可定义变化下的动态复杂性
Dynamic Complexity under Definable Changes
摘要：
本文在Patnaik和Immerman的DynFO框架下研究了可定义变更操作下的动态复杂性。它示出的变化定义的参数自由的一阶公式，所有（统一）$AC^1$查询可以保持一阶动态程序。进一步地，许多单元组变化的维护结果被扩展到更强大的变化操作：（1）无向图的可达性查询在单元组变化和一阶定义插入下是一阶维护，同样地，有向无环图的可达性查询在无量词插入下也是一阶维护。(2)上下文无关语言是在$Σ_1$定义的变化下的一阶维护。这些结果是补充了几个不可表达的结果，例如，可达性查询不能保持下定义的，无量词删除的无量词程序。
英文摘要：
This paper studies dynamic complexity under definable change operations in the DynFO framework by Patnaik and Immerman. It is shown that for changes definable by parameter-free first-order formulas, all (uniform) $AC^1$ queries can be maintained by first-order dynamic programs. Furthermore, many maintenance results for single-tuple changes are extended to more powerful change operations: (1) The reachability query for undirected graphs is first-order maintainable under single tuple changes and first-order defined insertions, likewise the reachability query for directed acyclic graphs under quantifier-free insertions. (2) Context-free languages are first-order maintainable under $Σ_1$-defined changes. These results are complemented by several inexpressibility results, for example, that the reachability query cannot be maintained by quantifier-free programs under definable, quantifier-free deletions.
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