Automaton Theory and Modeling of Biological Systems by M.L. Tsetlin (Eds.)

By M.L. Tsetlin (Eds.)

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In the final part of this section we shall give an example of a system whose functioning is largely similar to the behavior of automata in stationary random media and possesses a naturally defined expediency. Suppose there are n books K , , K2,. . , K,, lying in a stack on a desk. The books in the stack may be used in various ways-for example, one can choose the desired book and put it back in the same location after use. It is, however, often preferred not to search for the original location of the book chosen, but to simply put it on top of the stack.

We set x = x1 ix, , y = y, iy,, where x, , y, 2 0. We intend to find constants c > 0 and 0 < M < 1, such that if x1 5 c or x, 5 c, then I1ILM. + + 40 Automaton Theory It will be noted that = sin 2a ez1 + e-21 = 2 sin 2a cosh xl. Similarly, I 1I 5 sin 28 cosh y , . Taking this into consideration, we select c > 0 in such a way that cosh c < min{ 1 /sin 2%; 1 /sin 2 8 ) . We can do this since u, # n/4. r cosh c, sin 28 cosh c} < 1 . I 1, I + 1, the cases in which x1 5 c or x2 I c, do not, as we have shown, have t o be considered.

X - I; v = I,. . , n , I)s(~+l’gaij(f, s(t))(l - O i j ( t , s ( t ) ) ) , s ( t ) ) - (-l)sct+l’gaik(t,s(t))aij(t, ~ ( t ) ) , j f k. (33) In these formulas 0 5 g< 1 ; it is not hard to check that the matrix remains stochastic. The behavior of automata with an evolving structure in random media can be described by a nonhomogeneous Markov chain. For automata whose structures evolve as described by Eqs. (33), there exist stationary values of the transition probabilities, and one can speak of the final probabilities of the states of the system.

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