Polynomial automaticity, context-free languages, and fixed points of morphisms (Extended abstract)

Automaticity I: Properties of a Measure of Descriptional Complexity

“…Since n 1 Q n converges, by the Borel Cantelli lemma, with probability 1 at most finitely many of the events N L (n) q occur. Thus for almost all L, we have N L (n)>(1&=) k nÂ2 Â-k&1 for all sufficiently large n. K It can be shown using the results in [24] that if |7| =k, then the language L=[w # 7*: w=w R ] has nondeterministic automaticity 0(k nÂ2 ).…”

“…Then, as in Example 2, A L2 (n)=0(n 2 ). In [24], it is shown that N L2 (n)=0(n 2 ). However, for any language L, f L (n) is clearly bounded above by 1 plus the total number of distinct prefixes of length n of elements of L. Hence f L2 (n)=O(n).…”

“…Then, since L 2 contains at most one string of every length, it is easy to see that da L2 (n)=O(n). On the other hand, in [24] a general technique is developed which proves that N L2 (n)=0(n 2 ). K Theorem 24.…”

“…See, for example, [1]. Roughly speaking, we say that a sequence Note that the input to the machine M is the representation of m in base-k. We will assume that the machine reads the least significant digit first; unfortunately, this detail actually proves necessary to specify, since there exist languages with low automaticity whose reversal has high automaticity; see [24]. We will also assume that M always gives the proper response when the input is any of the``nonstandard'' base-k representations of m (i.e., those containing leading zeroes or more properly, trailing zeroes, since we are using the reversed representation).…”

Automaticity III: Polynomial automaticity and context-free languages

“…thesis of the rst author Glaister 1995 . A preliminary version was presented at the 21st International Symposium on Mathematical Foundations of Computer Science 1996 in Krak ow, Poland as Glaister & Shallit 1996 . Jean-Paul Allouche and Jonathan Buss read a draft of this paper and o ered several corrections.…”