Nondeterministic automatic complexity of overlapfree and almost squarefree words. Shallit and Wang studied deterministic automatic complexity of words. They showed that the automatic Hausdorff dimension I(mathbf t) of the infinite Thue word satisfies 13le I(mathbf t)le 23. We improve that result by showing that I(mathbf t)ge 12. For nondeterministic automatic complexity we show I(mathbf t)12. We prove that such complexity A_N of a word x of length n satisfies A_N(x)le b(n):lfloor n2rfloor  1. This enables us to define the complexity deficiency D(x)b(n)A_N(x). If x is squarefree then D(x)0. If x almost squarefree in the sense of Fraenkel and Simpson, or if x is a strongly cubefree binary word such as the infinite Thue word, then D(x)le 1. On the other hand, there is no constant upper bound on D for strongly cubefree words in a ternary alphabet, nor for cubefree words in a binary alphabet. The decision problem whether D(x)ge d for given x, d belongs to NPcap E.

Complexity of SuffixFree Regular Languages A sequence ((L_k,L_k1 dots )) of regular languages in some class (mathcal C), where n is the state complexity of (L_n), is called a stream. A stream is most complex in class (mathcal C) if its languages together with their dialects (that is, languages that differ only very slightly from the languages in the stream) meet the state complexity bounds for boolean operations, product (concatenation), star, and reversal, have the largest syntactic semigroups, and have the maximal numbers of atoms, each of which has maximal state complexity. It is known that there exist such most complex streams in the class of regular languages, and also in the classes of right, left, and twosided ideals. In contrast to this, we prove that there does not exist a most complex stream in the class of suffixfree regular languages. However, we do exhibit one ternary suffixfree stream that meets the bound for product and whose restrictions to binary alphabets meet the bounds for star and boolean operations. We also exhibit a quinary stream that meets the bounds for boolean operations, reversal, size of syntactic semigroup, and atom complexities. Moreover, we solve an open problem about the bound for the product of two languages of state complexities m and n in the binary case by showing that it can be met for infinitely many m and n.