Slicewise Definability in First-Order Logic with Bounded Quantifier Rank

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

Structural techniques in descriptive complexity

In 2017, Abramsky, Dawar, and Wang published a paper which gave a comonadic characterisation of pebble games, tree-width, and k-variable logic, a key trio of related concepts in Finite Model Theory. In 2018, Abramsky and Shah expanded upon this to give an analogous comonadic characterisation of Ehrenfeucht Fraisse games, tree-depth, and bounded quantifier rank logic. A key feature of these papers is the connection between two previously distinct subfields of logic in computer science; Categorical Semantics, and Finite Model Theory. This thesis applies the ideas and techniques in these papers to give a categorical account of some cornerstone results of Finite Model Theory, including Rossman’s Equirank Homomorphism Preservation Theorem, Courcelle’s Theorem (on the model-checking properties of structures of bounded tree-width), and Gaifman’s Locality Theorem