Some considerations on Hydra groups and a new bound for the length of words

Abstract

ABSTRACT. After a survey on some recent results of Riley and others on Ackermann functions and Hydra groups, we make an analogy between DNA sequences, whose growth is the same of that of Hydra groups, and a musical piece, written with the same algorithmic criterion. This is mainly an aesthetic observation, which emphasizes the importance of the combinatorics of words in two different contexts. A result of specific mathematical interest is placed at the end, where we sharpen some previous bounds on deterministic finite automata in which there are languages with hairpins. Hydra groupsThe combinatorics of words is increased significatively in the last years, since it is a powerful tool for the description of several processes in pure and applied sciences. A classic reference on the topic is [24] and will be used for the basic notions. We recall that a nonempty set A (which will be always assumed to be finite) is said to be an alphabet and its elements are called letters. By A + we denote the free semigroup generated by A, i.e. the set of all finite sequences of letters with the operation of concatenation. Elements of A + will be called words. It is well known that each word can be written in a unique way as a sequence of letters, so it is possible to define for each word w its length |w|:It is useful (though not necessary) to introduce the formal empty word ε: that is a word of length 0 such that2010 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: Primary 57M50; Secondary 68R05, 68Q15. K e y w o r d s: isoperimetric functions, finitely presented groups, lengths of words, counterpoint, Myhill-Nerode's theorem.