Embedding Turing machine [closed]

Let $\mathcal{M}$ be a class of binary functions acting on strings in $\Sigma^*$, along with a "size" function $|\cdot|:\mathcal{M}\to\{1,2,\ldots\}$ with the property that there are only finitely many $M\in\mathcal{M}$ of a given size. Turing machines and finite automata are of this form, where size can be taken as the number of states. Then you can define the following symmetric positive definite kernel on $\Sigma^*$: $$ K_n(x,x') = \sum_{M\in\mathcal{M}, |M|\le n} M(x)M(x'). $$ In words, this counts the number of $M$ on at most $n$ states which accept both $x$ and $x'$. This kernel then embeds $\Sigma^*$ into a corresponding Reproducing Kernel Hilbert Space. One can ask what languages can be defined by linear separators under this kernel. This topic was explored in https://www.sciencedirect.com/science/article/pii/S0304397508004581 and http://www.jmlr.org/papers/volume10/kontorovich09a/kontorovich09a.pdf .