文章基本信息

标题：Slicewise Definability in First-Order Logic with Bounded Quantifier Rank
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作者：Yijia Chen ; J{\"o}rg Flum ; Xuangui Huang 等
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2017
卷号：82
页码：19:1-19:16
DOI：10.4230/LIPIcs.CSL.2017.19
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：For every natural number q let FO_q denote the class of sentences of first-order logic FO of quantifier rank at most q. If a graph property can be defined in FO_q, then it can be decided in time O(n^q). Thus, minimizing q has favorable algorithmic consequences. Many graph properties amount to the existence of a certain set of vertices of size k. Usually this can only be expressed by a sentence of quantifier rank at least k. We use the color coding method to demonstrate that some (hyper)graph problems can be defined in FO_q where q is independent of k. This property of a graph problem is equivalent to the question of whether the corresponding parameterized problem is in the class para-AC^0. It is crucial for our results that the FO-sentences have access to built-in addition and multiplication (and constants for an initial segment of natural numbers whose length depends only on k). It is known that then FO corresponds to the circuit complexity class uniform AC^0. We explore the connection between the quantifier rank of FO-sentences and the depth of AC^0-circuits, and prove that FO_q is strictly contained in FO_{q+1} for structures with built-in addition and multiplication.
关键词：first-order logic; quantifier rank; parameterized AC^0; circuit depth