Kernel functions for vectors in discrete spaces

Question

There are lots of choices for kernel functions $k(\textbf{x}, \textbf{y})$ in continuous spaces (e.g., $\textbf{x}, \textbf{y}\in \mathbb{R}^d$), such as the RBF kernel, etc. However, what are some common kernel functions for vectors on discrete spaces (e.g., $\textbf{x}, \textbf{y} \in\{0,1\}^d$, or even $\textbf{x}, \textbf{y} \in\{1,\dots,K\}^d$)?