Mathematical Notes

, Volume 92, Issue 1–2, pp 3–15

# On a method for proving exact bounds on derivational complexity in thue systems

Article

## Abstract

In this paper, the following system of substitutions in a 3-letter alphabet
$$\sum { = \left\langle {\left. {a,b,c} \right|a^2 \to bc,b^2 \to ac,c^2 \to ab} \right\rangle }$$
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 Σ stated in the literature. The maximal possible number of steps in derivation sequences starting from a given word W is denoted by D(W). The maximum of D(W) for all words of length |W| = l is denoted by D(l). It is proved that the function D(W) on wordsW of given length |W| = m+2 reaches its maximum only on words of the form W = c 2 b m and W = b m a 2. For these words, the following precise estimate is established: where ⌌3m 2/2⌍ for odd |m| is the round-up of 3m 2/2 to the nearest integer.

## Keywords

word rewriting system derivational complexity Thue system polynomial upper bound left (right) divisibility of a word

## References

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