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Finite automata are considered in this paper a s instruments for classifying finite tapes. Each onetape automaton defines a set of tapes, a two-tape automaton defines a set of pairs of tapes, et cetera. The structure of the defined sets is studied. Various generalizations of the notion of an automaton are introduced and their relation to the classical automata is determined. Some decision problems concerning automata are shown to be solvable by effective algorithms; others turn out to be unsolvable by algorithms. 7. J. C. Shepherdson, "The reduction of two-way automata to one-way automata,"

Abstract. Binary trees are very useful tools in computer science for estimating the running time of so-called comparison based algorithms, algorithms in which every action is ultimately based on a prior comparison between two elements. For two given algorithms A and B where the decision tree of A is more balanced than that of B, it is known that the average and worst case times of A will be better than those of B, i.e., T A(n) ≤ T B (n) and T W A (n) ≤ T W B (n). Thus the most balanced and the most imbalanced binary trees play a main role. Here we consider them as semilattices and characterize the most balanced and the most imbalanced binary trees by topological and categorical properties. Also we define the composition of binary trees as a commutative binary operation, *, such that for binary trees A and B, A * B is the binary tree obtained by attaching a copy of B to any leaf of A. We show that (T, * ) is a commutative po-monoid and investigate its properties.2000 AMS Classification: 06A12, 06F05, 16B50.

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