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Kernels dictionaries

In this section we introduce two conditional kernels plKernelMap and plKernelTable. The particularity of these conditional kernels is that they are used to represent a directory of kernels rather than a family of them.

Definition 20   Lets $\rho_i$ be a kernel over $\Omega_s$. Given $\Omega_s$, $\Omega_k$, $\rho_0$ and a set $S$ of pairs $\{p_1,p_2,...,p_m\}$ of the form $p_i=(\omega_{k_i},\rho_i)$ where no two pairs on the set have the same $\omega_i$. A kernel dictionary $\varphi$ on $\Omega_s$ and $\Omega_k$ given $\rho_0$ and $S$ is a conditional kernel such that

$$instantiate(\omega_k) = \rho_\ast$$ (3.16)

and

$$compute(\omega_s \cup \omega_k) = \rho_\ast.compute(\omega_s)$$ (3.17)

where

$$\rho_\ast = \left\{ \begin{array}{ll} second(p_i) & \mbox{if } \omega_k=first(p_i) \mbox{ with } p_i \in S\\ \rho_0 & \mbox{otherwise} \end{array}\right.$$ (3.18)

A kernel dictionary is provided of the methods push, push_default and find. The functionalities of these methods are described below

• push inserts a new kernel with an specified key value $\omega_k \in \Omega_k$.
• push_default inserts a default kernel for those key values $\omega_k \in \Omega_k$ non previously inserted with push.
• find returns true if an element with an specified key value $\omega_k \in \Omega_k$ has been inserted with push.

In short, a kernel dictionary is a collection of kernels on $\Omega_s$ where each element is accessed by a key $\omega_k \in \Omega_k$.

In the following we give the definition of these two kernel dictionaries.