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Automatic Structures and their Generalizations

Student: Gorchakov Timofei

Supervisor: Lev D. Beklemishev

Faculty: Faculty of Mathematics

Educational Programme: Mathematics (Bachelor)

Final Grade: 8

Year of Graduation: 2018

Automata structures are those structures whose base is a regular language, and also each relation of which can be formulated through finite automata. The structures are convenient in that they possess useful computational properties. About automata linear orders, a particular case of automaton structures, we know the upper bound on the condi- tions of condensability. Condensation is a transition from one linear order to another, no more complex, combining the intervals in which between any two elements a finite number of elements, into one element. If the order can not come for a finite number of applications of such an operation in an unchangeable form, then this order is not automatic. At the same time, it is known that there are orders that shrink in just two iterations, but are not automatic. The task of this paper is to investigate the orders of this kind and to reveal what restrictions on automatonness exist for such rather simple condenser-structurally stable ones. In the first part we will introduce the concept of finite automaton, mathematical structure, and show when the structure is automata. At the end of the first part, we show the main technical means of automaton functions by Theorems 5 and 6: the solvability of the elementary theory for automaton structures and the possibility of replenishing the theory with an additional predicate "there are infinitely many" with preservation of solvability. In the second part, we will acquaint ourselves with the classical method of ranking linear orders, We briefly present the already known results for automaton linear orders, and then we will analyze what we have obtained about orders of rank 2.

Full text (added June 3, 2018)

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