Word Length Controlled DT0L Systems and Slender Languages

We introduce a new controlled DT0L system, called a word length controlled DT0L system, or a wlcDT0L system for short. A wlcDT0L system is a DT0L system with a control function which maps from the set of nonnegative integers to the set of tables. A wlcDT0L system derives exactly one word from a given word by iterating the table which is the value of the control function of the length of the given word. Thus a wlcDT0L system generates a sequence of words which starts from the axiom. We prove that every wlcPDT0L system generates a slender language. We also prove that there is a wlcDT0L language which is not slender. Since a slender language is applicable to cryptography, the family of wlcPDT0L languages may be useful for cryptography.