### The use of petal finite automata to verify the fulfillment of a special case of the Zyu hypothesis (for a given finite language)

#### Abstract

In this paper, we continue to study the properties of a special binary equivalence relation at infinity. Before, we described one hypothesis of the theory of formal languages, which we called the Zyu hypothesis; one of some its equivalent formulations can be expressed as follows. For two finite languages A and B without empty words, we can write the necessary and sufficient condition that the iteration of any of these languages belongs to the set of prefixes of the second language as follows: there is some alphabet (different from the alphabet over which A and B are given), over it there are two certain maximal prefix codes (generally speaking, different ones) and two languages containing these maximal prefix codes as subsets, as well as some morphism between the two considered alphabets. Then, applying this morphism to the last two languages, we obtain the given languages A and B. We consider an equivalent formulation of this hypothesis using not two languages over the given alphabet, but only one; at the same time, it is important to note that such a variant of the Zyu hypothesis consists in fulfilling some condition for any finite language. Using the above formulation, the subject of this article can be briefly described as follows: how can a similar version of the Zyu hypothesis be tested not for any finite language, but for one given specific language A. At the same time, we do not strive for the polynomial algorithm of such a check: such a polynomiality would follow also the polynomiality of the verification algorithm of equivalence languages of two given nondeterministic finite automata. The paper presents the formulations of a variant of the Zyu hypothesis using a petal nondeterministic automaton constructed according to some given finite language A. After that, the formulation of another variant of this hypothesis is given; it uses a canonical automaton equivalent to such a petal automaton. All variants of the hypothesis are illustrated by examples in which the same language is considered; at the same time, certainly, the examples are given only for one language, while hypotheses are formulated for the whole set of finite languages

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