Hierarchical Type Classes and Their Entropy Functions

Abstract

For each $j \geq 1$, if $T_j$ is the finite rooted binary tree with $2^j$ leaves, the hierarchical type class of binary string $x$ of length $2^j$ is obtained by placing the entries of $x$ as label son the leaves of $T_j$ and then forming all permutations of $x$according to the permutations of the leaf labels under all isomorphisms of tree $T_j$ into itself. The set of binary strings of length $2^j$ is partitioned into hierarchical type classes, and in each such class, all of the strings have the same type $(n_0^j, n_1^j)$, where $n_0^j, n_1^j$ are respectively the numbers of zeroes and ones in the strings. Let $p(n_0^j, n_1^j)$ be the probability vector $(n_0^j/2^j, n_1^j/2^j)$belonging to the set ${\cal P}_2$ of all two-dimensional probability vectors. For each $j \geq 1$, and each of the $2^j+1$ possible types $(n_0^j, n_1^j)$, a hierarchical type class ${\cal S}(n_0^j, n_1^j)$is specified. Conditions are investigated under which there will exist a function $h:{\cal P}_2\to [0, \infty)$ such that for each $p\in {\cal P}_2$, if $\{(n_0^j, n_1^j):j\geq 1\}$ is any sequence of types for which $p(n_0^j, n_1^j) \to p$, then the sequence $\{2^{-j}\log_2({\rm card}({\cal S}(n_0^j, n_1^j))):j \geq 1\}$converges to $h(p)$. Such functions $h$, called hierarchical entropy functions, play the same role in hierarchical type class coding theory that the Shannon entropy function on ${\cal P}_2$ does in traditional type class coding theory, except that there are infinitely many hierarchical entropy functions but only one Shannon entropy function. One of the hierarchical entropy functions $h$ that is studied is a self-affine function for which a closed-form expression is obtained making use of an iterated function system whose attractor is the graph of $h$.