A deterministic L system generates an infinite word  α  if each word in its derivation sequence is a prefix of the next, yielding  α  as a limit. We generalize this notion to arbitrary L systems via the concept of prefix languages. A prefix language is a language  L    such that for all   x,y∈L     x  ,  y  ∈  L       ,  x  is a prefix of  y  or  y  is a prefix of  x . Every infinite prefix language determines a single infinite word. Where  C    is a class of L systems (e.g. 0L, DT0L), we denote by   ω(C)     ω  (  C  )        the class of infinite words determined by the prefix languages in  C   . This allows us to speak of infinite 0L words, infinite DT0L words, etc. We categorize the infinite words determined by a variety of L systems, showing that the whole hierarchy collapses to just three distinct classes of infinite words:   ω(PD0L)     ω  (  PD0L  )       ,   ω(D0L)     ω  (  D0L  )       , and   ω(CD0L)     ω  (  CD0L  )       . Our results are obtained with the help of a pumping lemma which we prove for T0L systems.
['Tim Smith']
On infinite words determined by L systems