Kernels of Context-Free Languages

While the closure of a language family $\mathcal{ℒ}$ under certain language operations is the least family of languages which contains all members of $\mathcal{ℒ}$ and is closed under all of the operations, a kernel of $\mathcal{ℒ}$ is a maximal family of languages which is a sub-family of $\mathcal{ℒ}$ and is closed under all of the operations. Here we investigate properties of kernels of general language families and operations defined thereon as well as kernels of (deterministic) (linear) context-free languages with a focus on Boolean operations. While the closures of language families are unique, this uniqueness is not obvious for kernels. We consider properties of language families and of operations that yield unique and non-unique, i.e. a set, of kernels. For the latter case, the question whether the union of all kernels coincides with the language family, or whether there are languages that do not belong to any kernel is addressed. Additionally, languages that are mandatory for each (Boolean) kernel and languages that are optional for (Boolean) kernels are studied. That is, we consider the intersection of all Boolean kernels as well as their union. The expressive capacities of these families are addressed leading to a hierarchical structure. Further closure properties are considered. Furthermore, we study descriptional complexity aspects of these families, where languages are represented by context-free grammars with proofs attached. It turns out that the size trade-offs between all families in question and deterministic context-free languages are non-recursive. That is, one can choose an arbitrarily large recursive function $f$, but the gain in economy of description eventually exceeds $f$ when changing from the latter system to the former.