### A polynomial algorithm for constructing the optimal inverse morphism

#### Abstract

In this paper, we continue the theme of some our previous works. In this topic, we consider two similar problems of the formal languages theory: the problem of extracting the root of a given degree and the problem of constructing an optimal inverse morphism, where optimality can be defined as the length of the maximum word of a language that is an inverse morphic image. Instead of considering all subsets of the set of so-called potential roots, which always leads to exponential algorithms, we obtained some various polynomial algorithms in these publications. In order to formulate the second of these problems in a way convenient for constructing polynomial algorithms, we have considered some equivalent versions of the special hypothesis of the formal languages theory in previous papers. One of its equivalent formulations can be expressed as follows. For two finite languages not containing the empty word, we can consider the necessary and sufficient condition that the iteration of any of both languages belongs to the set of prefixes of the other language, and write it as follows. There is some new alphabet, different from the alphabet over which the source languages are set, and above this alphabet there are two maximum prefix codes (generally speaking, different ones) and two languages containing these maximum prefix codes as subsets. In addition, there is some special morphism, and the source languages should be formed by applying this morphism to the languages containing these maximal prefix codes as subsets. In this paper, we show that when this hypothesis is fulfilled, a polynomial algorithm for constructing an optimal inverse morphism is possible. Moreover, simplifying the situation a little, we can say that such an algorithm is a special combination of several polynomial algorithms considered in our previous papers, or simple modifications of such algorithms.

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