Slicewise Definability in First-Order Logic with Bounded Quantifier Rank.

DOI10.4230/LIPICS.CSL.2017.19zbMATH Open1434.03019arXiv1704.03167OpenAlexW2964221993MaRDI QIDQ5111186

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Publication date: 26 May 2020

Abstract: For every $qinmathbbN$ let $extrm{FO}_q$ denote the class of sentences of first-order logic FO of quantifier rank at most $q$. If a graph property can be defined in $extrm{FO}_q$, 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 $extrm{FO}_q$ 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 $extrm{para-AC}^0$. It is crucial for our results that the FO-sentences have access to built-in addition and multiplication. It is known that then FO corresponds to the circuit complexity class uniform $extrm{AC}^0$. We explore the connection between the quantifier rank of FO-sentences and the depth of $extrm{AC}^0$-circuits, and prove that $extrm{FO}_{q}subsetneqextrm{FO}_{q+1}$ for structures with built-in addition and multiplication.

Full work available at URL: https://arxiv.org/abs/1704.03167

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