Counting of Teams in First-Order Team Logics

We study descriptive complexity of counting complexity classes in the range from $\#$P to $\#\cdot$NP. A corollary of Fagin's characterization of NP by existential second-order logic is that $\#$P can be logically described as the class of functions counting satisfying assignments to free relation variables in first-order formulae. In this paper we extend this study to classes beyond $\#$P and extensions of first-order logic with team semantics. These team-based logics are closely related to existential second-order logic and its fragments, hence our results also shed light on the complexity of counting for extensions of FO in Tarski's semantics. Our results show that the class $\#\cdot$NP can be logically characterized by independence logic and existential second-order logic, whereas dependence logic and inclusion logic give rise to subclasses of $\#\cdot$NP and $\#$P , respectively. Our main technical result shows that the problem of counting satisfying assignments for monotone Boolean $\Sigma_1$-formulae is $\#\cdot$NP-complete as well as complete for the function class generated by dependence logic.Peer reviewe

Parameterized complexity of weighted team definability

In this article, we study the complexity of weighted team definability for logics with team semantics. This problem is a natural analog of one of the most studied problems in parameterized complexity, the notion of weighted Fagin-definability, which is formulated in terms of satisfaction of first-order formulas with free relation variables. We focus on the parameterized complexity of weighted team definability for a fixed formula ϕ of central team-based logics. Given a first-order structure A and the parameter value k ∈ N as input, the question is to determine whether A, T |= ϕ for some team T of size k. We show several results on the complexity of this problem for dependence, independence, and inclusion logic formulas. Moreover, we also relate the complexity of weighted team definability to the complexity classes in the well-known W-hierarchy as well as paraNP

Unified Foundations of Team Semantics via Semirings

Semiring semantics for first-order logic provides a way to trace how facts represented by a model are used to deduce satisfaction of a formula. Team semantics is a framework for studying logics of dependence and independence in diverse contexts such as databases, quantum mechanics, and statistics by extending first-order logic with atoms that describe dependencies between variables. Combining these two, we propose a unifying approach for analysing the concepts of dependence and independence via a novel semiring team semantics, which subsumes all the previously considered variants for first-order team semantics. In particular, we study the preservation of satisfaction of dependencies and formulae between different semirings. In addition we create links to reasoning tasks such as provenance, counting, and repairs

Counting and enumerating in first-order team logics

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