On Infinite Words Determined by L Systems
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 is a prefix of y or y is a prefix of x. Every infinite prefix language determines an infinite word. Where C is a class of L systems (e.g. 0L, DT0L), we denote by 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(D0L), and ω(CD0L).
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