Logic attempts to describe the nature of sentences that have descriptions in some restricted format. Computational complexity investigates the number of steps needed on a computer to solve a given mathematical problem. Despite the seemingly disjoint nature of the topics, Logic and Computational Complexity have had a rich history of interactions in the past. Fagin's classical theorem giving a logical definition of NP is probably one of the first connections in the area. Other celebrated results in the nexus of finite logic and computational complexity include the Immerman-Szelepsenyi theorem showing non-deterministic logspace is closed under complementation; and Ajtai's work on certain formulae on finite structures, which shows that parity is not first-order definable (which turns out to be equivalent to Furst, Saxe, and Sipser's result that parity does not have constant depth, polynomial size circuits). These results have advanced logic as well as computational complexity.
This research project is inspired by recent work by the student investigators on this project, Swastik Kopparty and Ben Rossman who have unearthed new connections between these areas leading to several new results. This project outlines novel further questions in the intersection of logic and computational complexity and outlines methods that may be employed to make progress on these questions. Some specific questions include:
- Size lower bounds for uniform logarithmic depth circuits. - Explicit functions that are hard for first-order logic augmented with arbitrary modular counting quantifiers (modulo composites in particular). - Choiceless algorithms for solving linear systems. - Decidability of the containment problem for conjunctive queries under multiset semantics.
