Variants of finite automata corresponding to infinite iterative morphism trees. Part II

Boris Melnikov

Abstract


Quite briefly, the subject of this paper can be formulated as follows: in the previously considered infinite iterative morphism trees, we combine equivalent vertices, in fact, obtaining a deterministic finite automaton; after that, we investigate some properties of the resulting automaton. In addition, the article describes the possible connection of the automata we are considering with problems arising in other areas of the theory of formal languages. In more detail, we work with various variants of finite automata, each of which corresponds to some infinite iterative tree of the morphism under consideration. In this case, the automata constructed for different morphisms describe the main properties of the given morphisms. In addition, in each case (i.e., for each variant of the automaton), the following “inverse problem” also arises: to describe a morphism (or simply specify a pair of languages) for which some given automaton is obtained. In addition, the paper describes the possible connection of the material under consideration with problems arising in other areas of the theory of formal languages. Among the variants of automata corresponding to an infinite iterative morphism tree for a given ordered pair of finite languages, we first define the so­called primary automaton. It is deterministic, defined on sets of words, and each of these sets is a subset of the set of prefixes of the second of the given languages. Next, we consider some variants of nondeterministic automata corresponding to it. After that, we introduce a completely different object, i.e., a simplified primary automaton defined not on sets of words, but on words. Despite the significant difference with automata built on sets of words, all constructions for specific examples of languages can be performed using the same computer program. Next, we consider the features that appear when applying algorithms that form finite automata to pairs of matching languages. In conclusion, we briefly formulate the directions of further work related to the issues discussed in this paper. In this Part II, we consider primarily nondeterministic automata. The features that appear when applying algorithms that form finite automata to pairs of matching languages are also briefly considered. In addition, we consider an important property of the introduced automata, we define a special algebraic system, the properties of which is proposed to consider in more detail in following publications.

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References


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