On a Method for Proving Exact Bounds on Derivational Complexity in Thue Systems

In this paper, the following system of substitutions in a $3$-letter alphabet
$$ \mathbf\Sigma=\langle a,b,c\mid a^2 \to bc,\,b^2\to ac,\,c^2\to ab\rangle $$
is considered. A detailed proof of results that were described briefly in the author's paper [1] is presented. They give an answer to the specific question on the possibility of giving a polynomial upper bound for the lengths of derivations from a given word in the system $\mathbf\Sigma$ stated in the literature. The maximal possible number of steps in derivation sequences starting from a given word $W$ is denoted by $\mathbf D(W)$. The maximum of $\mathbf D(W)$ for all words of length $|W|=l$ is denoted by $\mathbf D(l)$. It is proved that the function $\mathbf D(W)$ on words $W$ of given length $|W|=m+2$ reaches its maximum only on words of the form $W=c^2b^m$ and $W=b^ma^2$. For these words, the following precise estimate is established:
$$ \mathbf D(m+2)=\mathbf D(c^2b^m)=\mathbf D(b^ma^2) =\biggl\rceil\frac{3m^2}{2}\biggr\lceil+m+1<\frac{3(m+1)^2}{2}, $$
where $\lceil{3m^2}/{2}\rceil$ for odd $|m|$ is the round-up of ${3m^2}/{2}$ to the nearest integer.