Team semantics is the mathematical basis of modern logics of dependence and independence. In contrast to classical Tarski semantics, a formula is evaluated not for a single assignment of values to the free variables, but on a set of such assignments, called a team. Team semantics is appropriate for a purely logical understanding of dependency notions, where only the presence or absence of data matters, but based on sets, it does not take into account multiple occurrences of data values. It is therefore insufficient in scenarios where such multiplicities matter, in particular for reasoning about probabilities and statistical independencies. Therefore, an extension from teams to multiteams (i.e. multisets of assignments) has been proposed by several authors. In this paper we aim at a systematic development of logics of dependence and independence based on multiteam semantics. We study atomic dependency properties of finite multiteams and discuss the appropriate meaning of logical operators to extend the atomic dependencies to full-fledged logics for reasoning about dependence properties in a multiteam setting. We explore properties and expressive power of a wide spectrum of different multiteam logics and compare them to logics with team semantics. We also study the relationship of logics with multiteam semantics with existential second-order logic for a specific class of metafinite structures. It turns out that inclusion-exclusion logic can be characterised in a precise sense by the Presburger fragment of this logic, but for capturing independence, we need to go beyond it and add some form of multiplication. Finally we also consider multiteams with weights in the reals and study the expressive power of formulae by means of topological properties.

中文翻译：

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多团队语义的逻辑

团队语义是现代依赖和独立逻辑的数学基础。与经典的 Tarski 语义相反，公式不是针对自由变量的单个值赋值，而是针对一组此类赋值进行评估，称为团队。团队语义适用于对依赖概念的纯逻辑理解，其中只有数据的存在或不存在才重要，但基于集合，它不考虑数据值的多次出现。因此，在这种多重性很重要的情况下，特别是对于概率和统计独立性的推理来说，这是不够的。因此，几位作者提出了从团队到多团队（即多组作业）的扩展。在本文中，我们旨在系统地开发基于多团队语义的依赖和独立逻辑。我们研究了有限多团队的原子依赖属性，并讨论了逻辑运算符的适当含义，以将原子依赖扩展到成熟的逻辑，以推理多团队设置中的依赖属性。我们探索各种不同多团队逻辑的属性和表达能力，并将它们与具有团队语义的逻辑进行比较。我们还研究了具有多组语义的逻辑与特定类型的元有限结构的存在二阶逻辑的关系。事实证明，包含-排除逻辑可以通过该逻辑的 Presburger 片段精确地表征，但是为了捕获独立性，我们需要超越它并添加某种形式的乘法。最后，我们还考虑了具有实数权重的多团队，并通过拓扑属性研究了公式的表达能力。