In this article, we study the complexity of weighted team definability for logics with team semantics. This problem is a natural analogue 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 phi of central team-based logics. Given a first-order structure A and the parameter value k as input, the question is to determine whether A,T models phi 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.

翻译：本文研究了具有团队语义的逻辑中加权团队可定义性的复杂度问题。该问题是参数复杂度领域最受关注的问题之一——加权Fagin可定义性（以自由关系变量满足一阶公式的形式定义）的自然类比。我们聚焦于固定中心团队逻辑公式φ的加权团队可定义性的参数复杂度。给定一阶结构A和参数值k作为输入，问题在于判断是否存在规模为k的团队T使得A,T ⊧ φ成立。我们针对依赖逻辑、独立逻辑和包含逻辑公式展示了该问题的若干复杂度结果。此外，还将加权团队可定义性的复杂度与著名的W-层次结构中的复杂度类以及paraNP类建立了关联。