New Abelian Square-Free DT0L-Languages over 4 Letters

In 1906 Axel Thue [34] started the systematic study of structures in words. Consequently, he studied basic objects of theoretical computer science long before the invention of the computer or DNA. In 1961 Paul Erdös [13] raised the question whether abelian squares can be avoided in infinitely long words. In 1992, we presented in [19], see also [20–23], an abelian square-free (a-2-free) endomorphism g85 on the four letter alphabet S4 = {a, b, c, d}. The size of g85 , i.e. |g85 (abcd)|, is equal to 4×85. Until now, all known methods for constructing arbitrarily long a-2-free words on S4 have been based on the structure of this g85 ; see Arturo Carpi [4–6]. In this paper, we report of a completely new endomorphism g98 of S4 , the iteration of which produces an infinite abelian square-free word. The size of g98 is 4×98, and the image words for letters are constructed, in part, differently from the case of g85 . For g85 they were directly obtained by permutating letters cyclically. The endomorphism g98 is not an a-2-free endomorphism itself, since it does not preserve the a-2-freeness of all words of length 7. However, g98 can be used together with g85 to produce a-2-free DT0L-languages of unlimited size. Here DT0L-languages mean deterministic context-independent Lindenmayer languages produced by using compositions of endomorphisms – so called tables; see [32, p.188]. Indeed, by using Carpi's algorithm [4] for prefixes of g85 (S) and g98 (S), and a modified version of this algorithm, one can check the following fact: for any a-2-free words w1 and w2 , where w1 does not contain a certain subword pattern, g98 (w1 ) and g85 (w2 ) are always a-2-free and avoid all undesirable patterns that would, in the case of g98 , lead to an abelian-square in the next iteration step. It is anticipated that later on this new result will lead to a sharper bound for the base number (~1.000021) found in [5], where Carpi showed that the number of a-2-free words over 4 letters of length n is ¥ 1.000021n . We explain extensive computer aided searches that have been carried out over 11 years to find new ways of constructing a-2-free words over 4 letters. In this process, we have encountered some unintuitive non-linear phenomena which, however, are in accordance with the complex behaviour of simple systems studied by Stephen Wolfram in his long-awaited book [35]. For example, two same looking initial values for prefixes and suffixes of image words for letters may yield 1000-fold running times when searching through all possibilities for proper infixes! We also analyse the structure of g85 in terms of subwords, and discuss some open problems.