A language L has the bounded g r owth property if (and only if) L is a singleton or 9k 2 IN = 8w 2 L ; 9w 0 2 L n f wg = jw 0 j 0 j wj k Now, it is easy to prove (see Appendix) that: Theorem 7 Any language of analogical strings veries the bounded g r owth property.
a language like fa 2 n =n 2 INg is not a language of analogical strings, as it does not have the bounded growth property.