Semiring semantics evaluates logical statements by values in some commutative semiring (K, +, ·, 0, 1). Random semiring interpretations, induced by a probability distribution on K, generalise random structures, and we investigate here the question of how classical results on first-order logic on random structures, most importantly the 0-1 laws of Glebskii et al. and Fagin, generalise to semiring semantics. For positive semirings, the classical 0-1 law implies that every first-order sentence is, asymptotically, either almost surely evaluated to 0 by random semiring interpretations, or almost surely takes only values different from 0. However, by means of a more sophisticated analysis, based on appropriate extension properties and on algebraic representations of first-order formulae, we can prove much stronger results.

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 being 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 second-order logic and to logics with team semantics. In many cases the results resemble what is known in team semantics, but there are also interesting differences. While in team semantics, the combination of inclusion and exclusion dependencies leads to a logic with the full power of both independence logic and existential second-order logic, independence properties of multiteams are not definable by any combination of properties that are downwards closed or union closed and thus are strictly more powerful than inclusion-exclusion logic. 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.

We present syntactic characterisations for the union closed fragments of existential second-order logic and of logics with team semantics. Since union closure is a semantical and undecidable property, the normal form we introduce enables the handling and provides a better understanding of this fragment. We also introduce inclusion-exclusion games that turn out to be precisely the corresponding model-checking games. These games are not only interesting in their own right, but they also are a key factor towards building a bridge between the semantic and syntactic fragments. On the level of logics with team semantics we additionally present restrictions of inclusion-exclusion logic to capture the union closed fragment. Moreover, we define a team based atom that when adding it to first-order logic also precisely captures the union closed fragment of existential second-order logic which answers an open question by Galliani and Hella.