Counting and enumerating in first-order team logics

Abstract

Descriptive complexity theory is the study of the expressibility of computational problems in certain logics. Most of the results in this field use (fragments or extensions of) first-order logic or second-order logic to describe decision complexity classes. For example the complexity class NP can be characterized as the set of problems that are describable by a dependence logic formula, in short NP = FO(=(...)). Dependence logic is a certain team logic, where a team logic is an extension of first-order logic by some new atoms, with new semantics, called team semantics. Compared to decision complexity where one is interested in the existence of a solution to an input instance, in counting complexity one is interested in the number of solutions and in enumeration complexity one wants to compute all solutions. Counting complexity has been less studied in terms of descriptive complexity than decision complexity, whereas there are no results for enumeration complexity in this field. The latter is because the concept of hardness in the enumeration setting was first introduced rather recently. In this thesis, we characterize counting and enumeration complexity classes with team logics and compare the results to the corresponding results for decision complexity classes. To study the framework of hard enumeration a bit more, we investigate further team logic based enumeration problems. In the counting setting we characterize the classes #P and #•NP as #P = #FOT and #•NP = #FO(⊥). Furthermore, we establish two team logic based classes #FO(⊆) and #FO(=(...)) which seem to have no direct counterpart in classical counting complexity, but contain problems that are complete under Turing reductions for #P and #•NP, respectively. To show the latter we identify a new #•NP-complete problem with respect to Turing reductions. We show that in the enumeration setting the classes behave very similarly to the corresponding classes in the decision setting. We translate the results P = FO(⊆) and NP = FO(=(...)) to the enumeration setting which results in DelP = DelFO(⊆) and DelNP = DelFO(=(...)). Furthermore, we identify several DelP and DelNP-complete problems which yield additional characterisations of DelP and DelNP. For one of the investigated problems we were only able to show Del+NP membership (and DelNP-hardness), a precise classification remains open.