous maintenance results to these more expressive change operations. More specifically, it is shown that the undirected reachability query is first-order maintainable under single-tuple changes and first-order defined insertions, likewise the directed reachability query for directed acyclic graphs is first-order maintainable under insertions defined by quantifier-free first-order queries. These results rely on bounded bridge properties , which basically say that, after an insertion of a defined set of edges, for each connected pair of nodes there is some path with a bounded number of new edges. While this bound can be huge, in general, it is shown to be small for insertion queries defined by unions of conjunctive queries. To illustrate that the results for this restricted setting could be practically relevant, they are complemented by an experimental study that compares the performance of dynamic programs with complex changes, dynamic programs with single changes, and with recomputation from scratch. The positive results are complemented by several inexpressibility results. For example, it is shown that—unlike for single-tuple insertions—dynamic programs that maintain the reachability query under definable, quantifier-free changes strictly need update formulas with quantifiers. Finally, further positive results unrelated to reachability are presented: it is shown that for changes definable by parameter-free first-order formulas, all LOGSPACE-definable (and even AC 1 -definable) queries can be maintained by first-order dynamic programs.

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可定义变化下的动态复杂性

在动态复杂性的设置中，动态程序的目标是维护一个固定查询的结果，该查询可能会使用额外的辅助关系，而该输入数据库可能会发生变化。换句话说，只要基本关系发生变化，动态程序就会更新物化视图。查询结果和辅助关系的更新使用一阶逻辑或等效的关系代数指定。Patnaik 和 Immerman 的原始框架仅考虑插入或删除单个元组的数据库更改。本文将设置扩展到可定义的变化，也由对数据库的一阶查询指定，并将以前的维护结果推广到这些更具表现力的更改操作。更具体地说，表明无向可达性查询在单元组更改和一阶定义的插入下是一阶可维护的，同样，有向无环图的有向可达性查询在由无量词定义的插入下是一阶可维护的-订单查询。这些结果依赖于有界桥属性，这基本上是说，在插入一组定义的边之后，对于每对连接的节点，都有一些具有有限数量的新边的路径。虽然这个界限可能很大，但一般来说，它对于由联合查询定义的插入查询显示很小。为了说明这种受限设置的结果可能具有实际相关性，它们辅以一项实验研究，该研究比较了具有复杂变化的动态程序、具有单一变化的动态程序以及从头开始重新计算的性能。一些不可表达的结果补充了积极的结果。例如，它表明——与单元组插入不同——在可定义的、无量词的变化下维护可达性查询的动态程序严格需要更新带有量词的公式。1-definable) 查询可以由一阶动态程序维护。