Slicewise Definability in First-Order Logic with Bounded Quantifier Rank

Abstract For every q ∈ N let FOq denote the class of sentences of first-order logic FO of quantifier rank at most q. If a graph property can be defined in FOq, then it can be decided in time O(n). 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 FOq 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-AC0. 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 AC0. We explore the connection between the quantifier rank of FO-sentences and the depth of AC0-circuits, and prove that FOq ( FOq+1 for structures with built-in addition and multiplication.