Infiniteness and Boundedness in 0L, DT0L, and T0L Systems
We investigate the boundary between finiteness and infiniteness in three types of L systems: 0L, DT0L, and T0L. We establish necessary and sufficient conditions for 0L, DT0L, and T0L systems to be infinite, and characterize the boundedness of finite classes of such systems. First, we give a pumping lemma for these systems, proving that the language of a system is infinite iff the system is pumpable. Next, we show that the number of steps needed to derive any string in any finite 0L or DT0L system is bounded by a function depending only on the size of the alphabet, and not on the production rules or start string. This alphabet boundedness does not hold for finite T0L systems in general. Finally, we show that every infinite 0L system has an infinite D0L subsystem.
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