C-dictionaries
---------------------

A chain dictionary ${\mathsf{C}}^{\DcSize}_{\SySize,\pFfork}$ containing $\DcSize$ words is generated using a symbol list of length $\SySize$. It is an incremental process, based on a starting point consisting of an initial random symbol $\alpha_\circ$. Each new word is added iteratively: either by (i) with probability $1-\pFfork$, an existing word being chosen and a new random symbol appended to create the new word, which is only kept if the result is not already present; or (ii) with probability $\pFfork$, another (single) random symbol is chosen to be the new word, which is only kept if the result is not already present. In this way the “fork probability” $\pFfork$ determines how fast the dictionary accumulates words with different starting symbols. Although we should note that the starting symbol has no special role, a low $\pFfork$ means that the early-chosen symbols, and in particular $\alpha_\circ$, will dominate the symbol frequencies in the resulting dictionary.

For low word counts, and large enough symbol lists, some symbols might not be chosen for use in words in the dictionary.

![Scaling of innovation in ${\mathsf{C}}^{\DcSize}_{\SySize,\pFfork}$ chain dictionaries, with symbol number $\SySize$ varying on a log scale from $2$ to $32$, and dictionary size $\DcSize$ on a 